Logics of Dependence and Independence: The Local Variants
LLogics of Dependence and Independence: TheLocal Variants
Erich Grädel
RWTH Aachen University, [email protected]
Phil Pützstück
RWTH Aachen University, [email protected]
Abstract
Modern logics of dependence and independence are based on team semantics, which means thatformulae are evaluated not on a single assignment of values to variables, but on a set of suchassignments, called a team. This leads to high expressive power, on the level of existential second-order logic. As an alternative, Baltag and van Benthem have proposed a local variant of dependencelogic, called logic of functional dependence (
LFD ). While its semantics is also based on a team, theformulae are evaluated locally on just one of its assignments, and the team just serves as the supplyof the possible assignments that are taken into account in the evaluation process. This logic thusrelies on the modal perspective of generalized assignments semantics, and can be seen as a fragmentof first-order logic. For the variant of
LFD without equality, the satisfiability problem is decidable.We extend the idea of localising logics of dependence and independence in a systematic way,taking into account local variants of standard atomic dependency properties: besides dependence andindependence, also inclusion, exclusion, and anonymity. We study model-theoretic and algorithmicquestions of the localised logics, and also resolve some of the questions that had been left open byBaltag and van Benthem. In particular, we study decidability issues of the local logics, and prove thatsatisfiability of
LFD with equality is undecidable. Further, we establish characterisation theoremsvia appropriate notions of bisimulation and study the complexity of model checking problems forthese logics.
Keywords and phrases logics of dependence and independence, decidability, bisimulation.
Following work that has been initiated by Hodges [14] and Väänänen [18], modern logicsof dependence and independence are generally based on team semantics . This meansthat a formula ϕ ( x , . . . , x m ) in such a logic is not evaluated on a single assignment s : { x , . . . , x m } → A of values to the free variables, but for a set of such assignments, whichis called a team . In these logics, dependence and independence statements about variables,such as “ y depends on x ” or “ x and y are independent” are considered as atomic propertiesof teams . Besides dependence and independence atoms, further atomic team propertieshave been considered: inclusion, exclusion, anonymity, conditional independence, and others.A crucial feature of the logics with team semantics is that they manipulate second-orderobjects by first-order syntax. The evaluation of a formula on a team can be considered as adynamic process (or game) that modifies the team while moving from the formula through itssyntax tree to the atoms or, equivalently, as an annotation of the syntax tree of the formulaby teams. The expressive power of logics with team semantics is typically on the level ofpowerful fragments of existential second-order logic. More precisely, with a logic L with teamsemantics, which syntactically extends first-order logic by certain atomic team properties,one can associate a fragment F ⊆ Σ of existential-second-order, such that every formula ϕ (¯ x ) ∈ L (of vocabulary τ ) is equivalent to a sentence ψ ∈ F (of vocabulary τ ∪ { T } , where T is a predicate for the team), such that ϕ (¯ x ) is true in a structure M with a team T , if, a r X i v : . [ c s . L O ] F e b Logics of Dependence and Independence: The Local Variants and only if the expanded structure ( M , T ) is a model of ψ . Understanding the expressivepower of a logic with team semantics thus means to identify the fragment F ⊆ Σ such thatthese equivalences hold in both directions. Here are some of the most important results ofthis kind:Dependence logic FO (dep) and exclusion logic FO ( | ) are equivalent to the fragment ofΣ -sentences in which the predicate for the team appears only negatively [16].Inclusion logic FO ( ⊆ ) and anonymity logic FO (Υ) are equivalent to sentences of form ∀ ¯ x ( T ¯ x → ψ ( T, ¯ x )) in the posGFP-fragment of least fixed-point logic, such that T occursonly positively in ψ [9].Independence logic FO ( ⊥ ) and inclusion-exclusion logic FO ( ⊆ , | ) are equivalent with fullΣ (and thus can describe all NP-properties of teams) [8].In particular, all of these logics are much more expressive than classical first-order logic andare of course undecidable for satisfiability.Recently, Baltag and van Benthem [3] have proposed a different kind of dependencelogic, called logic of functional dependence LFD , which can be considered as a fragmentof first-order logic. Its semantics is also based on an underlying team of assignments, butthe formulae are evaluated locally on just one of these assignments, while the team servesas the supply of (or one might also say, the restriction for) the possible assignments thatare admitted in the evaluation process of the formula. This is the modal perspective ofgeneralized assignments semantics, where not all possible assignments of values to variablesare available but only a given set of them; these may be considered as the possible worlds ina Kripke style semantics, where neighbouring worlds are assignments that agree on somesubset of the variables. Thus, in an evaluation process of such a formula, the team remainsunchanged, but one moves around between different assignments in that team. Dependencecomes in by means of atoms D X y (where y is a variable and X is a set of variables), whichare true at a given assignment s if all assignments t in the underlying team that agree with s on all variables in the set X , also agree with s on y . The atom D X y can thus be read as a local dependence of y on X (on the assignments related to the current one by agreement on X ), whereas the standard dependence atom dep( X, y ) in Väänänen’s dependence logic [18]says that y globally depends on X in the team, in the sense that there is function which, for all assignments in the team, maps the values for X to the value for y . There is a furtherdependence operator in LFD , of form D X ϕ , with the meaning that ϕ is true at all assignmentsthat agree with the current one on X .Baltag and van Benthem provide a detailed study of many aspects of LFD . In particularthey prove that
LFD can be embedded into classical first-order logic (with the team ofpossible assignments as an additional predicate) and they view
LFD as a minimal logic offunctional dependence. They prove that the variant of
LFD without equality is decidablefor satisfiability, but leave open the problem whether this is also the case for
LFD = , theversion with equality. Towards the end of their paper, Baltag and van Benthem also studya local version of independence , by atoms Ind X ( Y ), saying that the values of the variablesin X at the current assignment s do not constrain the values of the variables in Y : for anyassignment t in the team there is one with the same X -values as s and the same Y -values as t . Interestingly, this local notion of independence is not symmetric: Ind X ( Y ) does not implyInd Y ( X ), contrary to the global independence atoms X ⊥ Y in the independence logic ofGrädel and Väänänen [10]. Baltag and van Benthem also observe that this local independencelogic is undecidable, even without equality. . Grädel and P. Pützstück 3 In this paper, we resolve some of the problems left open by Baltag and van Benthem. Inparticular, we prove that
LFD = is undecidable, and we establish a characterisation theoremvia an appropriate notion of bisimulation.Further, we study the idea of localising logics of dependence and independence in a moresystematic way. We consider local variants of all the standard atomic dependency notionsfor teams; beyond dependence and independence, this includes anonymity, inclusion, andexclusion. In this setting, it turns out that anonymity is just the negation of dependence, andinclusion is the negation of exclusion (which is not the case for the global team semanticalvariants of these atoms). This also suggests to look more closely at the role of negation.If, as in global logics of dependence and independence, we insist that formulae are writtenin negation normal form, and that negation is applied to classical atoms only, but not toatomic dependencies, then by considering local dependence atoms D X y and local anonymityatoms Υ X y ≡ ¬ D X y separately we obtain two even “more minimal” logics than LFD , whosecommon extension coincides with
LFD . A team T is a set of assignments s : V → M with a common domain V of variables and acommon set of values M , typically the universe of a structure. For a tuple ¯ x of variablesfrom V we write T (¯ x ) := { s (¯ x ) | s ∈ T } for the set of values of ¯ x in the team T . Whenever V is finite, it is convenient to fix some enumeration ¯ v of V so we can denote assignments s bytheir tuple of values ¯ s = s (¯ v ). Conversely, given some fitting tuple of values ¯ a , the notation¯ v ¯ a represents an assignment s with s (¯ v ) = ¯ a . This allows us to view T as a relation inthe usual sense.For the purpose of this paper, we call the standard team semantical atoms such asdependence, independence, anonymity, inclusion and exclusion global atoms . A global atom α , over a set of variables V ( α ) defines, for every set M of values, the extension [[ α ]] M of allteams T such that M | = T α . We require that the domain of T contains at least V ( α ), andthat the truth of α in T only depends on the variables in V ( α ), i.e., M | = T α if, and only if, M | = T (cid:22) V ( α ) α . The most important global atoms are: Dependence: M | = T dep(¯ x, y ) : ⇐⇒ ( ∀ s ∈ T )( ∀ t ∈ T ) s (¯ x ) = t (¯ x ) → s ( y ) = t ( y ); Inclusion: M | = T ¯ x ⊆ ¯ y : ⇐⇒ ( ∀ s ∈ T )( ∃ t ∈ T ) s (¯ x ) = t (¯ y ); Exclusion: M | = T ¯ x | ¯ y : ⇐⇒ ( ∀ s ∈ T )( ∀ t ∈ T ) s (¯ x ) = t (¯ y ); Anonymity: M | = T ¯ x Υ y : ⇐⇒ ( ∀ s ∈ T )( ∃ t ∈ T ) s (¯ x ) = t (¯ x ) ∧ s ( y ) = t ( y ); Independence: M | = T ¯ x ⊥ ¯ y : ⇐⇒ ( ∀ s, t ∈ T )( ∃ u ∈ T ) s (¯ x ) = u (¯ x ) ∧ t (¯ y ) = u (¯ y ).Notice that all these global atoms are defined by a universal expression of form ( ∀ s ∈ T ) β where β is a statement about equalities and inequalities between values of s and values ofother t ∈ T .A local team semantical atom β instead is evaluated on a local assignment s of some fixedunderlying team T . Thus, the extension [[ β ]] M,T of β on a set M of values and a team T ofassignments t : V → M , is the set of all s ∈ T such that ( M, T ) | = s β . We say that β is thelocal variant of the global atom α if, for any set of values M and any team T , we have that M | = T α ⇐⇒ ( M, T ) | = s β for all s ∈ T. The following example gives a local variant of dependence.
Logics of Dependence and Independence: The Local Variants (cid:73)
Example 1.
