Finite Model Theory of the Triguarded Fragment and Related Logics
FFinite Model Theory of the Triguarded Fragment and Related Logics
Emanuel Kiero´nski
Institute of Computer ScienceUniversity of WrocławEmail: [email protected]
Sebastian Rudolph
Computational Logic GroupTechnische Universit¨at DresdenEmail: [email protected]
Abstract —The Triguarded Fragment (TGF) is among the mostexpressive decidable fragments of first-order logic, subsumingboth its two-variable and guarded fragments without equality.We show that the TGF has the finite model property (providinga tight doubly exponential bound on the model size) and hencefinite satisfiability coincides with satisfiability known to beN2E XP T IME -complete. Using similar constructions, we alsoestablish 2E XP T IME -completeness for finite satisfiability of theconstant-free (tri)guarded fragment with transitive guards.
1. Introduction
Ever since first-order logic (FOL) was found to havean undecidable satisfiability problem, researchers have at-tempted to identify expressive yet decidable fragments ofFOL and pinpoint their complexity. Two of the most promi-nent fragments in this regard are FO (the two-variablefragment ) and GF (the guarded fragment ).For FO , decidability is retained through reducing thenumber of available variables to , essentially restrictingexpressivity to independent pairwise interactions betweendomain elements. Decidability of FO without equality wasalready established in the 1960s [13]; in the 1970s the resultwas extended to the case with equality [11]. NE XP T IME -completeness was established in the 1990s [7].GF, which owes its decidability to the restricted“guarded” use of quantifiers, originated in the late 1990s [1].Its satisfiability problem is 2E XP T IME -complete but dropsto E XP T IME -completeness when the maximum predicatearity or the number of variables is bounded [6], [15].Both FO and GF possess the finite model property (FMP), meaning that any satisfiable sentence has a finitemodel. As a consequence, finite-model reasoning coincideswith reasoning under arbitrary models for these fragments.For FO , existence of a finite model of only exponentialsize in the sentence was actually the path to establishingthe above mentioned complexity. For GF, the original FMPresult gave rise to a triply exponential bound on the modelsize [6], whereas a tight doubly-exponential bound wasestablished much more recently [2].In an attempt to unify FO and GF toward an evenmore expressive decidable FOL fragment, the triguardedfragment (TGF) was introduced [12], extending prior re-sults [8] as well as refining and correcting previous ideas related to “cross products” [3]. TGF relaxes the guardednessrestrictions of GF by allowing non-guarded quantificationof subformulas with up to two free variables. The priceto pay for retaining decidability is that equality needs tobe disallowed, or at least its use must be significantlyrestricted. TGF brings a new quality, as it allows one toexpress properties expressible in neither FO nor GF. Inparticular it embeds one of the most important prefix classes ,namely G¨odel’s class without equality, consisting of prenexformulas of the shape ∃ ¯ x ∀ y y ∃ ¯ zϕ . Indeed given such aformula we can translate it to TGF by eliminating the initialprefix of existential quantifiers, replacing the variables ¯ x by constants, and guarding the block of quantifiers ∃ ¯ z bya dummy guard G ( y , y , ¯ z ) . Along the same lines, TGFsettles an open question by ten Cate and Franceschet [15]about the decidability of formulas of the shape ∃ ¯ x ∀ y y ∃ ¯ zψ where ψ is a guarded formula. In fact, checking satisfiabilityof TGF is N2E XP T IME -complete, dropping to 2E XP T IME when disallowing constants – as opposed to FO and GF,where presence or absence of constants does not make adifference, complexity-wise – and to NE XP T IME if the arityof predicates is bounded.One central question left wide open in the original workon TGF [12] is if TGF has the FMP (and thus, if finite modelreasoning and the associated complexity is any differentfrom the arbitrary-model case). In that paper, it is notedthat neither technique used for establishing the FMP forFO and GF seems to directly lend itself for solving thequestion for TGF, yet it is conjectured that the FMP holds.Indeed one of this paper’s core contributions is to answerthis open question to the positive.An important, practically relevant and theoretically chal-lenging modelling feature is transitivity of a binary relation.Neither FO , nor GF, and also not TGF allow for axioma-tising transitivity. As a remedy, it has been suggested toprovide a set of dedicated binary predicate names whosetransitivity is “hard-wired” into the logic, that is, externallyimposed by the semantics. As it turned out, when doing so,one has to be very careful not to lose decidability. Unre-stricted use of transitive relations in GF is known to lead toundecidability [6], this even holds for GF ( = FO ∩ GF),the two-variable guarded fragment [5].One way out is to restrain the use of transitive relationsso that they only are allowed to occur in guards. Indeed, sat- a r X i v : . [ c s . L O ] J a n sfiability of GF+TG ( GF with transitive guards ) was shownto be decidable and, in fact, 2E XP T IME -complete [14], aswas – more recently – satisfiability of TGF+TG [9]. Resultsfor the finite model case are less extensive: so far, only finitesatisfiability of GF +TG was shown to be decidable and2E XP T IME -complete [10]. We note that GF +TG does nothave the FMP: indeed, a typical infinity axiom saying that,for a transitive relation T , every element has a T -successorbut is not related by T to itself is naturally expressible inGF +TG. We remark that all the results concerning logicswith TG assume the absence of constants. It is conjecturedthat adding constants to the picture is technically challengingbut generally possible without hazarding decidability.In this paper, we significantly advance the state of theart in finite model theory for the (tri)guarded fragment withand without transitive guards showing the following: • TGF (with and without constants) has the FMP, thusfinite satisfiability coincides with satisfiability knownto be N2E XP T IME -complete with and 2E XP T IME -complete without constants. • Finite satisfiability of constant-free GF+TG (withequality) is decidable and 2E XP T IME -complete. • Finite satisfiability of constant-free TGF+TG (withoutequality) is decidable and 2E XP T IME -complete. • All three results come with a tight upper bound on thesize of the finite model which is doubly exponential inthe formula length.The results are established through novel, rather elab-orate, carefully crafted model constructions coupled withmeticulous inspections of existing proofs toward the extrac-tion of tight bounds.
2. Logics
We work with signatures containing relation symbols ofarbitrary positive arity and, possibly, constant symbols. Werefer to structures using Fraktur capital letters A , B , C , . . . ,and to their domains using the corresponding Roman capi-tals A, B, C, . . . . Given a structure A and some B ⊆ A wedenote by A (cid:22) B the restriction of A to its subdomain B .We usually use a, b, . . . to denote domain elements ofstructures, ¯ a , ¯ b, . . . for tuples of domain elements, x , y, . . . for variables, ¯ x , ¯ y, . . . for tuples of variables, and c forconstants, all of these possibly with decorations. For atuple of variables ¯ x we use ψ (¯ x ) to denote that a formula(subformula) ψ has at most free variables from ¯ x . Whereconvenient, tuples of elements will be treated as sets builtout of its members.For a structure A , a formula ψ with free variables ¯ x , anda tuple ¯ a of elements of A of the same length as ¯ x , we willwrite A | = ψ [¯ a ] to denote that ψ (¯ x ) is satisfied in A underthe assignment ¯ x (cid:55)→ ¯ a .For (cid:96) > , an ( atomic ) (cid:96) - type over a finite signature σ is a maximal consistent set of atomic or negated atomicformulas over σ in (cid:96) variables x , . . . , x (cid:96) (we note that typescontain equalities/inequalities and occurrences of constantsif they are present in σ ). A type is an (cid:96) -type for some (cid:96) . We often identify a type with the formula obtained by takingthe conjunction over its elements. A type is guarded if itcontains a positive literal containing all its variables. Notethat all -types are guarded as they all contain the atom x = x .Let A be a structure, and let ¯ a be a tuple of its elements.We denote by tp A (¯ a ) the unique type realized in A by thetuple ¯ a , i.e. , the type α (¯ x ) such that A | = α [¯ a ] . We say that ¯ a is guarded in A if ¯ a is built out of a single element orthere is a tuple of elements ¯ b containing all the elements of ¯ a and a relation symbol P ∈ σ such that A | = P [¯ b ] ( i.e. , ¯ b realizes a guarded type).We will be particularly interested in types over signa-tures σ consisting of the relation symbols (and constants,if present) used in some given formula. A particularly im-portant role will be played by -types and -types. Observethat, in the absence of constants, the number of -types isbounded by a function which is exponential in | σ | , and hencealso in the length of the formula. This is because any -typejust corresponds to a subset of σ . On the other hand, whenat least one constant c is present, then the number of -types may be doubly exponentially large. This is becausea -type must completely describe the substructure on agiven element and the interpretation of c , and there are n relations of arity n on a pair of elements.A -type will be called non-degenerate if it contains x (cid:54) = x . The number of -types may be doubly exponentialin the length of the formula even in the absence of constants.Given a formula ϕ , its width is the maximal numberof free variables across all subformulas of ϕ , whereas fora signature σ , its width is the maximal arity among thesymbols in σ . Guarded fragment.
The set of GF formulas is defined asthe least set such that1) every atomic formula belongs to GF,2) GF is closed under the standard boolean connectives ∨ , ∧ , ¬ , ⇒ , ⇔ , and3) if ψ (¯ x, ¯ y ) ∈ GF then ∀ ¯ x ( γ (¯ x, ¯ y ) ⇒ ψ (¯ x, ¯ y )) and ∃ ¯ x ( γ (¯ x, ¯ y ) ∧ ψ (¯ x, ¯ y )) are in GF, where γ (¯ x, ¯ y ) is anatomic formula containing all the free variables of ψ .The atoms γ relativising quantifiers in point (3) of theabove definition are called the guards of the quantifiers.For convenience, we sometimes allow ourselves to leavequantifiers for subformulas with at most one free variableto be unguarded (formally speaking, they can be guardedby atoms x = x ; such guards cause no problems even inthose of our constructions in which equalities are generallyforbidden).In GF we admit the use of equality and constants, butfunction symbols of arity greater than zero are forbidden. Triguarded fragment.
