Hilbert's Tenth problem and NP-completeness of Boolean Syllogistic with unordered cartesian product
aa r X i v : . [ m a t h . L O ] J a n Hilbert’s Tenth problem and NP-completeness ofBoolean Syllogistic with unordered cartesianproduct ( BS ⊗ ) Domenico Cantone and Pietro Ursino
Dipartimento di Matematica e Informatica, Universit`a di CataniaViale Andrea Doria 6, I-95125 Catania, Italy.
E-mail: [email protected],[email protected]
January 5, 2021
Abstract
We relate the decidability problem for BS ⊗ with Hilbert’s Tenth prob-lem and prove that BS ⊗ is NP-complete. Introduction
The well-celebrated Hilbert’s Tenth problem (HTP, for short; see [Hilbert-02]),posed by David Hilbert at the beginning of last century, asks for a uniformprocedure that can determine in a finite number of steps whether any givenDiophantine polynomial equation with integral coefficients is solvable in in-tegers.In 1970, it was shown that no algorithmic procedure exists for the solv-ability problem of generic polynomial Diophantine equations, as was provedby the combined efforts of M. Davis, H. Putnam, J. Robinson, and Y.Matiyasevich (DPRM theorem, see [Rob, DPR61, Mat70]).In the early eighties, Martin Davis asked whether the decision problemsfor the theories
MLS × and MLS ⊗ can be reducible to HTP.By considering MLS × (resp., MLS ⊗ ) as set-theoretic counterpart of HTP,disjoint sets union and Cartesian product (unordered Cartesian product of MLS × is the acronym for MultiLevel Syllogistic with Cartesian product ( MLS ⊗ withunordered cartesian product), MLS , is the quantifier-free fragment of set theory involvingthe Boolean set operators and the equality and membership predicates; see [FOS80].
MLS ⊗ ) play in some sense the roles of integeraddition and multiplication, respectively, since | s ∪ t | = | s | + | t | , for anydisjoint sets s and t , and | s × t | = | s | · | t | , for any sets s and t (whereas | s ⊗ t | = | s | · | t | , for any disjoint sets s and t )(we will show in a future articlethat MLS ⊗ is decidable).This connection has been fully established by Cantone, Cutello and Poli-criti in [CCP90].Indeed, when MLS ⊗ and MLS × are extended with the two-place predi-cate | · | | · | for cardinality comparison, where | s | | t | holds if and onlyif the cardinality of s does not exceed that of t , their satisfiability problemsbecome undecidable, since Hilbert’s Tenth problem would be reducible tothem.We denote by BS ⊗ , the language MLS ⊗ without the use of membershipoperator.In the above cited reduction to HTP, membership operator plays no role[CU18], then, by extending BS ⊗ with the two-place predicate | · | | · | forcardinality comparison, you get again a problem reducible to HTP.Therefore the real set-theoretic counterpart of HTP is actually BS ⊗ .Moreover this language can force a model to be infinite, hence there isno way to prove small model property (Definition 1).Nevertheless, we prove in [CU14] that even theories which force a modelto be infinite can be proved to be decidable by using the technique of for-mative processes [CU18] and the small witness-model property (Definition2), which is a way to finitely represent the infinity.Observe that Boolean Syllogistic ( BS ) is NP-complete.In the present paper we prove that BS ⊗ is not only decidable but, ratherunexpectedly, NP-complete (Theorem 5).Actually, the real counterpart of HTP is BS ⊗ fin , the language BS ⊗ re-stricted to finite models, which is proved to enjoy small model property(Corollary 1).The language BS with cardinal inequalities is equivalent to a pure ex-istential presburger arithmetic, which is proved to be NP-complete in [Sca](anyway you can perform a straightforward calculation through our tools,just considering that BS is NP-complete and cardinal inequalities are poly-nomial time verifiable).Combining the NP-completeness of the pure existential presburger arith-metic and the main result of the present article, we can argue that unde-cidability of HTP arises from an interaction between unordered cartesian BS ⊗ is the acronym for Boolean Syllogistic with unordered Cartesian product. PRELIMINARIES BS ⊗ BS ⊗ is the quantifier-free fragment of set theory consisting of the proposi-tional closure of atoms of the following types: x = y ∪ z, x = y ∩ z, x = y \ z, x = y ⊗ z, x ⊆ y where x, y, z stand for set variables or the constant ∅ . BS ⊗ The semantics of BS ⊗ is defined in a very natural way. A set assignment M is any map from a collection V of set variables (called the variablesdomain of M and denoted dom ( M )) into the von Neumann universe V ofall well-founded sets.We recall that V is a cumulative hierarchy constructed in stages by