Consider the team over V = { x, y, z } represented by the following table: x y z .We say that x locally depends on y at the assignment (1 , , y to be itsvalue in this specific assignment , also fixes x to its value in said assignment. In other words,fixing y to 1 causes x to be fixed to 1. As another example, y locally depends on z at theassignments (1 , ,
1) and (2 , , z = 1 entails y = 2 in the above table. However, y does not depend on x at (1 , , y depends on x in the whole team if, and only if, y locally depends on x at everyassignment in the team.The local variants of the standard global dependency atoms are the following: Dependence: ( M, T ) | = s D ¯ x y : ⇐⇒ ( ∀ t ∈ T ) s (¯ x ) = t (¯ x ) → s ( y ) = t ( y ); Inclusion: ( M, T ) | = s ¯ x ∈ ¯ y : ⇐⇒ s (¯ x ) ∈ T (¯ y ); Exclusion: ( M, T ) | = s ¯ x / ∈ ¯ y : ⇐⇒ s (¯ x ) / ∈ T (¯ y ); Anonymity: ( M, T ) | = s Υ ¯ x y : ⇐⇒ ( ∃ t ∈ T ) s (¯ x ) = t (¯ x ) ∧ s ( y ) = t ( y ); Independence: ( M, T ) | = s Ind ¯ x (¯ y ) : ⇐⇒ ( ∀ t ∈ T )( ∃ u ∈ T ) s (¯ x ) = u (¯ x ) ∧ t (¯ y ) = u (¯ y ).Notice that there are some striking differences between the properties of local and globalatoms. For the local atoms, anonymity and dependence, as well as inclusion and exclusion,are directly related via negation, which is not the case for the global atoms. Further, localindependence is not symmetric in ¯ x and ¯ y . We now describe the logic of functional dependence
LFD , as introduced by Baltag and vanBenthem [3], and recall the main results they proved on this logic. We shall then propose aslightly different presentation of local logics on teams, of which
LFD is one special case.For a tuple ¯ a = ( a , . . . , a n ) ∈ A n , we denote by [¯ a ] := { a , . . . , a n } the set of its compon-ents. Given some function f with [¯ a ] ⊆ dom( f ), we also write f (¯ a ) := ( f ( a ) , . . . , f ( a n )). (cid:73) Definition 2.
A type ( τ, V ) consists of a relational vocabulary τ and a set V of variables.If both τ and V are finite, we say ( τ, V ) is a finite type. A dependence model ( M , T ) of type( τ, V ) consists of a τ -structure M with universe M , and a nonempty team T with domain V and co-domain M . Pointed dependence models ( M , T ) , s further distinguish a “current”assignment s ∈ T . (cid:73) Definition 3.
The syntax of formulae in
LFD ( τ, V ) is given by ϕ ::= R ¯ x | D X y | ¬ ϕ | ϕ ∧ ϕ | D X ϕ, where R ∈ τ is a relation symbol, ¯ x is a tuple of variables of appropriate length, X ⊆ V is finite , and y ∈ V . Further, LFD = is the extension of LFD by equality atoms x = y . We shallalso make use of the following notations:The dual of D X ϕ is defined as E X ϕ := ¬ D X ¬ ϕ . . Grädel and P. Pützstück 5 For the special case X = ∅ we use ∀ ϕ := D ∅ ϕ and ∃ ϕ := E ∅ ϕ .We use D ¯ x ϕ := D [¯ x ] ϕ and likewise for the other quantifiers.We refer to the D X and E X as dependence quantifiers or modalities of LFD , and also call ∀ and ∃ global modalities. As we will often deal with dependence on sets of variables, wewill use the notation s = X t whenever the assignments s, t ∈ T agree on the set of variables X ⊆ V , i.e. s ( x ) = t ( x ) for all x ∈ X . Note that = ∅ = T × T . (cid:73) Definition 4.
The semantics of
LFD and
LFD = on a dependence model ( M , T ) is definedby the usual rules for atomic formulae and connectives together with( M , T ) | = s D X y : ⇐⇒ t = X s implies t = y s for all t ∈ T , ( M , T ) | = s D X ϕ : ⇐⇒ ( M , T ) | = t ϕ for all t ∈ T with t = X s. Obviously, E X , ∃ , ∀ then have the expected semantics( M , T ) | = s E X ϕ ⇐⇒ ( M , T ) | = t ϕ for some t ∈ T with t = X s, ( M , T ) | = s ∃ ϕ ⇐⇒ ( M , T ) | = t ϕ for some t ∈ T , ( M , T ) | = s ∀ ϕ ⇐⇒ ( M , T ) | = t ϕ for all t ∈ T .
Of course, D X and E X are variants of the traditional quantifiers ∀ and ∃ , and we brieflywant to justify the use of these new quantifiers instead of the classical ones. The main pointis that the reasoning about free and bound variables becomes more transparent. Indeed, it isa desirable feature that the meaning of a formula should only depend on its free variables,in the sense that if s = free( ϕ ) t then it should be the case that ( M , T ) | = s ϕ if, and onlyif, ( M , T ) | = t ϕ . By defining free( D X y ) = free( D X ϕ ) = X , this is easily seen to be thecase. Using a naive interpretation of traditional quantifiers, saying that ∀ yϕ holds for theassignment s if, and only if ϕ holds for all assignments t that agree with s on all variablesexcept y , we would get the unwanted behaviour that ∀ yϕ may depend on variables notoccurring free in it. Indeed, it is easy to construct a team in which the formula ∀ yP y istrue at some assignments and false at others, although all assignments obviously coincideon the free variables of ∀ yP y (of which there are none). On the other side, Baltag and vanBenthem showed that such problems do not occur if one translates dependency quantifiersinto universal ones by D ¯ x ϕ ⇐⇒ ∀ ¯ zϕ , where ¯ z is an enumeration of V \ [¯ x ].The semantics of LFD formalizes a local notion of dependence, as in Example 1. It is alsothis locality, together with the semantics of the dependence quantifiers D X and E X , whichemphasizes the modal character of LFD . Indeed, notice the similarities to the modalities (cid:3) and ♦ of propositional modal logic ML ; the binary relations = X on teams can be viewed asthe accessibility relations of the modality D X and its dual E X on the team. In this sense,( M , T ) can be viewed as a Kripke structure that has T as its universe and accessibilityrelations (= X ) for X ⊆ V . Note that global functional dependence is expressible in LFD via ∀ D X y , which guarantees that X locally determines y at every assignment in the team,i.e. that X determines y in the whole team.We shall see that there are important differences between LFD and
LFD = . In particular,without equality, we can assume, without loss of generality, that the teams T that we considerare variable-distinguished, i.e. T ( x ) ∩ T ( y ) = ∅ for distinct variables x, y . Often called non-locality, although we do not want to get these two notions of locality mixed up.
Logics of Dependence and Independence: The Local Variants (cid:73)
Proposition 5.
Let ( M , T ) be a dependence model with universe M and variables V . Set N = M × V , T = { s | s ∈ T } where s ( x ) = ( s ( x ) , x ) , and construct N with universe N by adapting the relations so that N | = s R ¯ x ⇐⇒ M | = s R ¯ x . Then ( N , T ) is variable-distinguished and LFD -equivalent to ( M , T ) . To consider logics based on other local atoms than the local dependency atom, we generalisethe logic
LFD described in the last subsection. We will use a basic sublogic L of relationalatoms and boolean connectives. This is extended by our local atoms and the dependencequantifiers D X and E X introduced in the last section. The main difference is that we willallow negation only on relational atoms, to further differentiate between different local atoms.Now let Ω ⊆ { D, Υ , = , = , ∈ , , Ind , . . . } be a collection of atomic operators, which, appliedto appropriate tuples of variables from V , define local atoms β (¯ x ) such as x = y , x = y , D ¯ x y ,Υ ¯ x y , ¯ x ∈ ¯ y , ¯ x ¯ y , Ind ¯ x ¯ y , etc. We denote the collection of such atoms by Ω[ V ]. (cid:73) Definition 6.
The syntax of formulae in the local team logic L [Ω] of type ( τ, V ) is given by ϕ ::= R ¯ x | ¬ R ¯ x | β (¯ x ) | ϕ ∨ ϕ | ϕ ∧ ϕ | E X ϕ | D X ϕ where R ∈ τ is a relation symbol, X ⊆ V is finite, ¯ x is a tuple of variables in V of appropriatelength, and β (¯ x ) ∈ Ω[ V ]. Formulae are evaluated on dependence models ( M , T ) consisting of τ -structure M and a team T with domain V and values in M , locally at some assignment s ∈ T . The rules to determine whether ( M , T ) | = s ϕ , for ϕ ∈ L [Ω] extend the truth definitionsfor τ -literals R ¯ x and ¬ R ¯ x , and for local team atoms β (¯ x ) ∈ Ω[ V ] by the standard rules forboolean connectives and the rules for dependence quantifiers D X , E X as in Definition 4.The free variables of = , = are defined as usual. For the other atoms, we set free( D X y ) = X ,free(¯ x ∈ ¯ y ) = [¯ x ], and free(Ind ¯ x (¯ y )) = [¯ x ], where the negations of these atoms have the samefree variables.We shall also use the operators ∀ and ∃ explained above. Note that we can now state ourdefinition that some atom β is the local variant of a global atom α via M | = T α ⇐⇒ ( M, T ) | = ∀ β : ⇐⇒ ( M, T ) | = s β for all s ∈ T. We will thus allow ourselves to use α in L [ β ], with the intended semantics α ≡ ∀ β , e.g. wecan write dep( x, y ) instead of ∀ D x y within L [ D ].Obviously, the logics LFD and
LFD = of Baltag and van Benthem are equivalent to L [ D, Υ]and L [ D, Υ , = , =], respectively. A dependence model ( M , T ) of type ( τ, V ) with finite V , enumerated as V = { v . . . , v n } ,can be viewed as a structure of vocabulary τ ∪ { T } , expanding M by a predicate (of arity |V| = n ) for the team. It is not difficult to see that, for any of the logics L [Ω] describedabove, there is straightforward translation that associates with every formula ϕ ∈ L [Ω] oftype ( τ, V ) an equivalent formula ϕ ∗ ∈ FO of vocabulary τ ∪ { T } , which means that, for all( M , T ) and all s ∈ T , we have that( M , T ) | = s ϕ ⇐⇒ ( M , T ) | = ϕ ∗ ( s (¯ v ))Notice that the semantics on the left side is the semantics of L [Ω] whereas on the right sidewe have the standard Tarski semantics of first-order logic. . Grädel and P. Pützstück 7 For the case of
LFD and