TGF is an extension of equality-freeGF in which quantification for subformulas with at mosttwo variables need not be guarded. Formally, the set ofTGF formulas is defined by taking the three syntax rulesdefining GF formulas (substituting in them ’GF’ to ’TGF’)and adding the following rule:) if ψ ( x, y ) is in TGF, then ∃ xψ ( x, y ) and ∀ xψ ( x, y ) belong to TGF.For convenience, instead of TGF we will mostly workwith the equivalent logic GFU, the guarded fragment withuniversal role . We assume that signatures for GFU alwayscontain the distinguished binary relation symbol U . The setof GFU formulas is then defined precisely as the set ofGF formulas, but the set of admissible models is restrictedto those which interpret U as the universally true relation.Structures interpreting U in this way will be called U -biquitous structures .It should be clear that TGF and GFU have the sameexpressive power (modulo the presence of the extra predicate U ). For example, the TGF-formula ∀ xy ( P ( x ) ∧ Q ( y ) ⇒∃ zR ( x, y, z )) can be transformed to the (up to U ) equiv-alent GFU-formula ∀ xy ( U ( x, y ) ⇒ ( P ( x ) ∧ Q ( y ) ⇒∃ zR ( x, y, z ))) . In the opposite direction, GFU-formulas canbe equivalently translated to TGF just by appending to themthe conjunct ∀ xy U ( x, y ) , thereby axiomatising U .In our constructions, we will frequently interpret GFUformulas over non- U -biquitous structures. In this case, theyare treated as usual GF formulas. Logics with transitive guards.
The guarded fragment withtransitive guards , GF+TG, is the logic whose formulas areconstructed over purely relational signatures containing dis-tinguished binary symbols T , T , . . . . The syntax of GF+TGis defined as the syntax of GF, with the only difference that T , T , . . . can be used only as guards. The equality symbolis allowed. Regarding the semantics, we require admissiblestructures to interpret T , T , . . . as transitive relations.The triguarded fragment with transitive guards , denotedTGF+TG, is obtained from GF+TG, as expected, by elimi-nating equality, and allowing quantification for subformulaswith at most two free variables to be unguarded. As in thecase of TGF, instead of TGF+TG we will mostly work withthe equivalent logic GFU+TG, the guarded fragment withuniversal role and transitive guards , whose signatures con-tain the special binary symbol U . The syntax of GFU+TG isas the syntax of GF+TG, and the set of admissible modelsis restricted to U -biquitous ones interpreting T , T , . . . astransitive relations. Normal form.
We say that a GF (GFU, GF+TG, GFU+TG)formula is in normal form if it is of the shape (cid:94) i ∀ ¯ x ( γ i (¯ x ) ⇒ ∃ ¯ y ( γ (cid:48) i (¯ x, ¯ y ) ∧ ψ i (¯ x, ¯ y ))) ∧ (cid:94) j ∀ ¯ x ( γ j (¯ x ) ⇒ ψ j (¯ x )) (1)where the γ i and the γ (cid:48) i and γ j are guards and the ψ i , ψ j are quantifier-free. The conjuncts indexed by i will besometimes called ∀∃ -conjuncts, while the conjuncts indexedby j will be called ∀ -conjuncts. Note that in our normal formwe do not explicitly include purely existential conjuncts like ∃ ¯ y ( γ (¯ y ) ∧ ψ (¯ y )) , which sometimes appear in similar normalforms; nevertheless, we will occasionally allow ourselves touse them, as they can be always simulated by ∀∃ -conjuncts ∀ x ( x = x ⇒ ∃ ¯ y ( G ( x, ¯ y ) ∧ γ (¯ y ) ∧ ψ (¯ y )) , for a fresh G . Let ϕ be normal form formula, A | = ϕ , ζ i the i -th ∀∃ -conjunct of ϕ and ¯ a a tuple of elements of A such that A | = γ i [¯ a ] . We then say that a tuple ¯ b such that A | = γ (cid:48) i (¯ a, ¯ b ) ∧ ψ i (¯ a, ¯ b ) is a witness for ¯ a and ζ i .The following lemma will allow us, when dealing with(finite) satisfiability or analysing the size of minimal modelsof GF (GFU) or GF+TG (GFU+TG) formulas, to concen-trate on normal form sentences of the shape as in (1). Aproof of a very similar lemma can be found in [14] (seeLemma 2 there). Lemma 1.
Let ϕ be a GF (GFU, GF+TG, GFU+TG)formula over a signature σ . Then one can effectivelycompute a set ∆ = { ϕ (cid:48) , . . . , ϕ (cid:48) d } of normal form GF(GFU, GF+TG, GFU+TG) formulas over an extendedsignature σ = σ ∪ σ aux of size polynomial in | σ | suchthat all the ϕ (cid:48) i are of length polynomial in | ϕ | , d isat most exponential in | ϕ | , (cid:87) s ≤ d ϕ (cid:48) s | = ϕ and every A | = ϕ has a σ -expansion A (cid:48) | = (cid:87) s ≤ d ϕ (cid:48) s .We conclude this subsection with two simple obser-vations with straightforward proofs allowing one to buildbigger models from existing ones. Both of them are intendedto be used in the absence of constants. Their variants for thecase with constants will be presented in the Appendix.Let σ be a purely relational signature. Let ( A i ) i ∈I bea family of σ -structures having disjoint domains. Their disjoint union is the structure A with domain A = (cid:83) i ∈I A i such that A (cid:22) A i is equal to A i and for any tuple ¯ a containingelements from at least two different A i and any relationsymbol P ∈ σ of arity | ¯ a | we have A | = ¬ P (¯ a ) . Lemma 2.
Let ϕ be a GF or GF+TG normal form formulaover a purely relational signature. The disjoint union ofany family of its models is also its model.Let A − be a σ -structure. Its doubling is the structure A built out of two copies of A − . Formally, its domainis A := A − × { , } and for each P ∈ σ we set A | = P [( a , (cid:96) ) , . . . , ( a k , (cid:96) k )] iff A − | = P [ a , . . . , a k ] forall a i ∈ A − and (cid:96) i ∈ { , } . Lemma 3.
Let ϕ be a normal form GF, GFU, GF+TG,GFU+TG formula which does not use equality (or usesit only as trivial guards x = x ) and let A − be a modelof ϕ . Then its doubling A is still a model of ϕ . External constructions and procedures.
In our work wewill extensively use results on the complexity of guardedlogics and on the size of their minimal finite models. Wecollect the relevant results in this paragraph. Generally,bounds in the original papers are formulated in terms ofthe length of the input formula. We additionally give somemore specific estimations, implicit in the original works,obtained by careful yet routine analysis of the proofs. Somecomments concerning these estimations can be found in theAppendix.
Theorem 4 ([2]).
GF (with constants and equalities) hasthe finite model property. Every satisfiable formula hasa model of size bounded doubly exponentially in itslength. More specifically, the size of minimal modelsf normal form formulas is bounded exponentially inthe size of the signature and doubly exponentially in itswidth.
Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([10]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the ∀∃ -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([10]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its ∀∃ -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for TGF (GFU)
Let us fix a GFU sentence ϕ in normal form, withoutequality, over a purely relational signature σ (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ϕ . Our goal isto build a finite U -biquitous model A (cid:48) of ϕ . Let α be the set of -types realized in A . We constructa GF σ -sentence ϕ ∗ by appending to ϕ the followingconjuncts: ∀ x (cid:0) (cid:95) α ∈ α α ( x ) (cid:1) (2) (cid:94) α,α (cid:48) ∈ α ∃ xy (cid:0) α ( x ) ∧ α (cid:48) ( y ) ∧ U ( x, y ) ∧ U ( y, x ) (cid:1) (3) (cid:94) P ∈ σ ∀ ¯ x (cid:16) P (¯ x ) ⇒ (cid:94) ≤ i,j ≤| ¯ x | U ( x i , x j ) (cid:17) (4)saying, respectively, that only -types from α are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ϕ ∗ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C − | = ϕ ∗ , and let C beits doubling. As ϕ ∗ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ϕ ∗ . K Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without Figure 1. An example structure C . Different colours of nodes representdifferent -types. The black bidirectional edges depict U -connections; theorange connection represents a ternary atom; the violet connection – abinary one. While C needs not be U -biquitous, for any pair of node coloursthere is a pair of distinct nodes connected by U . Moreover, C has another convenient property. Let us callelements a, a (cid:48) ∈ C indistinguishable in C if for any relationsymbol P ∈ σ , any tuple ¯ a ⊆ C and any tuple ¯ a obtainedfrom ¯ a by replacing some occurrences of a by a (cid:48) and someoccurrences of a (cid:48) by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds (see Fig. 1): Claim 8.
For any pair of -types α, α (cid:48) ∈ α there is a pairof their distinct realizations a, a (cid:48) in C such that C | = U [ a, a (cid:48) ] ∧ U [ a (cid:48) , a ] . Moreover, if α = α (cid:48) , then we evenfind indistinguishable a, a (cid:48) with that property. Proof:
Let b, b (cid:48) be elements witnessing the corre-sponding conjunct from subsentence (3) of ϕ ∗ in C − . If α (cid:54) = α (cid:48) then b and b (cid:48) are distinct and we can take a = ( b, and a (cid:48) = ( b (cid:48) , . If α = α (cid:48) then we take a = ( b, and a (cid:48) = ( b, . By the construction of C , a and a (cid:48) have therequired property. Note in particular that all -types in C − contain U ( x, x ) as they are realized in a U -biquitous modelof ϕ . This implies that C | = U [ a, a (cid:48) ] ∧ U [ a (cid:48) , a ] .From this point on, the model C − will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ϕ ∗ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ϕ ∗ and the lastof them being a desired U -biquitous model A (cid:48) of ϕ ∗ (andthus also of ϕ ).The domains of all these structures will be identical. A i = B × { , . . . , K } × { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, (cid:96) ∈ { , . . . , K } we make A (cid:22) B × { k } × { (cid:96) } isomorphic to B (via thenatural projection ( b, k, (cid:96) ) (cid:55)→ b ). By Lemma 2 we have that A | = ϕ ∗ .It is helpful to think that each of the A i is organized ina square table of size K × K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfythe guard γ i in one of the ∀∃ -conjuncts and thus requirewitnesses. To provide such witnesses we connect the pair b , b to one substructure located in one of the cells in A i +1 .Thereby, the challenge is to design a strategy which willallow us to perform a process of this kind without causingconflicts regarding the newly assigned connections. We nowpropose such a strategy. Some notation.