trans-finite recursion over the class On of all ordinals. Specifically, V := S α ∈ On V α where, recursively, V α := S β<α pow( V β ), for every α ∈ On , where pow( · )denotes the powerset operator. Based on such construction, we can readilydefine the rank of any well-founded set s ∈ V , denoted rk s , as the leastordinal α such that s ⊆ V α . The collection of sets of finite rank, hencebelonging to V α for some finite ordinal α , forms the set HF of the hered-itarily finite sets . Plainly, HF = V ω , where ω is the first limit ordinal,namely the smallest non-null ordinal having no immediate predecessor.Given a set assignment M and a collection W ⊆ dom ( M ), we put M W := { M v | v ∈ W } . The set domain of M is defined as the set S M V = S v ∈ V M v . The rank of M is the rank of its set domain, namely, rk ( M ) := rk ( S M V ) (so that, when V is finite, rk ( M ) = max v ∈ V rk ( M v )). A setassignment M is finite , if so is its set domain.For x, y ∈ dom ( M ), we set M ( x ∪ y ) := M x ∪ M y, M ( x ∩ y ) := M x ∩ M y, M ( x \ y ) := M x \ M y,M ( x ⊗ y ) := M x ⊗ M y = (cid:8) { u, u ′ } | u ∈ M x, u ′ ∈ M y (cid:9) . PRELIMINARIES M ( x = y ⋆ z ) = true ←→ M x = M ( y ⋆ z ) , where ⋆ ∈ {∪ , ∩ , \} . Finally, we put recursively M ( ¬ Φ) := ¬ M Φ , M (Φ ∧ Ψ) := M Φ ∧ M Ψ ,M (Φ ∨ ψ ) := M Φ ∨ M Ψ , M (Φ → ψ ) := M Φ → M Ψ , etc.,for all BS ⊗ -formulae Φ and Ψ such that Vars(Φ) , Vars(Ψ) ⊆ dom ( M ).For a given BS ⊗ -formula Φ, a set assignment M defined over Vars(Φ)is said to satisfy Φ if M Φ = true holds, in which case we also write M | = Φ and say that M is a model for Φ. If Φ has a model, we say thatΦ is satisfiable ; otherwise, we say that Φ is unsatisfiable . Two BS ⊗ formulae Φ and Ψ are said to be equisatisfiable if Φ is satisfiable if andonly if so is Ψ, possibly by distinct models.The decision problem or satisfiability problem for BS ⊗ is theproblem of establishing algorithmically whether any given BS ⊗ formula issatisfiable or not by some set assignment.By applying disjoint normal form and the simplification rules illustratedin [CU18], the satisfiability problem for BS ⊗ can be reduced to the satisfi-ability problem for normalized conjunctions of BS ⊗ , namely conjunc-tions of BS ⊗ -atoms of the following restricted types: x = y ∪ z, x = y \ z, x = y ⊗ z, (1)where x, y, z stand for set variables. BS ⊗ ⊆ Strictly related to the satisfiability problem for BS ⊗ -normalized conjunc-tions is the satisfiability problem for the fragment BS ⊗ ⊆ consisting of theconjunctions of atoms of the following types: x = y ∪ z, x = y \ z, x ⊆ y ⊗ z. (2)Plainly, BS ⊗ ⊆ -conjunctions can be expressed in the theory BS ⊗ , so thedecidability of the satisfiability problem for BS ⊗ ⊆ will follow from that of BS ⊗ . However, whereas any satisfiable BS ⊗ ⊆ -conjunction always admits afinite model (in fact, a model of finite bounded rank), the same is not true PRELIMINARIES BS ⊗ -conjunctions. Consider for instance the BS ⊗ -conjunctionΦ ∞ := x = ∅ ∧ z = x ⊗ x ∧ z ⊆ x. Putting M ∗ x := HF and M ∗ z := HF ⊗ HF (so, M ∗ z is the collection of allnonempty hereditarily finite sets with at most two members), it is an easymatter to check that the set assignment M ∗ satisfies Φ ∞ . In addition, if a setassignment M satisfies Φ ∞ , then for every s ∈ M x we have { s } ∈ M z ⊆ M x ,and therefore { s } ∈ M x . For any set a , define the n -iterated singleton { a } n by putting recursively ( { a } = 0 { a } n +1 = (cid:8) { a } n (cid:9) , for n ∈ N . Thus, letting s be any member of M x (which exists since
M x = ∅ ), it followsthat M x contains as a subset the infinite set (cid:8) { s } n | n ∈ N (cid:9) , proving that M x is infinite, and in turn showing that the conjunction Φ ∞ is satisfied onlyby infinite models. Definition 1.
We say that a given quantifier-free subtheory T of set the-ory has the small model property if there exists a computable function c : N → N such that, for any satisfiable T -formula Ψ there is a set assign-ment M of rank at most c ( | Ψ | ) that is a model for Ψ . We shall prove that the theory BS ⊗ ⊆ enjoys small model property.This definition could seem useless in case of languages have not the smallmodel property. This is the case of the theory BS ⊗ . Indeed, there are sat-isfiable BS ⊗ -conjunctions admitting only models of infinite rank, thereforethe finite partition which imitates the original one cannot satisfy in any casea BS ⊗ -formula. However, we have showed in other works that even if thepartition does not generate any model for the given formula still the notionof imitation (see [CU18] and Section 1.5) makes sense. Indeed, it witnessesthe existence of a model. Definition 2.