LFD = this has been called the standard translation in [3], and itis a straightforward generalisation of the translations of modal logics into first-order logic.The rules of the translation are trivial for τ -literals and Boolean connectives. Quantifiers aretranslated via relativisation to the team predicate. Given a formula ϕ , X ⊆ V and a tuple ¯ z enumerating V \ X (in particular, ¯ z is a subtuple of ¯ v ), we put( E X ϕ ) ∗ := ∃ ¯ z ( T ¯ v ∧ ϕ ∗ ) and ( D X ϕ ) ∗ := ∀ ¯ z ( T ¯ v → ϕ ∗ ) . It remains to provide translations for the local team atoms β (¯ x ) ∈ Ω[ V ]. The translationpreserves their free variables. For ease of notation, we explicitly give the translations justfor atoms with two variables v i , v j from V . It is obvious that the translations generalise toarbitrary tuples. Let ¯ z, ¯ z be distinct copies of ¯ v = ( v , . . . , v n ). For j ≤ n , let ¯ z − j denote thetuple obtained by omitting z j , and let ¯ z [ j y ] = ( z , . . . , z j − , y, z j +1 , . . . , z n ). Then we set( v i ∈ v j ) ∗ := ∃ ¯ z ( T ¯ z ∧ z j = v i ) , ( D v i v j ) ∗ := ∀ ¯ z − i ∀ ¯ z i (( T ¯ z [ i v i ] ∧ T ¯ z [ i v i ]) → z j = z j ) , (Ind v i v j ) ∗ := ∀ ¯ z ( T ¯ z → ∃ ¯ z ( T ¯ z ∧ z i = v i ∧ z j = z j )) . For the negations of these atoms, we consider the negation of their translation, so for example( v i / ∈ v j ) ∗ = ¬ ( v i ∈ v j ) ∗ .Thus, the local logics of dependence and independence L [Ω] can all be considered asfragments of first-order logic. We remark that this restricted expressive power, comparedto the global logics of dependence and independence which are fragments of existentialsecond-order logic, is not just due to the localisation of the team atoms. Indeed, as we haveseen, the global dependency atoms are easily expressible in the local logics, via a furtheruniversal quantification with ∀ . But the localisation of the dependencies, together with theglobal restriction of the available assignments to a fixed team , permits to evaluate these logicsby first-order rules, as opposed to the inherent second-order operations in team semanticssuch as the decomposition of teams, and their Skolem extensions. In the next sections,we explore whether this more limited expressiveness is balanced by the benefits of betteralgorithmic manageability and convenient model-theoretic properties. We now want to discuss the question which of the logics L [Ω] are decidable for satisfiability.An obvious road for proving this is to show that the standard translation puts L [Ω] into aknown decidable fragment of FO . This works in some cases, but not always, and in particular, LFD seems to resist such an approach. By different methods, Baltag and van Benthem haveshown that
LFD is decidable, and have formulated the corresponding question for
LFD = asan open problem. We shall prove below that LFD = is undecidable. A known decidable fragment of FO is the guarded fragment GF . In general, guarded logicsarise as a natural generalization of modal logics. Consider the standard translation ϕ ϕ ∗ ( x )that identifies propositional modal logic ML with the modal fragment of FO , by rewritingmodal operators as relativised quantifiers:( (cid:3) ϕ ) ∗ = ∀ y ( Exy → ϕ ∗ ( y )) and ( ♦ ϕ ) ∗ = ∃ y ( Exy ∧ ϕ ∗ )) . Logics of Dependence and Independence: The Local Variants
The guarded fragment GF , introduced in [1], generalises this idea to a much more powerfulsetting of first-order logic with an arbitrary relational vocabulary and an arbitrary numberof variables. It lifts all restrictions of the modal fragment, except for the requirement that allquantifiers must be relativised (guarded) by some atom that contains all free variables of thequantified formula. More formally, GF is the smallest fragment of relational FO generatedfrom atomic formulae by propositional connectives and guarded quantification: if ϕ (¯ x, ¯ y ) isa formula in GF and α (¯ x, ¯ y ) is an atomic formula that contains all free variables of ϕ (¯ x, ¯ y ),then also ∀ ¯ y ( α (¯ x, ¯ y ) → ϕ (¯ x, ¯ y )) and ∃ ¯ y ( α (¯ x, ¯ y ) ∧ ϕ (¯ x, ¯ y ))are formulae of GF .The guarded fragment preserves, and to some degree explains, many of the good model-theoretic and algorithmic properties of modal logics. In particular, GF is decidable [1] andindeed has the finite model property [12]: every satisfiable formula of GF has a finite model.If we consider the standard translation of the logics L [Ω] into FO , we see that that thetranslations of the logical operators preserve guardedness, so the questions is just, whichof the local atoms β (¯ x ) can be rewritten as a guarded formula. Our previous first-ordertranslation of the local inclusion and exclusion atoms can be rewritten as( v i ∈ v j ) ∗ := ∃ ¯ z − j T ¯ z [ j v i ] and ( v i v j ) ∗ := ∀ ¯ z − j ( T ¯ z [ j v i ] → ⊥ )which are guarded formulae. (cid:73) Proposition 7.
The local variant of inclusion-exclusion logic L [= , = , ∈ , ] is a fragment of GF . In particular, it has the finite model property (and is therefore decidable for satisfiabilityand validity). Recall that the global inclusion-exclusion logic, instead, has the full power of independencelogic and Σ . The standard translations of local dependence, local anonymity, and localindependence are, however, not guarded. For local independence logic, a straightforwardargument shows that even without equality, one can enforce cartesian products within theassignment space, leading to the expressive power of usual first-order logic. Hence L [Ind] isalready undecidable, and can therefore not be embedded into GF . The relationship of LFD with the guarded fragment, or other guarded logics, is more difficult to analyse; we shalladdress this issue in Section 6. (cid:73)
Proposition 8. L [ D, Υ , = , / ∈ ] is decidable for satisfiability. Proof.
Given a dependence model ( M , T ) we denote by ( N , T ) the corresponding LFD -equivalent variable-distinguished (v.d.) model, as described in Proposition 5. We first provevia induction on ϕ ∈ L [ D, Υ , = , / ∈ ] that whenever ( M , T ) | = s ϕ , we also have ( N , T ) | = s ϕ .If ϕ ∈ { ¯ x = ¯ y, ¯ x / ∈ ¯ y } and ( M , T ) | = s ϕ , then we know that ¯ x = ¯ y as tuples ofvariables , so obviously ( N , T ) | = s ϕ . If otherwise ϕ ∈ L [ D, Υ] ≡ LFD , this follows from( M , T ) , s ≡ LFD ( N , T ) , s . The induction step for boolean connectives ∧ and ∨ is clear. Also,note that if s, t ∈ T , then we have s (¯ x ) = t (¯ x ) if and only if s (¯ x ) = t (¯ x ). Hence theinduction steps for the quantifiers E ¯ x ϕ and D ¯ x ϕ are straightforward: If ( M , T ) | = s D X ϕ and t ∈ T with t = X s , then also t = X s , so ( M , T ) | = t ϕ . By induction hypothesis we obtain( N , T ) | = t ϕ . Since t ∈ T with t = X s was arbitrary, we obtain ( N , T ) | = s D X ϕ . Theinduction step for E X is analogous. This concludes the induction. . Grädel and P. Pützstück 9 In the following, let [¯ x = ¯ y ] = > if ¯ x = ¯ y (as tuples of variables), and otherwise[¯ x = ¯ y ] = ⊥ . Then, since ( N , T ) is v.d., we have( N , T ) | = s ¯ x = ¯ y ⇐⇒ ( N , T ) | = s [¯ x = ¯ y ] . An analogue holds for ¯ x / ∈ ¯ y , so that over the class of v.d. dependence models, we have¯ x = ¯ y ≡ ¯ x / ∈ ¯ y ≡ [¯ x = ¯ y ], and thereby L [ D, Υ , = , / ∈ ] ≡ L [ D, Υ] ≡ LFD . Finally, since thev.d. model is
LFD -equivalent to the original, we see that
LFD -formulae are satisfiable if andonly if they have a v.d. model. Given ϕ ∈ L [ D, Υ , = , / ∈ ], let ϕ denote the LFD -formulaobtained from ϕ by replacing all occurrences of ¯ x = ¯ y or ¯ x / ∈ ¯ y . Then ϕ satisfiable ⇐⇒ ( M , T ) | = s ϕ for some pointed dependence model ( M , T ) , s ⇐⇒ ( N , T ) | = s ϕ ⇐⇒ ( N , T ) | = s ϕ ⇐⇒ ϕ satisfiable . Thus we have constructed a satisfiability-preserving reduction from L [ D, Υ , = , / ∈ ] to LFD .Since
LFD is known to be decidable for satisfiability [3], this concludes the proof. (cid:74)
We now solve a main open problem from [3] by proving that
LFD = is undecidable. In fact, wewill prove that L [ D, ∈ ] is undecidable and then show how to adapt the argument to L [ D, =],which is contained in L [ D, Υ , = , =] ≡ LFD = . The crucial reason for the undecidability isthat, in the presence of either equality or inclusion, no analogue of Proposition 5 can beestablished, and we can use these atoms to copy values between variables, while keepingcertain other values fixed. We demonstrate this idea for inclusion: (cid:73) Example 9.
Consider a dependence model ( M , T ) over variables V = { x, y, z } such that( M , T ) | = xyx ⊆ xyz , i.e. ( M , T ) | = ∀ ( xyx ∈ xyz ), meaning ( M , T ) | = s xyx ∈ xyz for all s ∈ T . Then, given some s ∈ T with s ( x, y, z ) = ( a, b, c ), there exists t ∈ T such that t ( x, y, z ) = ( a, b, a ). In this sense, we copied the value a of x to the variable z , while keepingthe value of the variables x and y fixed.We recall some basic notions concerning the classical decision problem for first-order logic.For details we refer to [6, Chapter 3.1]. A syntactic fragment F ⊆ FO is called a conservativereduction class if there exists a conservative reduction f : FO → F , i.e. a computable functionthat preserves satisfiability and finite satisfiability in both directions. If F is a conservativereduction class, then by Trakhtenbrot’s Theorem, the satisfiability and finite satisfiabilityproblems for F are undecidable.One of the classical conservative reduction classes is the Kahr-Class, denoted [ ∀∃∀ , ( ω, ∀ x ∃ y ∀ zϕ ( x, y, z ) where ϕ is quantifier-free, withoutequality, and which may only use a single binary relation but an unbounded number ofmonadic ones.We shall construct a conservative reduction from the Kahr-Class into fragments of L [ D, ∈ ]and L [ D, =]. We require only: four variables, one binary predicate and an unbounded numberof monadic ones, one dependence atom (used positively), and six inclusions/equalities (alsoused positively). (cid:73) Theorem 10.
The four-variable fragments of L [= , D ] and L [ D, ∈ ] are conservative reductionclasses. In particular, the satisfiability, validity and finite satisfiability problems for LFD = are undecidable. Proof.