To describe our strategy in detail, let usintroduce some further notation. We denote by B k,(cid:96)i thestructure in the cell ( k, (cid:96) ) of A i , that is the structure A i (cid:22) B ×{ k } × { (cid:96) } . We recall that B k,(cid:96) is isomorphic to B . We willsometimes say that an element ( b, k, (cid:96) ) is the b -th elementof B k,(cid:96)i . Further, for m = 0 , . . . , , we denote by C k,(cid:96),mi thestructure B k,(cid:96)i (cid:22) { mK + 1 , . . . , mK + K } × { k } × { (cid:96) } . Werecall that each C k,(cid:96),m is isomorphic to C . See Fig. 2. Entry elements and their use.
For any ≤ k, (cid:96) ≤ K ,let α k = tp B ( k ) and α (cid:96) = tp B ( (cid:96) ) . For each such pair k, (cid:96) we now choose a pair of entry elements for each of the fivestructures in the cell ( k, (cid:96) ) of A , that is for the structures C k,(cid:96),m ( m = 0 , , . . . , ).By Claim 8, there are distinct elements e , e ∈ C such that C | = α k [ e ] ∧ α (cid:96) [ e ] ∧ U [ e , e ] ∧ U [ e , e ] andif α k = α (cid:96) then e and e are indistinguishable in C . Wechoose the entry elements e k,(cid:96),m , e k,(cid:96),m to C k,(cid:96),m to bethe corresponding copies of e and e in each of C k,(cid:96),m (recalling that the domains of all the A i are the same, theelements e k,(cid:96),m , e k,(cid:96),m belong to C k,(cid:96),mi , for all i ). Theentry elements will serve as a template for connecting someexternal pairs of elements to C k,(cid:96),mi . This will be done bythe following construction.By + C k,(cid:96),mi we denote the structure with domain C k,(cid:96),mi ∪ { b , b } for some fresh elements b , b such that + C k,(cid:96),mi (cid:22) C k,(cid:96),mi = C k,(cid:96),mi and for each P ∈ σ andeach tuple ¯ a containing at least one of b , b we have + C k,(cid:96),mi | = P [¯ a ] iff C k,(cid:96),m | = P [ h (¯ a )] , where h is thefunction defined as h ( b ) = e k,(cid:96),m , h ( b ) = e k,(cid:96),m and h ( a ) = a for a ∈ C k,(cid:96),mi (we emphasize that in thisdefinition we copy the relations from C k,(cid:96),m and not from itspossibly modified version C k,(cid:96),mi ). In particular + C k,(cid:96),mi | = α k [ b ] ∧ α (cid:96) [ b ] ∧ U [ b , b ] ∧ U [ b , b ] . From A i to A i +1 . Assume now that the structure A i hasbeen defined, for some i ≥ . If A i is U -biquitous thenwe are done. Otherwise let b , b be a pair of elements in A i such that A i | = ¬ U [ b , b ] . For s = 1 , let k s , (cid:96) s , n s be such that b s is the n s -th element of B k s ,(cid:96) s i . Let uschoose t ∈ { , . . . , } such that C n ,n ,ti does not containthe k -th, (cid:96) -th, k -th or (cid:96) -th element of B n ,n i . Such a t must exist by the pigeon hole principle. We make the Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without k , ℓ Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without k , ℓ n n i i Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without n , n ,3 i Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without n , n ,1 i Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without n , n ,2 i Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without n , n ,4 i Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E
X P T I M E -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E X P T I M E -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P ( ¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without n , n i Theorem 5 ([6]).
The satisfiability problem for GF (withconstants and equalities) is 2E XP T IME -complete. Morespecifically, there is a procedure that, given a normalform formula, works in time bounded polynomially inthe length of the input, exponentially in the size of thesignature, and doubly exponentially in its width.
Theorem 6 ([11]).
Every finitely satisfiable GF +TG for-mula (without constants, with equalities) has a model ofsize bounded doubly exponentially in its length. Morespecifically, for normal form formulas, the size of theirminimal finite models is bounded exponentially in thenumber of the -conjuncts of the input and doublyexponentially in the size of the signature. Theorem 7 ([11]).
The satisfiability problem for GF +TG(without constants, with equalities) is 2E XP T IME -complete. More specifically, there is a procedure that,given a normal form formula, works in time boundedpolynomially in the length of its input, exponentially inthe number of its -conjuncts and doubly exponentiallyin the size of the signature.
3. Finite model construction for
TGF ( GFU ) Let us fix a GFU sentence ' in normal form, withoutequality over a purely relational signature (we will explainhow to cover the case of signatures containing constantslater) and let A be a U -biquitous model of ' . Our goal isto build a finite U -biquitous model A of ' . Let ↵ be the set of -types realized in A . We constructa GF -sentence ' ⇤ by appending to ' the followingconjuncts: x _ ↵ ↵ ↵ ( x ) (2) ^ ↵,↵ ↵ xy ↵ ( x ) ^ ↵ ( y ) ^ U ( x, y ) ^ U ( y, x ) (3) ^ P ¯ x ⇣ P (¯ x ) ) ^ i,j | ¯ x | U ( x i , x j ) ⌘ (4)saying, respectively, that only -types from ↵ are realized,every pair of -types has a realization both-ways connectedby U and every guarded pair of elements is connected by U . We can treat (2)–(4) as normal form conjuncts.It is clear that ' ⇤ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C | = ' ⇤ , and let C beits doubling. As ' ⇤ does not use equality (or, to be strict,needs it only for trivial guards x = x , omitted from (2)),we have by Lemma 3 that C | = ' ⇤ .Moreover, C has another convenient property. Let us callelements a, a C indistinguishable in C if for any relationsymbol P , any tuple ¯ a ✓ C and any tuple ¯ a obtained from ¯ a by replacing some occurrences of a by a and someoccurrences of a by a we have that C | = P [¯ a ] iff C | = P [¯ a ] . Then the following holds: Claim 8.
For any pair of -types ↵, ↵ ↵ there is a pairof their distinct realizations a, a in C such that C | = U [ a, a ] ^ U [ a , a ] . Moreover, if ↵ = ↵ , then we evenfind indistinguishable a, a with that property. Proof:
Let b, b be elements witnessing the corre-sponding conjunct from subsentence (3) of ' ⇤ in C . If ↵ = ↵ then b and b are distinct and we can take a = ( b, and a = ( b , . If ↵ = ↵ then we take a = ( b, and a = ( b, . By the construction of C , a and a have therequired property. Note in particular that all -types in C contain U ( x, x ) as they are realized in a U -biquitous modelof ' . This implies that C | = U [ a, a ] ^ U [ a , a ] .From this point on, the model C will not play anyrole. However, it will be convenient to build, using Lemma2, yet another model B | = ' ⇤ , this time as the disjointunion of five copies of C . Letting K = | C | , we assumethat the domain of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structure on { mK + 1 , . . . , mK + K } isisomorphic to C . U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ' ⇤ and the lastof them being a desired U -biquitous model A of ' ⇤ (andthus also of ' ).The domains of all these structures will be identical. A i = B ⇥ { , . . . , K } ⇥ { , . . . , K } . The initial structure A is defined as the disjoint union of (5 K ) copies of B . Namely, for each k, ` , . . . , K } we make A B ⇥ { k } ⇥ { ` } isomorphic to B (via thenatural projection ( b, k, ` ) b ). By Lemma 2 we have that A | = ' ⇤ .It is helpful to think that each of the A i is organized ina square table of size K ⇥ K . In particular every cell of A contains a copy of B (which itself is a 5-fold copy of C ), and in A , there are no connections whatsoever betweenelements from different cells. Outline of the construction.
The whole process may beseen as a careful saturation of the initial model A with U -connections. In the passage from A i to A i +1 we take apair of distinct domain elements b , b not connected by U yet. By Claim 8, we can find in C a pair of distinctelements a , a that have the same -types as b , b , but,in addition, are connected by U . We want to make theconnection between b and b isomorphic to the connectionbetween a and a , but after this, b , b may start to satisfysome of the guards i in one of the -conjuncts and thusrequire witnesses. To provide such witnesses we connectthe pair b , b to one substructure located in one of thecells in A i +1 . Thereby, the challenge is to design a strategywhich will allow us to perform a process of this kind without n , n ,0 i k ℓ ℓ k Figure 2. A single step of U -saturation. The green and blue elements in B n ,n i are the entry elements (to the C n ,n ,m ). The orange and violetconnections, as well as the black U -connections inside C n ,n , i are alreadypresent in C n ,n , (isomorphic to C ). They give rise to the orange, violetand black connections joining various cells of the table. For transparency,not all newly arising connections are shown. structure A i +1 (cid:22) C n ,n ,ti ∪ { b , b } isomorphic to + C n ,n ,ti .The rest of the structure A i remains untouched. Fig. 2illustrates the described step. Note that the orange ternaryatom from C n ,n , i is inherited from C n ,n , . Indeed aquick inspection shows that our construction never addsnew local ternary atoms, where by a local atom we meanan atom in one of the substructures C k,(cid:96),mi . Since C n ,n , satisfies (4) also the black connections shown in C n ,n , i are present already in C n ,n , . Later we will explain thatour construction never modifies guarded types, so also theviolet connection is present already in C n ,n , . We argue that for all i we have A i | = ϕ ∗ . Note first thatour construction never modifies the -types of elements. Claim 9.
For every i we have that tp A i ( a ) = tp A ( a ) . Proof:
The proof goes by induction. Assume that forall a and some i we have that tp A i ( a ) = tp A ( a ) . In thepassage from A i to A i +1 we modify only some substructure A i (cid:22) C k,(cid:96),mi ∪ { b , b } , where b is the k -the element of itscell and b is the (cid:96) -the element of its cell. By the inductiveassumption they retain in A i their -types from A whichare, α k and α (cid:96) , respectively. In this step, we do not modify C k,(cid:96),mi at all, so in particular its elements retain their -types. The -type of b ( b ) is set to be equal to the type ofthe first (second) entry element of C k,(cid:96),m which is, by ourefinition, of type α k ( α (cid:96) ). So also b and b do not changetheir -types.The following claim is crucial for the correctness of ourconstruction. Claim 10.