We first prove the claim for L [ D, ∈ ], and then show how to adapt the proof to L [ D, =].Since the Kahr-Class is a conservative reduction class, it suffices, by transitivity, to exhibit aconservative reduction ψ ψ ∗ from [ ∀∃∀ , ( ω, L [ D, ∈ ].Notice that, for ψ = ∀ x ∃ y ∀ zϕ ( x, y, z ) in the Kahr-Class, the quantifier-free part ϕ is inthe base logic L . We define ψ ∗ := ∀ ϕ ( x, y, z ) ∧ dep( x, y ) ∧ ^ i =0 ϑ i . The subformulae ϑ i allow us to copy the value of a variable to another, while keepingcertain other variables fixed , as demonstrated in the example above. This is used to enforcea cartesian product included in T ( x, z ) for the dependence models ( M , T ), allowing usto retrieve a classical model from them. We require an extra variable v as an additional“temporary storage” to copy values between variables. So while we keep ϕ to contain onlythe variables x, y, z , overall we will need four variables, whose order we fix to x, y, z, v . The ϑ i are defined as: ϑ := xx ⊆ xz (copy x to z , keeping x ) ,ϑ := yyv ⊆ yzv (copy y to z , keeping y, v ) ,ϑ := xzx ⊆ xzv (copy x to v , keeping x, z ) ,ϑ := yzy ⊆ yzv (copy y to v , keeping y, z ) ,ϑ := zzv ⊆ xzv (copy z to x , keeping z, v ) ,ϑ := vzv ⊆ xzv (copy v to x , keeping z, v ) . Since dep( x, y ) ≡ ∀ D x y and analogously ¯ x ⊆ ¯ y ≡ ∀ (¯ x ∈ ¯ y ) it follows that ψ ∗ indeed is inthe four-variable fragment of L [ D, ∈ ].We have to prove the following two claims: A (finite) model of ψ induces a (finite) dependence model of ψ ∗ . A (finite) dependence model of ψ ∗ induces a (finite) model of ψ .To prove Claim 1, assume that we have a model A | = ψ with universe A . Thus thereexists a function f : A → A such that A | = ϕ ( a, f a, b ) for all a, b ∈ A . We construct thedependence model ( A , T ) with team T given by T := { ( a, f a, b, c ) | a, b, c ∈ A } . Remember that we denote assignments by their tuple of values, so ( a, f a, b, c ) represents theassignment ( x, y, z, v ) ( a, f a, b, c ). It is clear that y globally depends on x , i.e. ( A , T ) | =dep( x, y ), and that by the choice of T we also have ( A , T ) | = ∀ ϕ ( x, y, z ). The ϑ i are satisfiedin ( M , T ), since T ( x, z, v ) = A is a cartesian product of the whole universe. Overall, weobtain ( A , T ) | = ψ ∗ . Clearly, if A is a finite model, then ( A , T ) is finite as well. This completesthe proof of Claim 1.For the converse, Claim 2, suppose that we have a dependence model ( M , T ) such that( M , T ) | = ψ ∗ . Because of the global dependence ( M , T ) | = dep( x, y ) there exists a function f : T ( x ) → T ( y ) such that t ( y ) = f ( t ( x )) for all t ∈ T . Note that T ( y ) ⊆ T ( x ), since ϑ and ϑ allow us to copy values from y to z and from there to x . Hence we have f : T ( x ) → T ( x ),i.e. we can iterate f on values of x . Fix some arbitrary s ∈ T and set 0 := s ( x ), as well as i := f i i ∈ N . . Grädel and P. Pützstück 11 We construct a model for ψ by A := M (cid:22) A where A := { i | i ∈ N } . The function f (cid:22) A : A → A plays the role of the Skolem function for y in the quantification ∀ x ∃ y ∀ z of ψ .We need to ensure that ϕ ( a, f a, b ) holds in M (and thus in A ) for all a, b ∈ A .Since ( M , T ) | = ∀ ϕ ( x, y, z ) we know that M | = ϕ ( t ( x ) , f ( t ( x )) , t ( z )) for all t ∈ T . Henceit suffices to show that A × A ⊆ T ( x, z ) . (1)In the following we write ∗ as placeholder for not further specified elements of M . Theexpression t ϑ i −→ u for some t ∈ T denotes that the existence of u ∈ T follows by applyingthe “copy-rule” which ϑ i represents. Notice that for all t ∈ T with t ( x ) = i we have t ( y ) = f ( i ) = i + 1. In particular, keeping the value of x implies keeping the value of y . (0 , ∈ T ( x, z ):We know that s looks like (0 , , ∗ , ∗ ). Since s ∈ T and s = (0 , , ∗ , ∗ ) ϑ −→ (0 , , , ∗ ) =: t (copy x to z , keeping x, y )we see that t ∈ T with t ( x, z ) = (0 , If (0 , j ) ∈ T ( x, z ), then also (0 , j + 1) ∈ T ( x, z ):By assumption we have t = (0 , , j, ∗ ) ∈ T . Together with the derivation t = (0 , , j, ∗ ) ϑ −→ (0 , , j,
0) (copy x to v , keeping x, y, z ) ϑ −→ ( j, j + 1 , j,
0) (copy z to x , keeping z, v ) ϑ −→ ( ∗ , j + 1 , j + 1 ,
0) (copy y to z , keeping y, v ) ϑ −→ (0 , , j + 1 ,
0) =: u (copy v to x , keeping z, v )we obtain u ∈ T with u ( x, z ) = (0 , j + 1). If ( i, j ) ∈ T ( x, z ) then also ( i + 1 , j ) ∈ T ( x, z ):By assumption we have t = ( i, i + 1 , j, ∗ ) ∈ T . Together with the derivation t = ( i, i + 1 , j, ∗ ) ϑ −→ ( ∗ , i + 1 , j, i + 1) (copy y to v , keeping y, z ) ϑ −→ ( i + 1 , i + 2 , j, i + 1) =: u (copy v to x , keeping z, v )we obtain u ∈ T with u ( x, z ) = ( i + 1 , j ).Now the inclusion (1) follows via induction. By the above argument this proves that A | = ϕ ( a, f a, b ) for all a, b ∈ A and hence A | = ∀ x ∃ y ∀ zϕ ( x, y, z ). Again it is clear thatif ( M , T ) is finite, then so is A . This concludes the proof of Claim 2, showing that thefour-variable fragment of L [ D, ∈ ] is a conservative reduction class.For the case of L [ D, =], note first that for any dependence model ( M , T ) with s ∈ T andnon-empty tuples of variables ¯ x, ¯ y, ¯ z with [¯ y ] ⊆ [¯ x ], we have( M , T ) | = s ¯ x ¯ y ∈ ¯ x ¯ z ⇐⇒ ( M , T ) | = s E ¯ x ¯ y (¯ y = ¯ z ) ⇐⇒ ( M , T ) | = s E ¯ x (¯ y = ¯ z ) . Thus, if [¯ y ] ⊆ [¯ x ], we have ¯ x ¯ y ⊆ ¯ x ¯ z ≡ ∀ E ¯ x (¯ y = ¯ z ). In our case, all inclusion atoms can bewritten in this form. Hence, we can define equivalent formulae in L [ D, =]: ϑ := ∀ E x ( x = z ) (copy x to z , keeping x ) ,ϑ := ∀ E yv ( y = z ) (copy y to z , keeping y, v ) ,ϑ := ∀ E xz ( x = v ) (copy x to v , keeping x, z ) ,ϑ := ∀ E yz ( y = v ) (copy y to v , keeping y, z ) ,ϑ := ∀ E zv ( z = x ) (copy z to x , keeping z, v ) ,ϑ := ∀ E zv ( v = x ) (copy v to x , keeping z, v ) . The proof then works in the exact same way as before. (cid:74)
Consider the lattice of the local logics using the atoms D, Υ , = , = , ∈ , / ∈ , Ind. Because we oftenwant Ω to be closed under negation, we may also consider ¬ Ind. In the above discussions,we have shown that L [ D, ∈ ] , L [ D, =] and L [Ind] are minimal undecidable extensions of L inthis lattice. FOL [ D, Υ , = , = , ∈ , / ∈ , Ind , ¬ Ind] L [ D, ∈ ] LFD = L [ D, =] L [Ind] GFL [= , = , ∈ , / ∈ ] L [ D, Υ , = , / ∈ ] LFD L
With this, many of the extensions of L by local atoms Ω ⊆ { D, Υ , = , = , ∈ , / ∈ , Ind , ¬ Ind } havebeen classified by decidability. The main question left is how negated local independence ¬ Ind fits into this picture, and how anonymity and inclusion / equality affect each other. (cid:73)
Open Problem.
Classify the remaining L [Ω] where Ω ⊆ { D, Υ , = , = , ∈ , / ∈ , Ind , ¬ Ind } isnot a subset of { D, Υ , = , / ∈} or { = , = , ∈ , / ∈} and not a superset of { D, ∈} , { D, = } or { Ind } .Of the extensions fitting this description L [Υ , =] , L [Υ , D ] and L [ ¬ Ind] are minimal, whereas L [ D, Υ , = , / ∈ , ¬ Ind] and L [Υ , = , = , ∈ , / ∈ , ¬ Ind] are maximal. . Grädel and P. Pützstück 13
We define a notion of bisimulation for local dependence logics L [Ω] and in particular for LFD and
LFD = in such a way that we obtain an analogue of the classical Ehrenfeucht-FraïsséTheorem in Section 4.2, and later an analogue of van Benthems’s Theorem in Section 5. Inthe following, many results will require that Ω and hence L [Ω] is closed under negation. Weshall also consider infinitary variants of these logics. (cid:73) Definition 11 (Bisimulation) . Let ( M , T ) and ( N , T ) be two dependence models of thesame type ( τ, V ). A binary relation Z ⊆ T × T is an L [Ω]-bisimulation between ( M , T ) and( N , T ) if for all ( s, s ) ∈ Z : ( M , T ) , s and ( N , T ) , s agree on the atoms of L [Ω]: a. For all R ∈ τ and ¯ x ∈ V ar( R ) we have ( M , T ) | = s R ¯ x ⇐⇒ ( N , T ) | = s R ¯ x . b. For all local atoms β ∈ Ω[ V ] we have ( M , T ) | = s β ⇐⇒ ( N , T ) | = s β . (back) For all t ∈ T and all finite X ⊆ t ∩ s := { x ∈ V | t = x s } there is some t ∈ T with ( t, t ) ∈ Z and t = X s . (forth) For all t ∈ T and all finite X ⊆ t ∩ s := { x ∈ V | t = x s } there is some t ∈ T with ( t, t ) ∈ Z and t = X s .We restrict ourselves to finite sets X ⊆ t ∩ s because L only allows finite sets within ourmodalities D X and E X . Whenever ( M , T ) , ( N , T ) and L [Ω] are clear from context, we write s ∼ s if there exists an L [Ω]-bisimulation Z between ( M , T ) and ( N , T ) with ( s, s ) ∈ Z . (cid:73) Definition 12 (Ordinal approximations to bisimulation) . We write s ∼ s and say that s and s are α -bisimilar if ( M , T ) , s and ( N , T ) , s agree on L [Ω]-atoms. Now define s ∼ α s for ordinals α ∈ On by induction; when defining s ∼ α +1 s , we require the conditions( α + 1)-back: For all t ∈ T and all finite X ⊆ t ∩ s there exists some t ∈ T with t ∼ α t and t = X s .( α + 1)-forth: For all t ∈ T and all finite X ⊆ t ∩ s there exists some t ∈ T with t ∼ α t and t = X s .For limit ordinals λ , we say that s ∼ λ s if s ∼ α s for all α < λ , so essentially ∼ λ = \ α<λ ∼ α . (2)As usual, α -bisimilarity implies β -bisimilarity for all β < α . Furthermore, full bisimilarity isnow simply given by ∼ = T α ∈ On ∼ α .We want to emphasize that the back and forth conditions of our bisimulations do notrequire the regarded assignments t or t to actually agree on any variables with s or s respectively, i.e. t ∩ s and t ∩ s may be empty. The reason for this is that L has the globalmodalities ∀ = D ∅ and ∃ = E ∅ and that we want bisimilarity to correspond to logicalequivalence (we say that some modality is global if its corresponding accessibility relation isthe all-relation, i.e. contains all possible pairs of objects, as is the case for = ∅ on teams). Asan example, let V be finite and ψ := ∃ ( Rx ) ∧ ^ v ∈V ¬ E v Rx.
Then ( M , T ) | = s ψ means that there is some t ∈ T with ( M , T ) | = t Rx , but that there is no u ∈ T with u = x s having this property, and thus t ∩ s = ∅ . Since L is able to witness this t ,it is natural to require a bisimilar t ∈ T for every t ∈ T , and not just for those t that agreewith s on some variable. Likewise for the back condition.This forces every bisimulation to be global , meaning that every assignment in T isbisimilar to at least one assignment in T , and vice versa. This is a common consequenceof having global modalities; in the context of, say, ML with an explicitly added globalmodality, often denoted ML ( ∀ ), the canonical bisimulation is just the global version ofordinary ML -bisimulation [7].Since we already defined bisimulation for infinite ordinals, the corresponding step for ourlogics is to consider their infinitary variants. We briefly give some routine definitions whichwe need in what follows. (cid:73) Definition 13 (L [Ω] ∞ ) . The infinitary extension L [Ω] ∞ of L [Ω] allows conjunction anddisjunction over arbitrarily large sets of L [Ω] ∞ -formulae, with the obvious semantics. (cid:73) Definition 14 (Quantifier Rank) . The quantifier rank qr( ϕ ) of some ϕ ∈ L [Ω] ∞ is arecursively defined ordinal. We define qr( β ) = 0 for atoms β (including negated relationalatoms and the local atoms), as well as qr( D X ϕ ) = qr( E X ϕ ) = qr( ϕ ) + 1. Lastly, we setqr (cid:0)V i ∈ I ϕ i (cid:1) = qr (cid:0)W i ∈ I ϕ i (cid:1) = sup i ∈ I qr( ϕ i ). (cid:73) Definition 15 (Equivalence) . We use the usual symbols ≡ L [Ω] ( ≡ α L [Ω] ) for equivalence ofpointed dependence models in the logic L [Ω] (up to quantifier rank α ). We also use the shortform s ≡ α L [Ω] s whenever ( M , T ) , s and ( N , T ) , s are clear from context. The infinitary case L [Ω] ∞ is defined analogously, but we often write ≡ ∞ L [Ω] instead of ≡ L [Ω] ∞ . Given a logic, a common goal is to find a correspondence between logical indistinguishabilityand behavioural equivalence in some structural form, often as a relation akin to bisimulation,a collection of partial isomorphisms, or a winning strategy of certain two-player games. For FO we have back-and-forth systems and Ehrenfeucht-Fraïssé games (cf. [15, Chapter 3.3]),whereas for ML one has ordinary bisimulation and the corresponding bisimulation games(cf. [5, Chapter 2.2]). The following results and in particular Theorem 18 show that thebisimulations defined above fulfill such a role for L [Ω] whenever Ω is closed under negation. (cid:73) Lemma 16.
Let ( M , T ) , s be a dependence model of some finite type ( τ, V ) and Ω befinite and closed under negation. For every k ∈ N there exists a formula χ ks ∈ L [Ω]( τ, V ) ofquantifier rank k that defines the ∼ k -class of s , so that for all suitable ( N , T ) , s ( N , T ) | = s χ ks ⇐⇒ s ∼ k s Up to L [Ω] -equivalence, the number of such χ ks is finite and depends only on τ, V , Ω and k . Proof.
Since τ, V and Ω are finite, there are up to equivalence only finitely many formulae ϕ ∈ L [Ω]( τ, V ) with qr( ϕ ) = 0, which allows us to define χ s := ^ { ϕ ∈ L [Ω]( τ, V ) | qr( ϕ ) = 0 , ( M , T ) | = s ϕ } . One proceeds inductively by defining χ k +1 s := ϕ k +1back ∧ ϕ k +1forth , where ϕ k +1back := ^ X ⊆V D X _ t ∈ Tt = X s χ kt and ϕ k +1forth := ^ t ∈ T ^ X ⊆V t = X s E X χ kt . . Grädel and P. Pützstück 15 By the induction hypothesis, these are well-defined L [Ω]( τ, V )-formulae and correspondprecisely to the ( k + 1)-back and ( k + 1)-forth conditions as given in Definition 12. (cid:74)(cid:73) Lemma 17.
Let ( M , T ) , s be a dependence model of some type ( τ, V ) and Ω be closedunder negation. For every α ∈ On there exists a formula χ αs ∈ L [Ω]( τ, V ) of quantifier rank α that defines the ∼ α -class of s , so that for all suitable ( N , T ) , s ( N , T ) | = s χ αs ⇐⇒ s ∼ α s. Proof.
For α = 0 and successor ordinals, we use analogous definitions to the ones above,except that we have to explicitly consider only finite X ⊆ V as they are used within thequantifiers D X and E X . For limit ordinals λ , we set χ λs := V α<λ χ αs , which corresponds todefinition of ∼ λ , see Equation (2) in Definition 12. Conclude via transfinite induction. (cid:74)(cid:73) Theorem 18 (Ehrenfeucht-Fraïssé and Karp theorems for L [Ω] ) . Let Ω be closed under negation. If Ω is finite and ( M , T ) , s and ( N , T ) , s are dependencemodels of the same finite type, then s ∼ k L [Ω] s ⇐⇒ s ≡ k L [Ω] s , k ∈ N . As a consequence we obtain that under those same conditions s ∼ ω L [Ω] s ⇐⇒ s ≡ L [Ω] s . For Ω not necessarily finite and arbitrary types we obtain s ∼ α L [Ω] s ⇐⇒ s ≡ α L [Ω] ∞ s , α ∈ On and therefore s ∼ L [Ω] s ⇐⇒ s ≡ ∞ L [Ω] s . Proof.
The proof is a routine induction. One shows that k - L [Ω]-bisimilarity entails L [Ω]-equivalence up to quantifier rank k by using the characteristics of our bisimulation, whereasthe converse implication is immediate from the above lemmas. The infinitary case is handledanalogously. (cid:74) This allows us to show undefinability of some property of (pointed) dependence models,by finding two such models that are bisimilar but differ on said property. By the abovetheorem, we then know that these models are logically indistinguishable in L [Ω], so theconsidered property cannot be defined in the respective logic. (cid:73) Example 19.
Let Ω = { D, Υ , = , = } . We show that the inclusion T ( x ) ⊆ T ( y ) is not L [Ω] ∞ -definable. It suffices to show this for τ = ∅ and V = { x, y } , because the examplebelow can be adapted accordingly. Consider dependence models ( M , T ) , ( N , T ) of type( ∅ , { x, y } ) with teams given by T := { ( a, b ) , ( b, a ) } and T := { (1 , , (2 , } . Note that T ( x ) ⊆ T ( y ), but T ( x ) T ( y ). Now let Z be the binary relation on T × T definedby ( a, b ) Z (1 ,
2) and ( b, a ) Z (2 , Z is a full L [Ω]-bisimulation andhence s ≡ ∞ L [Ω] s by Theorem 18, for all ( s, s ) ∈ Z . Indeed, note that Υ ∅ x, Υ ∅ y, D x y, D y x and x = y hold at all assignments in both teams, so the pairs of assignments agree on atoms.Furthermore, in both teams we see that the two assignments do not agree on any variables,so evidently the only choice we have at the back and forth clauses (every assignment standsin Z -relation to exactly one other) always works out. We now prove a characterisation theorem for L [Ω]. The theorem is an analogue of vanBenthem’s Theorem, which states that ML , via its standard translation into FO , is preciselythe bisimulation-invariant fragment of FO over the class of pointed Kripke structures. It wasfirst formulated in [19] and [20]. We adapt a well known proof using saturated structures byfollowing the exposition in [5, Chapter 2.6]. We assume that the reader is familiar with basicmodel-theoretic notions such as elementary extensions and ω -saturated structures.Given a dependence model ( M , T ) , s of finite type ( τ, V ), we have already seen that wecan interpret ( M , T ) as a τ ∪ { T } -structure, where T is a |V| -ary relation symbol; this is doneby fixing an enumeration ¯ v of V and viewing T as the |V| -ary relation T (¯ v ) = { s (¯ v ) | s ∈ T } over ( M , T ). In the following, we also write ¯ s := s (¯ v ) for the |V| -ary tuple correspondingto s ∈ T . For clarity of presentation we denote the evaluation of the first-order formula ϕ (¯ x ) ∈ FO ( τ ∪ { T } ) in the structure ( M , T ) at the tuple ¯ s under classical Tarski-semanticsby ( M , T ) | = FO ϕ (¯ s ). In Section 2.4 we gave a first-order translation L [Ω] → FO , ϕ ϕ ∗ such that for all ϕ ∈ L [Ω] and every fitting dependence model ( M , T ) , s we have( M , T ) | = s ϕ ⇐⇒ ( M , T ) | = FO ϕ ∗ (¯ s ) . In the following, let ( τ, V ) be a finite type with an enumeration ¯ v of V , and Ω ⊆ { D, Υ , = , = , ∈ , / ∈ , Ind , · · · } a finite set of local atoms that is closed under negation and to which we mayextend the above first-order translation. (cid:73) Definition 20.
For dependence models ( M , T ) , s and ( N , T ) , s letTh L [Ω] (( M , T ) , s ) := { ϕ ∗ | ϕ ∈ L [Ω] , ( M , T ) | = s ϕ } ⊆ FO ( τ ∪ { T } ) . We use Th L [Ω] ( s ) if ( M , T ) is clear from context. Note that Th L [Ω] ( s ) generally containsformulae with free variables among ¯ v , not just sentences. (cid:73) Lemma 21.
Let ( M , T ) , s and ( N , T ) , s be pointed dependence models. Then ( M , T ) | = FO Th L [Ω] ( s )(¯ s ) ⇐⇒ Th L [Ω] ( s ) = Th L [Ω] ( s ) ⇐⇒ s ≡ L [Ω] s . (cid:73) Definition 22.