Let i > and assume A i | = ϕ ∗ . Then everyguarded tuple ¯ a of domain elements in A i (includingthe tuples guarded by U ) retains its type in A i +1 , thatis: tp A i +1 (¯ a ) = tp A i (¯ a ) . Proof: A i +1 is obtained from A i by making changesonly in the substructure with domain C n ,n ,ti ∪ { b , b } (where n , n , t, b , b are as in the description of the con-struction of A i +1 ). The substructure C n ,n ,ti itself is nottouched at all.By the conjunct (4) of ϕ ∗ we have that the pair b , b cannot be guarded in A i . Thus any tuple guarded in A i which could potentially change its type in A i +1 must con-tain exactly one of b , b and (possibly) some elements of C n ,n ,ti . Consider one such tuple ¯ a .If ¯ a is built exclusively from b or exclusively from b then the claim follows from Claim 9.Consider the case when ¯ a contains exactly one of b , b and at least one other element. Then ¯ a contains eitherelements from two different cells of the K × K table,or elements from two different substructures C n ,n ,mi inthe cell n , n (the substructure containing b /b and thesubstructure with m = t ); in both cases it is not guarded in A . So, its type had to be modified in the passage from A j to A j +1 for some j < i − (it is also possible that it wasdefined in several such passages; in this case assume that j → j + 1 is the last of them).W.l.o.g. assume that out of b , b the tuple ¯ a contains b . Thus the two cells which contain the elements of ¯ a are ( n , n ) which contains C n ,n ,ti , and ( k , l ) whichcontains b (as we noted, it is possible that this is actuallythe same cell). Moreover, in the structure B n ,n i from thecell ( n , n ) its k -th and l -th elements are not membersof ¯ a , which is ensured by our choice of t . Hence, by ourstrategy, none of the elements of ¯ a \ { b } was a memberof a pair of elements which was connected to C k ,l ,ms forany s , and thus it must be the case that the element b ,together with some other element b (having the same -type as b ), were connected to C n ,n ,tj when forming A j +1 .If the -types α n , α n of b and, resp., b are different,then the truth values of the atoms containing b in A j +1 were defined in accordance with the truth values of thetuples containing the entry element e n ,n ,t in the structure A , exactly as they are defined in A i +1 . So, there are noconflicts in this case. If α n = α n , it may happen thatthe atoms containing b in A j +1 are defined in accordancewith the truth values of the tuples containing the entryelement e n ,n ,t . In this case however there are also noconflicts since the entry elements of the structures C n ,n ,t are indistinguishable when α n = α n .By straightforward induction we get: Claim 11.
Every guarded tuple of elements in A retains itstype in A i +1 . We are ready to show that A i | = ϕ ∗ implies A i +1 | = ϕ ∗ . Claim 12. If A i | = ϕ ∗ then A i +1 | = ϕ ∗ . Proof:
Let us observe first that A i +1 | = ϕ . Forthis consider any ∀∃ -conjunct of ϕ : ζ = ∀ ¯ x ( γ i (¯ x ) ⇒∃ ¯ y ( γ (cid:48) i (¯ x, ¯ y ) ∧ ψ i (¯ x, ¯ y ))) and assume A i +1 | = γ i [¯ a ] for sometuple ¯ a . We consider two cases:(a) A i | = γ i [¯ a ] . Since A i | = ϕ , we have in particular that A i | = γ (cid:48) i [¯ a, ¯ b ] ∧ ψ i [¯ a, ¯ b ] for some tuple ¯ b . As γ (cid:48) i is an atomicformula, the tuple ¯ a ¯ b is guarded in A i and by Claim 10 itretains its type in A i +1 . Hence A i +1 | = γ (cid:48) i [¯ a, ¯ b ] ∧ ψ i [¯ a, ¯ b ] . Itfollows that A | = ζ .(b) A i (cid:54)| = γ i [¯ a ] . In this case the fact γ i [¯ a ] appeared firstin A i +1 . Recall the construction of A i +1 and the notationused there. Let h : { b , b } ∪ C n ,n ,ti → C n ,n ,ti be thefunction returning e n ,n ,ts (for s = 1 , ) for b s and returning a for a ∈ C n ,n ,ti . By the definition of A i +1 we have that C n ,n ,t | = γ i [ h (¯ a )] . As C n ,n ,t | = ζ there is a tuple ¯ b ( = h (¯ b ) ) in C n ,n ,t such that C n ,n ,t | = γ (cid:48) i [ h (¯ a ) , h (¯ b )] ∧ ψ i [ h (¯ a ) , h (¯ b )] . Since γ (cid:48) i is an atom, the tuple h (¯ a ) h (¯ b ) isguarded in A . For any tuple ¯ a ⊆ ¯ a ¯ b not containing any of b , b we have that ¯ a = ( h (¯ a ) ) ⊆ h (¯ a ) h (¯ b ) , so the type of ¯ a = ( h (¯ a ) from A is retained in A i +1 by Claim 11. Fortuples ¯ a ⊆ ¯ a ¯ b containing b and/or b , their type in A i +1 is the same as the type of h (¯ a ) in A , by the definition of A i +1 . In both cases, the type of ¯ a in A i +1 is the same as thetype of h (¯ a ) in A . It follows that A i +1 | = γ (cid:48) i [¯ a, ¯ b ] ∧ ψ i [¯ a, ¯ b ] ,and thus A i +1 | = ζ .The reasoning for the ∀ -conjuncts is similar but simpler(actually, ∀ -conjuncts are special case of ∀∃ -conjuncts).As we noted the conjuncts (2)–(4) are normal formconjuncts and thus we can argue about them exactly as aboutthe conjuncts of ϕ .That our construction terminates follows from Claim 10.Indeed, it implies that all pairs of elements connected by U in A i remain connected by U in A i +1 . On the other handat least one new U -connection appears in A i +1 : the onebetween the elements b and b . As the number of elementsin the domain of our structures is fixed and finite, after afinite number of steps we end up in a structure A f in whichany two elements are connected by U . Since A | = ϕ ∗ ,Claim 12 implies, by induction, that A f | = ϕ ∗ and inparticular A f | = ϕ . Thus, we may take A (cid:48) := A f as thedesired U -biquitous model of ϕ . The proof of the FMP for GFU (TGF) presented abovecan be extended without major problems to the case ofsignatures containing constants. Here we outline the basicidea, for a more detailed description of the construction seeAppendix B.Given a structure A interpreting a signature with con-stants we call the subset ˆ A ⊆ A consisting of the interpreta-tions of all constants the named part of A . We set ˇ A := A \ ˆ A and call it the unnamed part of A .We proceed as previously. We take a satisfiable normalform formula ϕ , expand it to ϕ ∗ and take a (not necessarily -biquitous) finite model C − | = ϕ ∗ . In the absence ofconstants we extensively used constructions building biggermodels out of many copies of some existing ones (Lemmas2, 3). As this time we cannot reproduce the named part ofmodels, those constructions have to be replaced by oneswhich multiply only their unnamed parts. That is, whenproducing C we double only ˇ C − , when producing B weform the disjoint union of five copies of ˇ C , and whenproducing A we form the disjoint union of K × K copies of ˇ B . In each of the above steps, all the copiesof the unnamed part of the input model are attached to asingle, shared copy of its named part, in such a way thatthe restriction of the resulting structure to the union of anycopy of the unnamed part and the copy of the named part isisomorphic to the input model. In effect, the named part of A is inherited from the initial model C − . It is not difficultto show that after each of the above steps we still have amodel of ϕ ∗ . In particular A | = ϕ ∗ .Next, we perform the U -saturation process. Generally,it goes as previously: we find a pair of elements b , b notconnected by U , join them by U and connect them to theappropriate cell of the table to provide necessary witnesses.We note only, that this time, this step involves defining thetruth values of relations on tuples consisting of the b i , theelements from the cell to which the b i are connected and,possibly, the interpretations of constants. The process leadseventually to a U -biquitous model A (cid:48) | = ϕ ∗ . We now estimate the size of finite models that can beproduced by a use of our construction.Assume we want to construct a finite model of a satis-fiable formula ϕ over a signature σ . We first convert ϕ into a disjunction of normal form formulas as guaranteed byLemma 1, and choose a satisfiable normal form disjunct ϕ (over an extended signature σ ). We take an arbitrary model A | = ϕ . Next we append to ϕ the auxiliary conjunctsobtaining a normal form formula ϕ ∗ , send ϕ ∗ to a blackbox producing a finite but generally non- U -biquitous model C − | = ϕ ∗ , form models C , B , A and saturate A toget finally an U -biquitous model A (cid:48) . By Lemma 1, | ϕ | ispolynomial in | ϕ | . | ϕ ∗ | is exponential in | ϕ | in the casewithout constants and doubly exponential in the case withconstants. This follows from the fact that | ϕ ∗ | contains theconjuncts (2) and (3) whose size is polynomial in the numberof -types over σ . So, we need to be careful and avoidestimating the size of A only in terms of the length of ϕ ∗ .As the external black box procedure we can use anyprocedure constructing a finite model of a satisfiable GFformula. Let as assume that we use the model producedby the construction from [2]. By Thm. 4 the size of thismodel is bounded exponentially in the size and doublyexponentially in the width of the signature of ϕ ∗ , which isthe same as the signature of ϕ , σ . As the size and the widthof σ are bounded by | ϕ | which, by Lemma 1, is polynomialin | ϕ | , eventually our bound on the size of C is doublyexponential in the size of the input formula | ϕ | . Recall that | C | = 2 | C | , | B | = 5 | C | = 10 | C | , and forall i : | A i | = | B | · | B | = (10 | C | ) which is still doublyexponential in | ϕ | . Thus we get: Theorem 13.
Every satisfiable TGF (GFU) formula ϕ (withor without constants) has a finite model of size boundeddoubly exponentially in the length of ϕ .This bound is essentially optimal, since even in GFwihtout constants and equality one can construct a family ofsatisfiable formulas ϕ i , each of them of length polynomialin i , but having only models of size at least i . This isimplicit in [6].The finite model property of TGF (GFU) implies thatits finite satisfiability problem is equal to its satisfiabilityproblem and thus it is 2E XP T IME -complete in the absenceof constants and N2E XP T IME -complete with constants asshown in [12].