Write n = |V| = | ¯ v | and consider tuples of variables ¯ x = ( x , . . . , x n ) and ¯ y = ( y , . . . , y n ) . With respect to the ordering of V given by ¯ v , we identify X ⊆ V with a setof indices I X ⊆ { , . . . , n } . Now define ϑ X (¯ x, ¯ y ) ∈ FO ( τ ∪ { T } ) by ϑ X (¯ x, ¯ y ) := T ¯ x ∧ T ¯ y ∧ ^ i ∈ I X x i = y i . (cid:73) Lemma 23.
Let ( M , T ) be a dependence model, X ⊆ V and ¯ s, ¯ t tuples over M of length |V| . Write s, t for the induced assignments with s (¯ v ) = ¯ s and t (¯ v ) = ¯ t . Then ( M , T ) | = FO ϑ X (¯ s, ¯ t ) ⇐⇒ s, t ∈ T and s = X t. (cid:73) Lemma 24.
Let ( M , T ) and ( N , T ) be two ( τ, V ) dependence models, so that their cor-responding ( τ ∪ { T } ) -structures are ω -saturated. If ( M , T ) , s ∼ ω L [Ω] ( N , T ) , s , then already s ∼ L [Ω] s . Proof.
For ease of presentation we write ∼ and ∼ ω , omitting the subscript L [Ω]. Let ( M , T )and ( N , T ) be as described above. By our Ehrenfeucht-Fraïssé Theorem, it follows from s ∼ ω s that s ≡ L [Ω] s . Hence it suffices to show that Z := { ( s, s ) ∈ T × T | ( M , T ) , s ≡ L [Ω] ( N , T ) , s } . . Grädel and P. Pützstück 17 is a bisimulation between ( M , T ) and ( N , T ). Given ( s, s ) ∈ Z , we clearly have s ≡ L [Ω] s ,so s and s agree on atoms. We proceed by checking the forth condition.Let t ∈ T and X ⊆ t ∩ s = { x ∈ V | t = x s } be finite. Set p (¯ y ) := { ϑ X (¯ y, s ) } ∪ Th L [Ω] ( t )(¯ y ) . We want to show that p is a type of ( N , T ) with parameters ¯ s , i.e. that p together with thefirst-order theory Th FO (( N , T ) , ¯ s ) is satisfiable. For a compactness argument we consider afinite Φ (¯ y ) ⊆ Th L [Ω] ( t )(¯ y ) and define ϕ (¯ x ) := ∃ ¯ t ( ϑ X (¯ t , ¯ x ) ∧ ^ Φ (¯ t )) . Now there exists some finite Ψ ⊆ L [Ω] with Φ (¯ y ) = { ϕ ∗ (¯ y ) | ϕ ∈ Ψ } . Since our translationcommutes with ∧ , we obtain ( V Ψ ) ∗ = V Φ . We claim that ϕ ≡ ( E X V Ψ ) ∗ . Clearly any( τ ∪ { T } )-structure can be interpreted as the corresponding structure to a ( τ, V ) dependencemodel ( A , T ). For such a ( A , T ) and any a ∈ T we have( A , T ) | = FO ϕ (¯ a ) ⇐⇒ there exists ¯ b over A with ( A , T ) | = FO ϑ X (¯ b, ¯ a ) and ( A , T ) | = FO (cid:16)^ Ψ (cid:17) ∗ (¯ b ) ⇐⇒ there exists b ∈ T with b = X a and ( A , T ) | = b ^ Ψ ⇐⇒ ( A , T ) | = a E X ^ Ψ ⇐⇒ ( A , T ) | = FO (cid:16) E X ^ Ψ (cid:17) ∗ (¯ a ) . Thus, as claimed, we have ϕ ≡ ( E X V Ψ ) ∗ . From ( M , T ) | = t V Ψ and t = X s weobtain ( M , T ) | = s E X V Ψ . Since s ≡ L [Ω] s we get ( N , T ) | = s E X V Ψ and therefore( N , T ) | = FO ϕ ( ¯ s ). Hence { ϑ X (¯ y, ¯ s ) } ∪ Φ (¯ y ) ∪ Th FO (( N , T ) , ¯ s ) is satisfiable. It follows bycompactness that p (¯ y ) is a type with finitely many parameters (namely ¯ s ) over ( N , T ).By ω -saturatedness we obtain some tuple ¯ t in N with ( N , T ) | = FO p (¯ t ). From thedefinition of ϑ X we see that ¯ t = t (¯ v ) for some t ∈ T with t = X s . Furthermore( N , T ) | = FO Th L [Ω] ( t )(¯ t ), so t ≡ L [Ω] t and hence ( t, t ) ∈ Z , which proves the forth condition.The back condition is shown analogously. We conclude that Z is a bisimulation. (cid:74)(cid:73) Theorem 25 (Expressive Completeness) . For any ϕ ∈ FO ( τ ∪ { T } ) the following areequivalent: ϕ is L [Ω] -bisimulation-invariant, i.e. for all ( M , T ) , s and ( N , T ) , s of type ( τ, V ) s ∼ L [Ω] s = ⇒ ( M , T ) | = FO ϕ (¯ s ) iff ( N , T ) | = FO ϕ ( ¯ s ) . ϕ ≡ ψ ∗ for some ψ ∈ L [Ω]( τ, V ) , so that for all ( M , T ) , s of type ( τ, V )( M , T ) | = FO ϕ (¯ s ) ⇐⇒ ( M , T ) | = s ψ. As this result holds a fixed Ω and arbitrary finite types ( τ, V ) , we write FO / ∼ L [Ω] ≡ L [Ω] . Proof.
As in the previous proof, we omit L [Ω] from ∼ and ∼ ω , and say bisimulation instead of L [Ω]-bisimulation. The implication “(2) ⇒ (1)” is clear by our Ehrenfeucht-Fraïssé Theorem.Now assume that ϕ is bisimulation-invariant and consider the set of its L [Ω]-consequences: C ( ϕ )(¯ x ) := { ψ ∗ (¯ x ) | ψ ∈ L [Ω]( τ, V ) and ϕ | = ψ ∗ } . We claim that it suffices to show C ( ϕ ) | = ϕ . Indeed, using compactness this yields a finitesubset C ⊆ C ( ϕ ) with C | = ϕ and therefore V C ≡ ϕ . But C = { ψ ∗ | ψ ∈ Ψ } forsome (finite) Ψ ⊆ L [Ω]( τ, V ), so by setting ψ = V Ψ, we obtain the desired result that ϕ ≡ V C = ψ ∗ for some ψ ∈ L [Ω]( τ, V ).If C ( ϕ ) is unsatisfiable, C ( ϕ ) | = ϕ holds vacuously. Hence let ( M , T ) , s be a dependencemodel of type ( τ, V ) with ( M , T ) | = FO C ( ϕ )(¯ s ). We need to show ( M , T ) | = FO ϕ (¯ s ). IfTh L [Ω] ( s ) ∪ { ϕ } were unsatisfiable, by compactness there would be some finite Φ ⊆ Th L [Ω] ( s )such that Φ ∪ { ϕ } is unsatisfiable. This implies ϕ | = ¬ V Φ . Since Ω is closed undernegation, L [Ω] is too, and we obtain ¬ V Φ ∈ C ( ϕ ). This contradicts ( M , T ) | = FO C ( ϕ )(¯ s )and Φ ⊆ Th L [Ω] ( s ).Therefore Th L [Ω] ( s ) ∪ { ϕ } has some model ( N , T ) , s . Since ( N , T ) | = FO Th L [Ω] ( s )( ¯ s )we obtain s ≡ L [Ω] s . Now take ω -saturated elementary extensions ( M , T ) (cid:22) ( M + , T + )and ( N , T ) (cid:22) ( N + , T + ). By elementary extension we have ( M + , T + ) , s ≡ L [Ω] ( M , T ) , s andlikewise for ( N + , T + ) , s . It follows that( M + , T + ) , s ≡ L [Ω] ( M , T ) , s ≡ L [Ω] ( N , T ) , s ≡ L [Ω] ( N + , T + ) , s . Our Ehrenfeucht-Fraïssé Theorem yields ( M + , T + ) , s ∼ ω ( N + , T + ) , s . But now we can applyLemma 24 to find ( M + , T + ) , s ∼ ( N + , T + ) , s . Since ϕ ∈ FO , we infer ( N + , T + ) | = FO ϕ ( ¯ s )from ( N , T ) | = FO ϕ ( ¯ s ) by elementary extension. Moreover, ϕ is bisimulation-invariant, so weobtain ( M + , T + ) | = FO ϕ (¯ s ). Again by elementary extension, we arrive at ( M , T ) | = FO ϕ (¯ s ).With this we showed that whenever ( M , T ) | = FO ϕ (¯ s ) we also have ( M , T ) | = FO C ( ϕ )(¯ s )for an arbitrary ( M , T ) , s of type ( τ, V ). Hence C ( ϕ ) | = ϕ . We discussed above how thisconcludes the proof of the theorem. (cid:74) FO We have seen that some of the considered local logics can be embedded into GF , the guardedfragment of first-order logic. However, it was unclear whether this is also the case for LFD , which features the local dependence atom. We now discuss the relationship of localdependence with GF and other guarded fragments of FO . Arguably one of the most naturalgeneralizations of GF within FO is the clique-guarded fragment CGF , introduced in [12].Recall that the Gaifman graph of a relational τ -structure A has as its universe the universeof A , and an edge between two distinct elements a = b if these coexist in some atomic fact of A , i.e. they occur together in some ¯ c ∈ R A for some R ∈ τ . Obviously, guarded tuples in arelational structure A induce a clique in the Gaifman graph of A . Moreover, for each finiterelational τ and k ∈ N , there is a positive, existential first-order formula clique ( x , . . . , x k )which is satisfied at a tuple ¯ a of some τ -structure A if and only if ¯ a induces a clique in theGaifman graph of A . CGF is then defined in an analogous way to GF , but always uses clique of the right arity as a guard, in the sense of ∀ ¯ y ( clique (¯ x ¯ y ) → ϕ (¯ x ¯ y )) and ∃ ¯ y ( clique (¯ x ¯ y ) ∧ ϕ (¯ x ¯ y )) . Obviously ML (cid:40) GF (cid:40) CGF (cid:40) FO . Guarded bisimulations between structures A , ¯ a and B , ¯ b are defined as sets I of partial isomorphisms between A and B such that (¯ a ¯ b ) ∈ I and I is closed under suitable back and forth conditions. This notion naturally extends toclique-guarded bisimulation for CGF , as described in [12]. For a survey of various notionsof bisimulation and their uses for understanding expressive power, model-theoretic andalgorithmic properties of modal and guarded logics, we refer the reader to [13]. . Grädel and P. Pützstück 19Other first-order translations
Apart from the standard translation discussed until now, one can also consider other first-order translation of L [Ω]. In [3], Baltag and van Benthem emphasized the modal perspectiveof LFD ≡ L [ D, Υ], and presented a modal semantics for
LFD over so-called standard relationalmodels. The semantics of local dependence and local independence only require knowledgeabout structure of the team with respect to the ”agreement”-relations = X for X ⊆ V . In thissense, one introduces abstract equivalence relations ∼ X which allow us to abstract away theactual values of assignments while preserving the intended semantics. A dependence modelinduces such a standard relational model in a straightforward way: the new universe is theteam, the relations become monadic and we add the equivalences ∼ x for x ∈ V . Formulae inthe base logic L are then translated as tr s ( R ¯ x ) := R ¯ x s and tr s ( D X ϕ ) := ∀ t ^ x ∈ X t ∼ x s → tr t ( ϕ ) ! , where tr commutes with boolean connectives. For the local atoms, we set tr s ( D x y ) := ∀ t ( t ∼ x s → t ∼ y s ) and tr s (Ind x y ) := ∀ t ∃ u ( s ∼ x u ∧ u ∼ y t ) . If we restrict the considered class of structures to those that are induced by dependence modelsunder this correspondence (so in particular they have to interpret the ∼ x as equivalencerelations) then this translation shares many of the nice characteristics of the standardtranslation. Indeed, using the modal translation in the definition of Th L [Ω] and setting ϑ X ( s, t ) = V x ∈ X s ∼ x t , the proof of the characterisation theorem in the last section alsoworks in this modal context, with minimal adaptions. Expressive Incomparability
In the following we want to prove that under these translations, local dependence is inherentlyincompatible with the clique-guarded fragment of first-order logic.Let τ be a relational vocabulary and A , B two τ -structures with disjoint domains A, B .Their disjoint union is the τ -structure C = A ] B with domain C = A ] B so that C (cid:22) A and C (cid:22) B are isomorphic to A and B respectively, and for all tuples ¯ c over C which containelements from both A and B we have C | = ¬ R ¯ c for every R ∈ τ . (cid:73) Proposition 26.