4. Finite satisfiability of GF+TG and GFU+TG
Let us recall that in case of logics with transitive guards,we work with signatures containing no constants, however,in GF+TG we permit equality. Still, our decidability resultsfor GFU+TG will be obtained in the absence of equality,since, as we said, already GFU with equality is undecidable.For convenience, we first slightly enhance our normalform. Given a normal form GF+TG or GFU+TG formulaas in (1) we split its ∀∃ -conjuncts into those in which γ (cid:48) i isa non-transitive symbol and those in which it is transitive.Moreover, for the latter, we assume that the guard γ i hasonly one variable. If this is not the case – that is we have aconjunct of the form ∀ ¯ x (cid:16) γ i ( x , . . . , x k ) ⇒ ∃ y (cid:0) γ (cid:48) i ( x j , y ) ∧ ψ i ( x j , y ) (cid:1)(cid:17) with k > and γ (cid:48) i using a transitive symbol – we replace itby ∀ ¯ x (cid:0) γ i ( x , . . . , x k ) ⇒ G ji ( x j ) (cid:1) ∧ ∀ x j (cid:16) G ji ( x j ) ⇒ ∃ y (cid:0) γ (cid:48) i ( x j , y ) ∧ ψ i ( x j , y ) (cid:1)(cid:17) where G ji is a fresh unary symbol.Further, we assume that all the guards γ i in the ∀∃ -conjuncts are non-transitive. If this is not the case, that iswe have a transitive guard γ i , say of the form T ( x, y ) , thenwe replace it by G ( x, y ) , for a fresh, non-transitive symbol G , and append the ∀ -conjunct ∀ xy (cid:0) T ( x, y ) ⇒ G ( x, y ) (cid:1) .Finally, for convenience, we append to normal form for-mulas a conjunct saying that every guarded pair of elementsis connected by Aux , where
Aux is a fresh binary symbol.This auxiliary conjunct does not affect satsfiability of theformula.o, we will assume that normal form formulas forGF+TG and GFU+TG are of the shape: (cid:94) h ∀ ¯ x (cid:16) γ h (¯ x ) ⇒ ∃ ¯ y (cid:0) ϑ h (¯ x, ¯ y ) ∧ ψ h (¯ x, ¯ y ) (cid:1)(cid:17) ∧ (cid:94) i ∀ x (cid:16) γ i ( x ) ⇒ ∃ y (cid:0) θ i ( x, y ) ∧ ψ i ( x, y ) (cid:1)(cid:17) ∧ (cid:94) j ∀ ¯ x (cid:16) γ j (¯ x ) ⇒ ψ j (¯ x ) (cid:17) ∧ (cid:94) P ∈ σ ∀ ¯ x (cid:16) P (¯ x ) ⇒ (cid:94) ≤ i,j ≤| ¯ x | Aux ( x i , x j ) (cid:17) (5)where the γ h and γ i are non-transitive guards, γ j is a guard(transitive or non-transitive), the ϑ h are non-transitive guardsand the θ i are transitive guards. We recall that the transitivesymbols appear in none of ψ h , ψ i and ψ j . The conjunctsindexed by h will be called ∀∃ ntr -conjuncts, the conjunctsindexed by i will be called ∀∃ tr -conjuncts, the conjunctsindexed by j , together with the conjuncts speaking about Aux , will be called ∀ -conjuncts. Let us fix a finitely satisfiable normal form GF+TGformula ϕ over a purely relational signature σ , of the shapeas in (5). Equalities are allowed in ϕ . Let A be a finite modelof ϕ . We plan to construct a finite model A (cid:48) | = ϕ of sizebounded doubly exponentially in | ϕ | . Let α be the set of -types realized in A . Let β be the set of non-degenerateguarded -types realized in A . For β ∈ β let β − denote theset of formulas obtained from β by removing T ( x , x ) , T ( x , x ) , ¬ T ( x , x ) , ¬ T ( x , x ) , for all transitive T , ifthey are present in β . Note that β − still contains the literalsspeaking about T ( x , x ) and T ( x , x ) . β − will be calledthe transitive-free reduction of β . Also, let A − denote thestructure obtained from A by removing all facts T [ a, b ] fora transitive T and a (cid:54) = b . That is, in A − the only transitivefacts may be of the form T [ a, a ] for some a ∈ A . Constructing B ∗ and C ∗ . We now construct two auxiliaryformulas out of ϕ . Let ϕ B := (cid:94) h ∀ ¯ x (cid:16) γ h (¯ x ) ⇒ ∃ ¯ y (cid:0) ϑ h (¯ x, ¯ y ) ∧ ψ h (¯ x, ¯ y ) (cid:1)(cid:17) ∧ (cid:94) j ∀ ¯ x (cid:16) γ j (¯ x ) ⇒ ψ j (¯ x ) (cid:17) ∧ (cid:94) P ∈ σ ∀ ¯ x (cid:16) P (¯ x ) ⇒ (cid:94) ≤ i,j ≤| ¯ x | Aux ( x i , x j ) (cid:17) ∧ (cid:94) T s ∀ xy (cid:16) T s ( x, y ) ⇒ x = y (cid:17) ∧ (cid:94) β ∈ β ∃ xy β − ( x, y ) ∧ (cid:94) α ∈ α ∃ x α ( x ) ∧ ∀ x (cid:95) α ∈ α α ( x ) That is, ϕ B contains all the ∀∃ ntr -conjuncts of ϕ , all its ∀ -conjuncts, plus the conjuncts saying that the transitiverelations do not connect distinct elements, for every non-degenerate guarded -type realized in A its transitive-freereduction is realized (note that this is a guarded formula,since β − remains guarded as it contains Aux ( x, y ) ), andthat a -type is realized iff it is realized in A . We remarkthat the conjuncts speaking about α and β may containtransitive atoms outside guards, but these may only beatoms of the form T ( x, x ) or T ( y, y ) for some transitive T . We can replace them by P ( x ) or, resp., P ( y ) , forsome fresh P , and add normal form conjuncts ensuringthat ∀ x ( P ( x ) ⇔ ∃ y ( T ( x, y ) ∧ x = y )) , obtaining thisway formulas in which T is used only in guard positions.Moreover, as the restriction imposed by ϕ B on the transitiverelations makes their transitivity irrelevant, we will treat ϕ B as a GF formula.Further, let ϕ C := (cid:94) i ∀ x (cid:16) γ i ( x ) ⇒ ∃ y (cid:0) θ i ( x, y ) ∧ ψ i ( x, y ) (cid:1)(cid:17) ∧ (cid:94) j : γ j transitive ∀ ¯ x (cid:16) γ j (¯ x ) ⇒ ψ j (¯ x ) (cid:17) ∧ (cid:94) P ∈ σ (cid:94) ¯ x ∈S P ∀ xy (cid:16) P (¯ x ) ⇒ Aux ( x, y ) (cid:17) ∧ ∀ xy (cid:16) Aux ( x, y ) ⇒ ( x (cid:54) = y ⇒ (cid:95) β ∈ β β − ( x, y )) (cid:17) ∧ (cid:94) α ∈ α ∃ xα ( x ) ∧ ∀ x (cid:95) α ∈ α α ( x ) where S P is the set of tuples of length equal to the arity of P built out of variables x and y , and containing at least oneoccurrence of each of them.That is, ϕ C contains all the ∀∃ tr -conjuncts and those ∀ -conjuncts of ϕ that do not speak about Aux , plus theconjuncts saying that for every guarded tuple built out of twoelements these two elements are connected by
Aux , everyguarded pair of distinct elements satisfies the transitive-freereduction of a guarded -type from A , and that a -typeis realized iff it is realized in A . Note that the ∀ -conjunctswe include here have transitive γ j , so they use at most twovariables; we may assume that they are x, y . The remainingconjuncts also use only variables x and y , so ϕ C is aformula belonging to GF +TG (again after the appropriateadjustments concerning the use of transitive relations in eachof the α and β ).Note that both ϕ B and ϕ C are finitely satisfiable, as theformer is satisfied in A − and the latter in A . Treating ϕ B asa GF formula we take its small finite model B as guaranteedby [2]. Similarly, treating ϕ C as a GF +TG formula we takeits small finite model C , as guaranteed by [10]. We remarkhere that while [10] considers explicitly only signatures withrelation symbols of arity and , a routine inspection showsthat the constructions there work smoothly even if symbolsof arity greater than are allowed, which is the case in ourwork. Indeed, the presence of relations of arity greater than could be important in [10] only when -types are assignedo pairs of elements. However, those -types are read offfrom a pattern model, and whether they contain higher arityrelations or not is not relevant. Since ϕ C is a two-variableformula, we may assume that C contains no fact with morethan two distinct elements. We note that C happens to satisfyall the ∀ -conjuncts of ϕ . The conjuncts with transitive γ j areincluded explicitly in ϕ C ; for those with non-transitive γ j assume that C | = γ j [¯ a ] for some tuple ¯ a . By our assumptionon C the tuple ¯ a is built out of at most two elements. So,either it uses only one element and then its -type is realizedin A (by the conjunct of ϕ C speaking about α ), or it usestwo elements, and then the transitive-free reduction of their -type is the same as the reduction of some -type realizedin A (by the conjunct speaking about β ). Since A | = ϕ ,and in particular A | = ∀ ¯ x ( γ j (¯ x ) ⇒ ψ j (¯ x )) , and recallingthat ψ j does not contain any transitive relations, it followsthat C | = ψ j [¯ a ] .By the conjuncts of ϕ B and ϕ C speaking about α , weknow that the sets of -types realized in B and C are equal(concretely, they are equal to α ). We now construct models B ∗ | = ϕ B and C ∗ | = ϕ C such that the number of realizationsof α in B ∗ is equal to the number of its realizations in C ∗ ,for all α .Let m be the maximal number of realizations of a -type α in B (over all α ∈ α ). Let C ∗ be the disjoint union of m copies of C . By Lemma 2 we have that C ∗ | = ϕ C . Notethat for every α the number of realizations of α in B isless than or equal to the number of its realizations in C ∗ . Tomake these numbers equal we successively adjoin additionalrealizations of the appropriate -types to B . This is simple:to adjoin a realization b of a -type α we choose a patternelement a of type α in B and make the structure on (cid:0) B \{ a } (cid:1) ∪{ b } isomorphic to B ; we add no facts containing both a and b to the structure. After each such step the resultingstructure is still a model of ϕ B . This way we eventually getthe desired B ∗ . Constructing D . Let K = | B ∗ | = | C ∗ | . Let b , . . . , b K − be any enumeration of the elements of B ∗ and let α i =tp B ∗ ( b i ) , for all ≤ i < K . We create a new structure D , with domain D = { , . . . , K − } × { . . . , K − } . Weset tp D ( k, (cid:96) ) := α k + (cid:96) mod K . Viewing in the natural way D as a square table, we see that from its every row and itsevery column one can construct bijections into B ∗ and C ∗ preserving the -types. Without modifying the -types, wecan thus define the structure of D on every row and everycolumn in such a way that they become isomorphic copiesof B ∗ , and C ∗ , respectively. This completes the definition of D , that is, we add no further facts to it. Note that transitiverelations cannot connect elements from different columns.Call a guarded tuple of elements of D vertical ( hori-zontal ) if it belongs to a single column (row) of the table.Observe that the tuples built out of a single element areboth vertical and horizontal and every guarded tuple in D is either vertical or horizontal by the definition of D .Note that D satisfies the ∀∃ tr -conjuncts of ϕ . Indeed,any element a satisfying some γ i [ a ] has the required witnessin its column, since the relevant conjunct is a member of ϕ C . Also the ∀ -conjuncts are satisfied: we have explainedthat they are satisfied in C , so it follows that they are alsosatisfied in C ∗ , and thus in every column; on the otherhand ϕ B includes them explicitly, so they are safisfied inevery row. Concerning the ∀∃ ntr -conjuncts, note that anyhorizontal guarded tuple ¯ a satisfying any of γ h [¯ a ] has therequired witnesses in its row (since its row satisfies ϕ B ).The only problem is that some vertical guarded tuplesmay not have witnesses for some ∀∃ ntr -conjuncts. Notethat every such tuple is built out of precisely two elements.Indeed, by our assumption about C ∗ there are no verticalguarded tuples containing three or more distinct elementsthere, and on the other hand any tuple built out of a singleelement has its witnesses in its row.We will fix the above problem by taking an appropriatenumber of copies of D and adjoining every vertical guardedpair of elements to a row in some different copy of D . Thiswill be done in a circular way, reminiscent of the smallmodel construction for FO from [7]. We emphasise thatthis process is much simpler than the U -saturation processfrom Section 3.2 since this time we do not need to deal withpairs of elements from different copies of our basic buildingblock D . Let us turn to details. Building a small model of ϕ . Assume that D and D are two copies of D . Consider a vertical guarded pair ofdistinct elements b , b in D . As the column of this pair isa model of ϕ C we know that it satisfies Aux [ b , b ] and thusalso it satisfies β − [ b , b ] for some β ∈ β . (We recall that β − does not mention any transitive connection between b , b but such connections may be present in D .) As any row E of D is a model of ϕ B it follows that this row containsa pair of distinct elements a , a such that E | = β − ( a , a ) .(Here, there are no transitive connections between a and a as ensured by ϕ B .)We now describe the procedure to which we will laterrefer by saying: we connect the pair b , b to the row E (using a , a as a template) . For any tuple ¯ a containingat least one of b , b and some elements of E \ { a , a } and any non-transitive relation symbol P ∈ σ of arity | ¯ a | we add the fact P [¯ a ] iff E | = P [ h (¯ a )] where h is thefunction returning a s for b s ( s = 1 , ) and a for all a ∈ E .In other words, we connect b , b with the elements of E \ { a , a } exactly as a , a are connected with theseelements in E . We add no other facts. (In particular inthis procedure we add no transitive connections.) This waythe guarded tuples containing any of b , b (or both) andpossibly some elements of E , have all the required witnessesfor the ∀∃ ntr -conjuncts since the structure we have definedon (cid:0) E \ { a , a } (cid:1) ∪ { b , b } is isomorphic to E when thetransitive relations are not taken into account (and recallthat the ∀∃ ntr -conjuncts do not mention transitive relationsat all).Now, let the structure A (cid:48) be the disjoint union of K copies of D . More specifically its domain is A (cid:48) = { , , }×{ , . . . , K − } × D and A (cid:48) (cid:22) { i } × { j } × D is isomorphicto D for all i, j . Denote A i = { i } × { , . . . , K − } × D ,for i = 0 , , .or every vertical guarded pair of distinct elements b , b in A i chose a row in a copy of D contained in A ( i +1) mod 3 such that this row has not yet been used by any other pairfrom the column of b and b and connect b , b to this row(using an appropriate template). As the number of pairs ofelements in the row of b , b is smaller than K and thereare K copies of D in each of the A i , and each of them has K rows, we have sufficiently many rows to perform thisstep.Using three sets A i and applying the above describedcircular strategy guarantee that the process can be performedwithout conflicts: if an element b is connected to a row E then no element of E is ever connected to the row of b .Now it should be clear that indeed the eventually ob-tained structure models ϕ . Size of models and complexity.
We now analyse the smallmodel construction described in the previous paragraphs andobtain the following (optimal) bound on the size of minimalfinite models for GF+TG. For further purposes in the secondpart of this theorem we formulate a more specific bound fornormal form formulas.
Theorem 14.
Every finitely satisfiable GF+TG formula with-out constants has a model of size bounded doubly expo-nentially in its length. For finitely satisfiable normal formformulas there are models of size bounded exponentiallyin the size of the signature and the number of their ∀∃ -conjuncts, and doubly exponentially in the width of thesignature. Proof:
Let us summarize the steps needed to producea small model of an input finitely satisfiable GF+TG sen-tence ϕ over a signature σ . We convert it into a disjunctionof normal form formulas over an extended signature σ as inLemma 1, and choose its finitely satisfiable normal formdisjunct ϕ and its finite model A . Let r be the numberof symbols in σ and w its width. As by Lemma 1 | ϕ | ispolynimial in | ϕ | , both r and w are polynomial in | ϕ | .Perform our small model construction for ϕ and let ϕ B , ϕ C , B , C , B ∗ , C ∗ , D and A (cid:48) be as in this construction.Recall that as B we take the small model for ϕ B (which is a normal form formula over the signature σ ),constructed as in [2]. By Thm. 4 the size of B is boundedexponentially in r and doubly exponentially in w , that isdoubly exponentially in | ϕ | .Concerning C , we take as it the small model for ϕ C (which is a normal form formula over the signature σ ),constructed by applying the small model construction from[10]. By Thm. 6, | C | is bounded exponentially in the numberof its ∀∃ -conjuncts (which are actually taken from ϕ ) anddoubly exponentially the size of the signature. So, it isbounded doubly exponentially in | ϕ | .Further, each of the structures B ∗ , C ∗ has size at most | B |·| C | , the structure D – at most | B ∗ | and the final model A (cid:48) – at most | B ∗ || D | = 3 | B ∗ | = 3 | B | | C | , which is stilldoubly exponential in | ϕ | .The second part of the theorem, concerning normal formformulas follows easily from the information on the size of C and B given above and from the observation that the finalestimation on | A (cid:48) | is polynomial in | B | and | C | .Thm. 14 immediately yields a N2E XP T IME -upperbound on the complexity of the finite satisfiability problemfor GF+TG: it suffices to guess a bounded size structure andverify that it is indeed a model of the input formula. To getthe optimal 2E XP T IME -upper complexity bound, instead ofguessing a model, we may construct ϕ B and ϕ C for varioussets α and β and test their satisfiability. We first maketwo observations, Lemma 15 and Lemma 16, the second ofwhich reduces the number of possible choices of β , whichwill be crucial for lowering the complexity. Lemma 15.
Let ϕ be a normal form GF+TG formula overa signature σ . Then ϕ is finitely satisfiable iff there aresets α , β of -types, and, resp., non-degenerate guarded -types over σ , such that (i) the formulas ϕ B and ϕ C constructed with such α and β have finite models, (ii)for every β ∈ β any two-element structure of type β satisfies all the ∀ -conjuncts of ϕ . Proof: ⇒ Assume ϕ has a finite model A . As α and β take the set of -types and, resp., non-degenerate guarded -types realized in A . Then (i) holds since ϕ B is satisfied in A − and ϕ C — in A (cf. the paragraph about the constructionof B ∗ and C ∗ ), and (ii) holds since the guarded types in β are taken from a model of ϕ . ⇐ Having α and β satisfying(i) and (ii) we can perform a finite model construction for ϕ exactly as we did in this section (with condition (ii) usedto ensure that C respects all the ∀ -conjuncts of ϕ ). Lemma 16. If ϕ C has a finite model then it has one in whichthe number of realized -types is bounded polynomiallyin the number of -types, the number of the ∀∃ -conjunctsof ϕ and the number of transitive relations. Proof:
Let C be a finite model of ϕ C . Recall that ϕ C is in GF +TG and thus, as previously, we may assume that C contains no facts with more than two distinct elements.Additionally, we may also assume that every pair of distinctelements in C is connected by at most one transitive relation(this condition is ensured in the finite model construction in[10]; models satisfying this condition are called ramified there). Let β be the set of -types realized in C . For a -type β , by β (cid:22) x ( β (cid:22) x ) we denote the subset of β consistingof those literals which use only variable x ( x ); similarlyby β (cid:22) σ tr we denote the subset of β consisting of thoseliterals which use a transitive symbol. Let β − be the resultof switching the variables in β .Let us introduce an equivalence relation ∼ on β asfollows: β ∼ β iff the following conditions hold (i) β (cid:22) x = β (cid:22) x , β (cid:22) x = β (cid:22) x , β (cid:22) σ tr = β (cid:22) σ tr , (ii) foreach conjunct ∀ x ( γ i ( x ) ⇒ ∃ y ( θ i ( x, y ) ∧ ψ i ( x, y ))) it holdsthat β | = θ i ( x, y ) ∧ ψ i ( x, y ) iff β | = θ i ( x, y ) ∧ ψ i ( x, y ) and β − | = θ i ( x, y ) ∧ ψ i ( x, y ) iff β − | = θ i ( x, y ) ∧ ψ i ( x, y ) .Observe that β ∼ β iff β − ∼ β − .In every equivalence class of ∼ we distinguish one of itsmembers. We do this in such a way that if β is distinguishedin its class then β − is also distinguished in its class. Forvery pair of elements a, b ∈ C , if its type tp C ( a, b ) is notdistinguished in its class, change this type to the one whichis distinguished there. The strategy of distinguishing alwaysboth β and β − allows us to perform this process withoutconflicts which could potentially arise when the types ofpairs a, b and b, a are defined.In so obtained structure C (cid:48) the interpretation of thetransitive symbols remains unchanged (so they all remaintransitive) and every element has precisely the same wit-nesses for every ∀∃ -conjunct as it has in C (even thoughtit may be connected to them by different -types). The ∀ -conjuncts are satisfied in C (cid:48) as all the types are importedfrom C which is a model of ϕ C . Thus, still C (cid:48) | = ϕ C .It is readily verified that the number of equivalenceclasses of ∼ is bounded polynomially in the number of -types, in the number of the ∀∃ -conjuncts, and in the numberof transitive relations (the latter follows from the fact that C is ramified). From this we get that C (cid:48) is as required. Theorem 17.