The relevant bisimulations for GF and CGF are compatible with disjointunions of bisimilar models. Specifically, let I be a clique-guarded bisimulation between A and B , and I be one between A and C . Then I ∪ I is a clique-guarded bisimulation between A and B ] C . In particular, from A , ¯ a ∼ CGF B , ¯ b and A , ¯ a ∼ CGF C , ¯ c we infer A , ¯ a ∼ CGF ( B ] C ) , ¯ b and A , ¯ a ∼ CGF ( B ] C ) , ¯ c. In this setting, if ϕ (¯ x ) ∈ CGF , then A | = ϕ (¯ a ) holds if and only if ( B ] C ) | = ϕ (¯ b ) ∧ ϕ (¯ c ) . This highlights a small but important difference between these guarded fragments and ourlogics of local dependence. Namely, an analogue for invariance under disjoint union cannothold for dependence models in presence of the local dependence atom. Indeed, we can defineconstancy of a variable x via D ∅ x , which is clearly not invariant under disjoint unions. (cid:73) Proposition 27.
The standard and modal translations do not embed L [ D ] into CGF . Morespecifically, there does not exist a
CGF -sentence ψ that is equivalent to either translation ofthe constancy atom D ∅ x . Proof.
In the context of the standard translation, it is clear that the disjoint union ofthe first-order structures corresponding to two dependence models is itself the first-orderstructure corresponding to the disjoint union of the dependence models. Unlike D ∅ x , CGF isinvariant under disjoint unions, so there cannot exist a ψ ∈ CGF with ( D ∅ x ) ∗ ≡ ψ .The argument for the modal translation is similar, and relies on the fact that the class offirst-order structures we consider is well-behaved with respect to disjoint union in the abovesense. (cid:74) There are certainly also notions expressible in GF but not in L [Ω]. For one, in thesetting of the standard translation it becomes obvious that L [Ω] cannot make statementsabout assignments and values outside of the team; we may have ( M , T ) | = FO ∃ xRx andsimultaneously ( M , T ) | = ∀ ¬ Rx . Even in the modal setting we can easily find formulae in GF that are not equivalent to any LFD ≡ L [ D, Υ] formula. (cid:73)
Proposition 28.
Let ( τ, V ) be a finite type with x, y, z ∈ V and let ϕ ( s ) := ∀ t ( t ∼ x s → ( t ∼ y s ∨ t ∼ z s )) ∈ GF . Then there exists no ψ ∈ L [ D, Υ] whose modal translation is equivalent to ϕ over the consideredclass of first-order structures. Proof.
We give two ( ∅ , { x, y, z } ) dependence models that are L [ D, Υ]-bisimilar, but wheretheir corresponding (modal) first-order structures disagree on ϕ . These are uniquely de-termined by the structure of their teams. The first team has = y -classes { t , s } , { t } and= z -classes { t } , { s, t } . The second team has = y -classes { t , s , s } , { t , t } and = z -classes { t } , { s , t } , { s , t } . In both teams all assignments agree on x . The bisimulation relates s to s and s , t to t , and t to t and t . Then s ∼ L [ D, Υ] s but ϕ holds at s , while itdoes not hold at s . The example is easily extended to larger types containing the variables x, y, z . (cid:74)(cid:73) Corollary 29.
In the context of the standard and modal translations,
LFD ≡ L [ D, Υ] isexpressively incomparable to the guarded fragment GF and even the clique-guarded fragment CGF . In this last section we study the complexity of the model checking problem, abbreviatedMC( L [Ω]), for local logics L [Ω]: Given a formula ψ ∈ L [Ω] and a fitting finite pointeddependence model ( M , T ) , s , we ask whether ( M , T ) , s | = ψ . We consider the combinedcomplexity , measured with respect to the size of all inputs. It turns that this complexity islargely influenced by how we encode the team T of ( M , T ).For comparison, we recall that the model checking for first-order logic ( FO ) is Pspace -complete in general, but
Ptime -complete for many interesting fragments of FO [11, Chapter3.1], including the modal fragment ML , the bounded variable fragments FO k (for any k ≥ GF [4].For the rest of this section ( τ, V ) denotes a finite type, Ω ⊆ { D, Υ , = , = , ∈ , / ∈ , Ind , ¬ Ind } ,is a collection of local atoms, and we study the model checking problem of L [Ω]( τ, V ).Before we discuss these technicalities of team encodings, we consider the special case offull models. We call dependence models full if their team consists of the whole assignmentspace, so ( M , T ) is full if T = M V . Over the class of all full dependence models, even the baselogic L = L [ ∅ ] is as expressive as relational first-order logic without equality; the quantifiers . Grädel and P. Pützstück 21 ∃ x and ∀ x then have the same semantics as E V x and D V x respectively, where V x := V \ { x } .For example, if ( M , T ) is full and V = { x, y } , then( M , T ) | = FO ∀ x ∃ yRxy ⇐⇒ ( M , T ) | = D y E x Rxy ⇐⇒ ( M , T ) | = ∀ E x Rxy.
This was already noted for
LFD ≡ L [ D, Υ] by Baltag and van Benthem in [3].It is therefore not surprising that when restricting attention to full dependence models, weobtain the same lower bounds on the complexity of MC( L [Ω]) as for first-order logic. Denoteby MC full ( L [Ω]) the restriction of model checking for L [Ω] to instances where the team isalways the full team, and write MC k full ( L [Ω]) for its k -variable restriction. Thus the inputsare of the form ( ψ, M , ¯ v, ¯ s ) where ¯ v is an enumeration of V and ¯ s = s (¯ v ) for s ∈ M V is anencoding of the current assignment at which ψ should be evaluated. By the same techniquesas for first-order logic, one immediately obtains the following hardness results. (cid:73) Proposition 30. MC full ( L ) is Pspace -hard and MC k full ( L ) is Ptime -hard for k ≥ . Since L is a sublogic of L [Ω] , these results also hold for L [Ω] with arbitrary Ω . Solving the model checking problem for L [Ω] . We consider a general method for solvingthe model checking problem for L [Ω] that abstracts away the specifics on how the team isencoded. We assume familiarity with alternating complexity classes as given in [2, Chapter3] or [17, Chapters 16.2 & 19.1]. The model checking problem for first-order logic model canbe solved in a standard way (cf. [11, Chapter 3.1]) by an alternating algorithm which, todetermine whether A | = ψ (¯ a ), requiresalternating space O (log | ψ | + r log | A | ), where r is the maximal number of free variablesin any subformula of ψ , andalternating time O ( | ψ | log | A | ).Together with the well-known facts that Alogspace = Ptime and
Aptime = Pspace one then obtains that MC( FO k ) ∈ Ptime , k ∈ N and MC( FO ) ∈ Pspace . It is straightfor-ward to adapt this alternating algorithm to our setting.
Algorithm 1 : Alternating model checking for L [Ω] ModelCheck( ψ, ( M , T ) , s )Input: a formula ψ ∈ L [Ω]( τ, V ) in negation normal form where ( τ, V ) is finite, anda finite pointed ( τ, V ) dependence model ( M , T ) , s . if ψ is a literal thenif ( M , T ) , s | = ψ then Accept else
Reject if ψ = η ∧ ϑ thenuniversally choose ϕ ∈ { η, ϑ } ModelCheck( ϕ, ( M , T ) , s )if ψ = η ∨ ϑ thenexistentially guess ϕ ∈ { η, ϑ } ModelCheck( ϕ, ( M , T ) , s )if ψ = D X ϕ thenuniversally choose t ∈ T with t = X s ModelCheck( ϕ, ( M , T ) , t )if ψ = E X ϕ thenexistentially guess t ∈ T with t = X s ModelCheck( ϕ, ( M , T ) , t ) It implements the usual model checking game between an existential and universal player,played on positions ( ϕ, t ) where ϕ is some subformula of ψ and t ∈ T an assignment. Thealgorithm Accept s if and only if the existential player has a winning strategy for this game,which is the case if and only if ( M , T ) , s | = ψ . How to encode the team.
Now we come back to the discussion on how to encode theteam. A first idea might be to simply list all assignments of the team in the input. Call thisvariant MC list ( L [Ω]). In many cases, this encoding is rather inefficient; given M , ψ , a tuple ¯ v enumerating V and a current assignment s ∈ T , encoding T as a list of all its assignmentsmay cause an exponentially longer input, as in the case of full models where T = M V . As aconsequence, one obtains a deceptively low complexity for MC list ( L [Ω]). (cid:73) Proposition 31.
The alternating algorithm can be implemented to decide instances ( ψ, M , ¯ v, ¯ s, T ) of MC list ( L [Ω]) with alternating workspace O (log | ψ | + log | T | ) . In particular,we obtain that MC list ( L [Ω]) ∈ Alogspace = Ptime . Proof.