The finite satisfiability problem for GF+TGwithout constants is 2E XP T IME -complete. For normalform formulas it works in time polynomial in the sizeof the input formula, exponential in the number of the ∀∃ -conjuncts, and doubly exponential in the size andwidth of the signature. Proof:
The lower bound is inherited from GF [6]or from GF +TG [10]. Let us justify the upper bound. Let ϕ be any GF+TG formula. Convert it into a disjunction ofnormal form formulas over a signature σ as in Lemma 1and test satisfiability of each of its disjuncts. To do thelatter, for a single normal form disjunct ϕ over a signature σ construct all possible sets of -types α , and all possible setsof guarded non-degenerate -types of cardinality bounded asin Lemma 16. For each pair of such α and β construct ϕ B and ϕ C , which are normal form formula, and test their finitesatisfiability using the algorithm from Thm. 5, and, resp.,Thm. 7. Our algorithm returns ’yes’ iff for some normalform disjunct ϕ and some choice of α and β both theexternal algorithms return ’yes’.The correctness of the algorithm follows from Lemmas1, 15 and 16.Denote h the number of the ∀∃ -conjuncts of ϕ , k thenumber of transitive relations, r the size of σ and w itswidth.Recall that the number of disjuncts in the normal formof ϕ is at most exponential in | ϕ | . Note that the numberof -types is r , so the number of possible choices for α is r . Due to Lemma 16 we can restrict attention to sets β ofsize bounded polynomially in r , h and k . Observing thatthe number of -types is O ( r · w ) we see that the numberof relevant choices of β is doubly exponential in r and w and singly exponential in h and k .By Thm. 5 the first of the external procedures worksin time polynomial in | ϕ B | , exponential in r and doublyexponential in w . By Thm. 7 the second procedure worksin time polynomial in | ϕ C | , exponential in h and doublyexponential in r . Regarding the size of ϕ B and ϕ C they are both boundedpolynomially in | ϕ | , and in the size of α and β , that is theyare doubly exponential in r and w and singly exponentialin h an k .Gathering the above, the claim for normal form formulasfollows. The upper bound for arbitrary formulas followsfrom the fact that | ϕ | is polynomial in | ϕ | , and thus allthe parameters r , w , h are polynomial in | ϕ | . Assume that ϕ is a finitely satisfiable GFU+TG normalform formula without equality and let A be its finite ( U -biquitous) model. We will explain how to construct a ( U -biquitous) model A (cid:48) of ϕ of size bounded doubly expo-nentially in | ϕ | . The whole construction is almost identicalto the construction of a small model of a satisfiable GFUnormal form formula in Section 3. There are only two,rather natural, differences: first, obviously, we use a differentblack box procedure; second, when constructing the + C k,(cid:96),mi structures we must properly handle the transitive relations.So, we first construct the auxiliary formula ϕ ∗ exactlyas in Section 3 (we only need the adjustment concerningtransitive atoms of the form T ( x, x ) used outside the guardsin the α ( x ) , similar to that for ϕ B and ϕ C ), and treating itas a GF+TG formula we take its small (not necessarily U -biquitous) model C − as guaranteed by Thm. 14. We proceedas in Section 3, building the doubling C of C − , a -fold copy B of C , and a K × K table A of copies of C , where K = | C | . All these structures are models of ϕ ∗ by Lemmas3 and 2.We employ the same notation as in the case of GFU.We choose the entry elements for the structures C k,(cid:96),m andproceed to the definition of the structures + C k,(cid:96),mi . As previ-ously, C k,(cid:96),mi is the structure with domain C k,(cid:96),mi ∪ { b , b } for some fresh elements b , b such that + C k,(cid:96),mi (cid:22) C k,(cid:96),mi = C k,(cid:96),mi . Concerning the connections involving the new ele-ments b , b , for each non-transitive P ∈ σ and each tuple ¯ a containing at least one of b , b we set that + C k,(cid:96),mi | = P [¯ a ] iff C k,(cid:96),m | = P [ h (¯ a )] , where h is the function definedas h ( b ) = e k,(cid:96),m , h ( b ) = e k,(cid:96),m and h ( a ) = a for a ∈ C k,(cid:96),mi . That is, as previously, we copy the relationsfrom C k,(cid:96),m , but only those non-transitive. In particularthe elements b , b remain not connected by any transitiverelation even if the entry elements are connected by somein C k,(cid:96),m .We then build successively the structures A , A , . . . exactly as in the case of GFU, obtaining finally a U -biquitous structure A f which we take as the desired A (cid:48) .The correctness of the construction can be proved as inthe case of GFU. The analogues of the Claims 9, 10 and11 can be proved with literally no changes. Also, the proofof Claim 12 is almost the same. As we have emphasized,in our process we do not add any transitive connections;we also do not modify any -types containing any transitiveconnections. So, the ∀ -conjuncts with transitive guards aresatisfied in all the A i . Note also that we never need newitnesses for the ∀∃ tr -conjuncts, as such witnesses arerequired only for tuples built out of a single element, andsuch tuples have the required witnesses already in A . Thus,we only need to take care of witnesses for the conjuncts notmentioning transitive relations, and for satisfaction for the ∀ -conjuncts with non-transitive guards. For such conjunctsthe fact that we do not copy from A the complete types oftuples, but rather their transitive-free parts is not relevant. Size of models and complexity.
The following theoremfollows from an analysis of the size of the A i structures,similar to that in the case of GFU, and a use of the estimationon the size of C − from the second part of Thm. 14. Theorem 18.
Every finitely satisfiable GFU+TG (TGF+TG)formula without constants has a model of size boundeddoubly exponentially in its length.Concerning the complexity, we first make the followingobservation.
Lemma 19.
A normal form TGF+TG formula ϕ withoutconstants is finitely satisfiable iff there exists a set of -types α such that the formula ϕ ∗ , treated as a GF+TGformula, has a finite (not necessarily U -biquitous) model. Proof: ⇒ If A is a finite model of ϕ then we take as α the set of -types realized in A , and note that A | = ϕ ∗ . ⇐ Given a finite (not necessarily U -biquitous) model C − | = ϕ ∗ we construct a U -biquitous model of ϕ ∗ (and thus also of ϕ ) as described above.Finally we get: Theorem 20.
The finite satisfiability problem for GFU+TG(TGF+TG) without constants is 2E XP T IME -complete.
Proof:
The lower bound is inherited from GF+TG.To justify the upper bound we design the following algo-rithm: For input ϕ we convert it into normal form over asignature σ and for each of its disjuncts ϕ and each possiblechoice of α construct ϕ ∗ and test its finite satisfiabilityusing the procedure for GF+TG. We answer ’yes’ iff at leastone of the tests is positive.The correctness of this algorithm follows from Lemmas1 and 19. Recall that the number of disjuncts in normal formof ϕ is exponential in | ϕ | and the number of choices of α is doubly exponential in | σ | and thus also in | ϕ | . ByThm. 17 a single finsat test for ϕ ∗ takes time polynomialin | ϕ ∗ | (exponential in | ϕ | ), exponential in the number ofthe ∀∃ -conjuncts and doubly exponential in the size andwidth of σ (which are polynomial in | ϕ | ). So, overall, thealgorithm is doubly exponential in | ϕ | .
5. Conclusion
Settling an open problem, we established the finitemodel property of the triguarded fragment (and conse-quently of guarded formulae preceded by a sequence ∃ ∗ ∀∀∃ ∗ of unguarded quantifiers [15]), even providing adoubly exponential upper bound on the model size. Us-ing similar ideas, we settled open problems concerningthe guarded and triguarded fragment extended by transitive guards, providing tight complexity bounds for their finitesatisfiability problem in the constant-free case.While, by definition, GFU and GFU+TG disallow equal-ity (and including unrestricted equality would lead to unde-cidability [12]), we note that adding equality statements ofthe form x = c to GFU and equalities guarded by transitiveguards to GFU+TG can be done at no computational costand would not affect our constructions at all. The aboveadditions nicely extend the expressive power of the logics.The first of them allows us, e.g., to express naturally the con-cept of nominals known from description or hybrid logics,while with the second we can say that a transitive relationis actually an equivalence (cf. [9], Section 5.1), which givesa chance to capture some scenarios from epistemic logics.As a central open problem, it remains to clarify the de-cidability status of GF+TG and GFU+TG in the presence ofconstants. We assume the resulting fragments will still be de-cidable. Obviously a lower complexity bound for GFU+TGwith constants, inherited from GFU, is N2E XP T IME andhence harder than the constant-free case (under standardcomplexity-theoretic assumptions).
Acknowledgments
E.K. is supported by Polish National Science Centregrant No 2016/21/B/ST6/01444. S.R. is supported by theEuropean Research Council through the ERC ConsolidatorGrant 771779 (DeciGUT).
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Appendix A.Comments on the external procedures
Thm. 4.
Inspecting the proof of Thm. 1.2 in [2], we seethat the size of a minimal finite model of a normal formformula over a signature σ can be bounded by | J | O ( w ) forsome structure J whose size is bounded by the number ofguarded atomic types over σ , where w is the width of σ .The number of atomic k -types over σ is O ( r ( k + u ) w ) ,where r is the number of relation symbols and u the numberof constants in σ . Of course, every guarded type is a k -type for some k ≤ w , so the number of guarded types is w · O ( r ( w + u ) w ) , which is O ( r ( w + u ) w ) . Thus the size ofmodel is O ( r ( w + u ) O ( w ) . As each of r, w, u is bounded by | σ | the claim follows. Thm. 5.
The algorithm from [6] (page 1731) is an alter-nating algorithm. The algorithm stores two guarded typesand a counter counting up the the total number possibleguarded types. That is it needs O ( r ( w + u ) w ) space. Usingthe classical simulation of alternating Turing machines bydeterministic ones from [4] we get an algorithm working intime O ( r ( w + u ) w ) · O ( n ) , where n is the length of the inputformula. Thm. 6.
Let ϕ be a normal form formula in GF +TG overa signature σ . Denote L the number of -types over σ ( L =2 | σ | ), h the number of the ∀∃ -conjuncts of ϕ and k thenumber of transitive relations in σ .In [10] an important role is played by the parameter M ϕ . It is defined on page 14 as M ϕ = 3 L | ϕ | , but a closerinspection of the proof of Lemma 3.5 (iii), where the roleof M ϕ is revealed, shows that it is sufficient to take M ϕ =3 Lh .Looking at page 27 of [10] we see that the domain ofthe small finite model for ϕ which is constructed there is ofsize bounded by (2 L + 1) · · h · k · T · F where T is the number of the so-called enriched - M ϕ - counting types and F is a bound on the size of small modelsfor the ’symmetric’ part of ϕ . T is bounded by L × L × L M ϕ +1 , as each enriched- M ϕ -counting type is determined by two subsets A and B of -types and a function which for each -type returns a number from the range { , . . . , M ϕ } (see Def. 6.1). Substituting Lh for M ϕ we see that T is bounded exponentially in L (so,doubly exponentially in | σ | ) and h .Concerning F , it is actually a bound on the size ofthe structure constructed in the proof of Lemma 5.7, thatis h ( XL ) k , where X is the maximal value of a vari-able in some minimal solution for the system of linearinequalities constructed on page 20. By Lemma 5.6 X ≤ N · ( N M ϕ ) N +1 , for N being the number of inequalitiesin the system, which is bounded by L . Summarizing, F isnot greater than h (3 L · (9 L h ) L +1 ) k .Gathering the above estimations we get that the size ofthe finite model constructed is exponential in the numberof -types (and thus doubly exponential in the size of thesignature), in h , and in k . Taking into account that k ≤ | σ | the claim follows. Thm. 7.
Again, the algorithm from [10] (page 30) is al-ternating. Assume that the notation is as in the previousparagraph. What the algorithm stores is: • a counter counting up to kT , • a collection of at most kL enriched- M ϕ -counting typesThat is, the total space required is polynomial in the lengthof the input formula and exponential in L and h . Simu-lating alternating Turing machines by deterministic ones aspreviously we get an algorithm for normal form formulasworking in time bounded polynomially in the size of theinput and exponentially in the number of -types (doublyexponentially in the size of the signature) and the ∀∃ -conjuncts. Appendix B.Details on the FMP for GFU with constants
Given a structure A interpreting a signature σ consistingof relation symbols and constants we call the subset ˆ A ⊆ A consisting of all the elements interpreting the constants of σ the named part of A . The unnamed part is defined as ˇ A := A \ ˆ A . A -type is named if contains x = c for someconstant c and unnamed otherwise.We first redefine the notions of disjoint unions and dou-blings of structures. Let ( A i ) i ∈I be a family of σ -structureshaving disjoint unnamed parts and sharing the same namedpart; call such a family harmonized. Their harmonized union is the structure A with domain A = ˆ A t ∪ (cid:83) i ∈I ˇ A i , where t is chosen as any element of I , such that for all i we havethat A (cid:22) ˆ A t ∪ ˇ A i is isomorphic to A i , and for any tuple ¯ a containing elements from at least two different ˇ A i and anyrelation symbol P ∈ σ of arity | ¯ a | we have A | = ¬ P (¯ a ) . Lemma 21.
Let ϕ be a GF normal form formula over asignature consisting of relation symbols and constants.The harmonized union of any harmonized family ofmodels of ϕ is also its model.Let A − be a σ -structure. Its harmonized doubling is the structure A with domain A := ˆ A − × { } ∪ ˇ A − × { , } in which for each P ∈ σ we have A | = [( a , (cid:96) ) , . . . , ( a k , (cid:96) k )] iff A − | = P [ a , . . . , a k ] for all a i ∈ A − and (cid:96) i ∈ { , } if a i ∈ ˇ A − and (cid:96) i = 0 if a i ∈ ˆ A − . Lemma 22.
Let ϕ be a normal form GF or GFU formulawhich does not use equality outside guards and let A − be a model of ϕ . Then its harmonized doubling A is stilla model of ϕ .Let us now fix a GFU sentence ϕ in normal form,without equality over a signature σ consisting of relationsymbols and constants and let A be a U -biquitous model of ϕ . Our goal is to build a finite U -biquitous model A (cid:48) of ϕ . B.1. Preparing building blocks
We construct ϕ ∗ precisely as in the case without con-stants. We remark that this time each α ∈ α fully specifiesthe substructure consisting of the given element and thenamed part of the structure.It is clear that ϕ ∗ , treated as a GF-formula, is satisfiable.In fact, A is its model. Thus, by the finite model propertyfor GF, it also has a finite (not necessarily U -biquitous)model. We take such a finite model C − | = ϕ ∗ , and let C beits harmonized doubling. As ϕ ∗ does not use equality, byLemma 22 we have that C | = ϕ ∗ .Recalling the definition of indistinguishable elements weadapt Claim 8 to our current setting. Claim 23.
For any pair of unnamed -types α, α (cid:48) ∈ α thereis a pair of their distinct realizations a, a (cid:48) in C such that C | = U [ a, a (cid:48) ] ∧ U [ a (cid:48) , a ] . Moreover, if α = α (cid:48) , then weeven find indistinguishable a, a (cid:48) with that property.As previously we build yet another model B | = ϕ ∗ ,as the harmonized union of five copies of C . Letting K = | ˇ C | , we assume that the unnamed part of B is B := { , . . . , K } ; and that for m = 0 , . . . , the structureon ˆ B ∪ { mK + 1 , . . . , mK + K } is isomorphic to C . B.2. U -saturation We now build a finite sequence of finite structures A , A , . . . , A f , each of them being a model of ϕ ∗ and the lastof them being a desired U -biquitous model A (cid:48) of ϕ ∗ (andthus also of ϕ ).The domains of all these structures will be identical. A i = ˆ B ∪ ( ˇ B × { , . . . , K } × { , . . . , K } ) . We will view each of the A i as a K × K tablecontaining unnamed parts plus the shared named part.The initial structure A is defined as the harmonizedunion of (5 K ) copies of B . Namely, for each k, (cid:96) ∈{ , . . . , K } we make A (cid:22) ˆ B ∪ ( ˇ B × { k } × { (cid:96) } ) isomorphicto B (via the isomorphism working as the identity on ˆ B and as the natural projection ( b, k, (cid:96) ) (cid:55)→ b on the unnamedelements). By Lemma 21 we have that A | = ϕ ∗ . Some notation.
We adapt our notation. For each k, l wedenote by B k,(cid:96)i the structure consisting of the commonnamed part and the unnamed part in the cell ( k, (cid:96) ) of A i , that is the structure A i (cid:22) ˆ B ∪ ( ˇ B × { k } × { (cid:96) } ) . We recall that B k,(cid:96) is isomorphic to B . Further, for m = 0 , . . . , , we denoteby C k,(cid:96),mi the structure B k,(cid:96)i (cid:22) ˆ B ∪{ mK +1 , . . . , mK + K }×{ k } × { (cid:96) } . We recall that each C k,(cid:96),m is isomorphic to C . Entry elements and their use.
This time only members ofthe unnamed parts are entry elements. For any ≤ k, (cid:96) ≤ K , let α k = tp B ( k ) and α (cid:96) = tp B ( (cid:96) ) . Note that α k and α (cid:96) are unnamed.For each such pair k, (cid:96) we now choose a pair of entryelements for each of the five structures with unnamed partsin the cell ( k, (cid:96) ) of A , that is for the structures C k,(cid:96),m ( m = 0 , , . . . , ).By Claim 23, there are distinct elements e , e ∈ C such that C | = α k [ e ] ∧ α (cid:96) [ e ] ∧ U [ e , e ] ∧ U [ e , e ] andif α k = α (cid:96) then e and e are indistinguishable in C . Wechoose the entry elements e k,(cid:96),m , e k,(cid:96),m to C k,(cid:96),m to be thecorresponding copies of e and e in each of C k,(cid:96),m .By + C k,(cid:96),mi we denote the structure with domain C k,(cid:96),mi ∪ { b , b } for some fresh unnamed elements b , b such that + C k,(cid:96),mi (cid:22) C k,(cid:96),mi = C k,(cid:96),mi and for each P ∈ σ and each tuple ¯ a containing at least one of b , b wehave + C k,(cid:96),mi | = P [¯ a ] iff C k,(cid:96),m | = P [ h (¯ a )] , where h isthe function defined as h ( b ) = e k,(cid:96),m , h ( b ) = e k,(cid:96),m and h ( a ) = a for a ∈ C k,(cid:96),mi In particular + C k,(cid:96),mi | = α k [ b ] ∧ α (cid:96) [ b ] ∧ U [ b , b ] ∧ U [ b , b ] . From A i to A i +1 . Assume now that the structure A i hasbeen defined, for some i ≥ , A i | = ϕ ∗ . If A i is U -biquitousthen we are done. Otherwise let b , b be a pair of elementsin A i such that A i | = ¬ U [ b , b ] . Note that b , b mustbe unnamed. Indeed, as in the case without constants, ourprocess does not modify the types of the guarded tuples. Inparticular, in each of the A i the -types are retained from A , where they are copied from the model C of ϕ ∗ . By theconjunct (2) of ϕ ∗ all the -types in A belong to α , theset of -types realized in the U -biquitous structure A . Thus,if, say, b would interpret a constant c then, as the -type of b must contain U ( x, c ) ∧ U ( c, x ) (since this -type belongsto α ), we would have that A i | = U [ b , b ] ∧ U [ b , b ] .For s = 1 , let k s , l s , n s be such that b s is the n s -th element of the unnamed part of B k s ,l s i . Let us choose t ∈ { , . . . , } such that C n ,n ,ti does not contain the k -th, l -th, k -th or l -th element of the unnamed part of B n ,n i . Such a t must exist by the pigeon hole principle.We make the structure A i +1 (cid:22) C n ,n ,ti ∪ { b , b } isomorphicto + C n ,n ,ti . The rest of the structure A i remains untouched. Correctness.
The correctness of the construction can beproved in the same vain as in the case without constants.In particular Claims 9, 10, 11 and 12 remain true withliterally no changes and their proofs require only routineadjustments concerning the division of the domains intonamed/unnamed parts. That is the whole process eventuallyends in U -biquitous model of ϕ ∗∗