We implement picking (existentially guessing or universally choosing) assignments t ∈ T by picking a pointer to some tuple in the list representing T . Such a pointer onlyrequires log | T | space, which allows us to decide whether ( M , T ) , s | = β with alternatingworkspace O (log | T | ), for any β ∈ Ω. This is clear for β ∈ { = , = } . For β = D X y , weuniversally choose t ∈ T with t = X s and Accept if and only if t = y s . If β = (¯ x ∈ ¯ y ), thenwe existentially guess t ∈ T and Accept if and only if s (¯ x ) = t (¯ y ). Given β = Ind ¯ x ¯ y , weuniversally choose u ∈ T and then existentially guess t ∈ T , accepting only in the case that s (¯ x ) = t (¯ x ) and u (¯ y ) = t (¯ y ). The atoms Υ , / ∈ , ¬ Ind are handled dually to their counterparts.A pointer of length log | ψ | suffices to specify the current subformula ϕ of ψ . Since ψ, M and T are never modified, the algorithm only needs to keep track of at most 3 assignmentsand 3 subformulae at any time. Together with the above analysis, this proves the claim. (cid:74) Comparing the above to our result that MC full ( L [Ω]) is Pspace -hard, we seem to havea contradiction to the common belief that
Ptime = Pspace . This is however not thecase, as the two problems differ on the length of their inputs. Indeed, there cannot exist apolynomial-time reduction from MC full ( L [Ω]) to MC list ( L [Ω]), because the size of the fullteam M V is exponential in the size | ψ | + | M | + |V| of the input of MC full ( L [Ω]).Nevertheless, this disparity shows that the approach of encoding T as a list may yieldunsatisfactory results. A different approach is to encode the team as a first-order formula ϕ T over the vocabulary τ ∪ M where all elements of M are added as constants interpretedby themselves in the corresponding expansion M M of M . We want that for all ¯ t ∈ M |V| : M M | = ϕ T (¯ t ) ⇐⇒ there is some t ∈ T with t (¯ v ) = ¯ t . (cid:73) Definition 32.
We encode the team by a first-order formula as described above. Morespecifically, define the problem MC formula ( L [Ω]) as follows. The inputs are tuples ( ψ, M , ¯ v, ¯ s, ϕ T ) where a. ( ψ, M , ¯ v, ¯ s ) have the same interpretation as before, with ψ ∈ L [Ω]( τ, V ), M a τ -structurewith universe M , ¯ v an enumeration of V and ¯ s ∈ M |V| encoding a current assignment. b. ϕ T ∈ FO ( τ ∪ M ) encodes the team T ⊆ M V as described above, with M M | = ϕ T (¯ s ). The task is to decide whether ( M , T ) , s | = ψ . The complexity is measured in the input size, so essentially with respect to | ψ | + | M | + |V| + | ϕ T | . . Grädel and P. Pützstück 23 (cid:73) Proposition 33.
We can implement the alternating algorithm so that a given instance ( ψ, M , ¯ v, ¯ s, ϕ T ) for MC formula ( L [Ω]) is solved requiring only alternating space O (log | ψ | + log | ϕ T | + ( |V| + r ) log | M | ) , where r is the maximal numberof free variables in any subformula of ϕ T , and alternating time O ( | ψ | · ( | ψ | + | M | + ( |V| + | ϕ T | ) log | M | )) .In particular, we obtain that MC formula ( L [Ω]) ∈ Aptime = Pspace . Proof.
Assignments t are encoded by their values ¯ t = t (¯ v ), thus taking O ( A ) := O ( |V| log | M | )space. We implement picking assignments t ∈ T by picking a tuple ¯ t ∈ M |V| and thenperforming first-order model checking on ( ϕ T , M M , ¯ t ). We know that | M M | is polynomial in | M | and hence log | M M | ∈ O (log | M | ). It follows from the complexity of first-order modelchecking listed in the last paragraph that picking an assignment in this way requires alternatingspace O ( P s ) := O ( A + log | ϕ T | + r log | M | ) and alternating time O ( P t ) := O ( A + | ϕ T | log | M | ).The space-analysis is parallel to the proof of Proposition 31. We store a subformula ϕ of ψ as a pointer of length log | ψ | . Then ψ, M and ϕ T are never modified, and at any time weneed at most 3 assignments and 3 subformulae in the workspace. This yields the claimedalternating space-complexity O ( P s + log | ψ | ), since we can always reuse the space requiredfor the first-order model checking.For the time analysis, note that checking something such as t (¯ x ) = s (¯ y ) for two assignments s, t in the workspace is possible within alternating time O ( A ), the size of the assignments. If ϕ is a relational literal we can evaluate whether ( M , T ) , s | = ϕ in alternating time O ( | ψ | + | M | + A ). For other local atoms β ∈ Ω or dependence quantifiers, the algorithm picks at most twonew assignments and then does a constant number of checks of the form s (¯ x ) = t (¯ y ). Inthe case of dependence quantifiers, it also updates the current subformula from D X ϕ or E X ϕ to ϕ . This can be accomplished in alternating time O ( P t + | ψ | ). Choosing some subformula at conjunctions and disjunctions takes only O ( | ψ | ) time, sincewe just have to move our pointer within ψ .Clearly we have at most | ψ | recursive calls, which yields the claimed alternating time-complexity O ( | ψ | ( P t + | ψ | + | M | )). (cid:74) We now want to show an analogue of MC( FO k ) ∈ Alogspace = Ptime , so we defineMC k formula ( L [Ω]) as the restriction of MC formula ( L [Ω]) to instances where |V| ≤ k . We nowwant the alternating space-complexity given in Proposition 33 to be logarithmic in theinput. Problematic is the occurrence of r , which describes the maximum number of freevariables in any subformula of ϕ T , and originates from the space-complexity of the first-ordermodel checking we perform when picking new assignments. Currently, we allow arbitrary ϕ T ∈ FO ( τ ∪ M ) to represent the team T in the input. To obtain our wanted analogue, weneed to bound r by some constant for all instances of MC k formula ( L [Ω]).Every team T ⊆ M V can be encoded by an FO ( τ ∪ M )-formula ϕ T that uses only thevariables in V . Indeed, we can just set ϕ T (¯ x ) = W t ∈ T ¯ x = t (¯ v ). This shows that it is verylenient to assume that there exists some global bound for r in all instances of MC k formula ( L [Ω]). (cid:73) Definition 34.
For B ≥ k ∈ N define MC B,k formula ( L [Ω]) as the restriction of MC formula ( L [Ω])to instances ( ψ, M , ¯ v, ¯ s, ϕ T ) where: |V| ≤ k , and every subformula of ϕ T has at most B free variables. (cid:73) Corollary 35. MC B,k formula ( L [Ω]) ∈ Alogspace = Ptime for all B ≥ k ∈ N . Proof.
From Proposition 33 and the above definition of MC
B,k formula ( L [Ω]) we see that we cansolve instances ( ψ, M , ¯ v, ¯ s, ϕ T ) of MC B,k formula ( L [Ω]) with the algorithm from Proposition 33requiring only alternating space O (log | ψ | + log | M | + log | ϕ T | ). (cid:74)(cid:73) Proposition 36.1. MC formula ( L [Ω]) is Pspace -complete. MC B,k formula ( L [Ω]) is Ptime -complete for all B ≥ k ≥ . Proof.
Since the full team is specified by ϕ M V = True, we obtain the following logspace-computable reductionMC full ( L [Ω]) → MC formula ( L [Ω]) , ( ψ, M , ¯ v, ¯ s ) ( ψ, M , ¯ v, ¯ s, True) , which shows that MC full ( L [Ω]) ≤ log MC formula ( L [Ω]). Via the same reduction we can showthat MC k full ( L [Ω]) ≤ log MC B,k formula ( L [Ω]) for all B ≥ k ∈ N . Hence the hardness-results followfrom Proposition 30. The rest was already discussed in Proposition 33 and Corollary 35. (cid:74) This shows that if the local atoms are efficiently checkable, the complexity of modelchecking L [Ω] depends mostly on how one encodes the team. Encoding the team as alist, as one would do for ordinary relations, we obtain Ptime -completeness for both thefinite-variable and the unconstrained variant. Encoding the team as a first-order formulais more efficient, and yields essentially the same complexity as that of first-order modelchecking;
Pspace -complete in general, but
Ptime -complete in restriction to k ≥ References H. Andréka, J. van Benthem, and I. Németi. Modal languages and bounded fragments ofpredicate logic.
Journal of Philosophical Logic , 27:217–274, 1998. J. Balcázar, J. Díaz, and J. Gabarró.
Structural Complexity II . Springer, 1990. A. Baltag and J. van Benthem. A simple logic of functional dependence.
Journal of PhilosopicalLogic , to appear, 2021. A preliminary version appeared in the ILLC pre-publication seriesPP-2020-06. D. Berwanger and E. Grädel. Games and model checking for guarded logics. In
Logic forProgramming, Artificial Intelligence, and Reasoning , pages 70–84. Springer, 2001. P. Blackburn, M. de Rijke, and Y. Venema.
Modal Logic . Cambridge University Press, 2001. E. Börger, E. Grädel, and Y. Gurevich.
The Classical Decision Problem . Springer, 1997. A. Dawar and M. Otto. Modal characterisation theorems over special classes of frames.
Ann.Pure Appl. Logic , 161:1–42, 10 2009. P. Galliani. Inclusion and exclusion dependencies in team semantics: On some logics ofimperfect information.
Ann. Pure Appl. Log. , 163:68–84, 2012. P. Galliani and L. Hella. Inclusion logic and fixed-point logic. In
Computer Science Logic2013 , volume 23, pages 281–295, 2013. E. Grädel and J. Väänänen. Dependence and independence.
Studia Logica , 101(2):399–410,2013. E. Grädel et al.
Finite Model Theory and Its Applications . Springer-Verlag, 2007. E. Grädel. Decision procedures for guarded logics. In
Automated Deduction — CADE-16 ,pages 31–51. Springer, 1999. E. Grädel and M. Otto. The freedoms of (guarded) bisimulation. In
Johan van Benthem onLogic and Information Dynamics , pages 3–31. Springer, 2014. W. Hodges. Compositional semantics for a language of imperfect information.
Log. J. IGPL ,5:539–563, 1997. W. Hodges.
A Shorter Model Theory . Cambridge University Press, 1997. J. Kontinen and J. Väänänen. On definability in dependence logic.
Journal of Logic, Language,and Information , 18:317–241, 2009. C. Papadimitriou.
Computational Complexity . Addison-Wesley, 1994. J. Väänänen.
Dependence logic: A new approach to independence friendly logic . CambridgeUniversity Press, 2007. J. van Benthem.
Modal Correspondence Theory . PhD thesis, University of Amsterdam, 1976. J. van Benthem.