We say that a given quantifier-free subtheory T of set the-ory has the small witness-model property if there exists a computablefunction c : N → N such that, for any satisfiable T -formula Ψ there is a setassignment M of rank at most c ( | Ψ | ) that certifies the existence of a modelfor Ψ PRELIMINARIES T implies thedecidability of the satisfiability problem for T .More specifically, for any given satisfiable BS ⊗ -conjunction Φ, any wit-ness model M for Φ will be a small model of the related relaxed con-junction ˘Φ obtained from Φ by replacing each of its literals x = y ⊗ z bythe literal x ⊆ y ⊗ z .A rather common strategy to solve decidability problems consists infinding for any model of the formula a finite bounded assignment whicheither satisfies the formula or witnesses the existence of a model for theformula. To this purpose, it becomes extremely useful finding finite boundedpartitions which imitates the original model.We shall review next the notions of satisfiability by partitions , partitionsimulations , ⊗ graphs (special graphs superimposed to the Venn partitionof a given model), together with some of their properties (see [CU18]). A partition is a collection of pairwise disjoint non-null sets, called the blocks of the partition. The union S Σ of a partition Σ is the domain ofΣ.
Let V be a finite collection of set variables and Σ a partition. Also, let I : V → pow(Σ). The map I induces in a very natural way a set assignment M I over V by putting M I v := S I ( v ) , for v ∈ V .We refer to the triple (Σ , V, I ) as a partition assignment . Definition 3.
Given a map I : V → pow(Σ) over a finite collection V of setvariables, with Σ a partition, for any BS ⊗ -formula Φ such that Vars(Φ) ⊆ V , the partition Σ satisfies Φ via the map I (or, equivalently, the partition assignment (Σ , V, I ) satisfies Φ), and we write Σ / I | = Φ, ifthe set assignment M I induced by I satisfies Φ. We say that Σ satisfies Φ, and write Σ | = Φ, if Σ satisfies Φ via some map I : V → pow(Σ).Thus, if an BS ⊗ -formula Φ is satisfied by some partition, then it issatisfied by some set assignment. The converse holds too. Indeed, let us PRELIMINARIES M | = Φ, for some set assignment M over a given collection V of set variables such that Vars(Φ) ⊆ V . Let Σ M be the Venn partition induced by M , namelyΣ M := n \ M V ′ \ [ M ( V \ V ′ ) | ∅ 6 = V ′ ⊆ V o \ (cid:8) ∅ (cid:9) . Thus, for any σ ∈ Σ M and v ∈ V , either σ ∩ M v = ∅ or σ ⊆ M v . Let I M : V → pow(Σ M ) be the map defined by putting I M ( v ) := { σ ∈ Σ M | σ ⊆ M v } , for v ∈ V .It is an easy matter to check that the set assignment induced by I M is just M . Thus Σ M / I M | = Φ, and therefore Σ M | = Φ, proving that Φ is satisfiedby some partition, in fact by a finite partition.Therefore the notions of satisfiability by set assignments and that ofsatisfiability by partitions coincide. We extensively treated the above argument in [CU18]. Here we provide ashort and, possibly, exhaustive resume of this.We start by observing that satisfiability of Boolean literals of type x = y ∪ z and x = y \ z by the set assignment M I depends solely on I , as shownin the following lemma. Lemma 1.
Let Σ be a partition and let I : V → pow(Σ) be a map over a(finite) set of variables V . Also, let M I be the set assignment induced by I over V . Then M I | = x = y ∪ z ←→ I ( x ) = I ( y ) ∪ I ( z ) M I | = x = y \ z ←→ I ( x ) = I ( y ) \ I ( z ) , for any x, y, z ∈ V .Proof. Let x, y, z ∈ V , and let ⋆ ∈ {∪ , \ } . Since Σ is a partition (andtherefore its blocks are nonempty and mutually disjoint), we have: M I | = x = y ⋆ z ←→ [ I ( x ) = [ I ( y ) ⋆ [ I ( z ) ←→ [ I ( x ) = [ (cid:0) I ( y ) ⋆ I ( z ) (cid:1) ←→ I ( x ) = I ( y ) ⋆ I ( z ) . PRELIMINARIES Remark 1.
By exploiting the fact that s ∩ t = s \ ( s \ t ), for any sets s and t , under the assumptions of Lemma 1 we have M I | = x = y ∪ z ←→ M I | = x = y \ ( y \ z ) ←→ M I ′ | = x = y \ x ′ ∧ x ′ = y \ z, where x ′ / ∈ V and I ′ extends I over V ∪ { x ′ } by letting I ′ ( x ′ ) := I ( y ) \ I ( z ).Hence, M I | = x = y ∩ z ←→ I ′ ( x ) = I ′ ( y ) \ I ′ ( x ′ ) ∧ I ′ ( x ′ ) = I ′ ( y ) \ I ′ ( z ) ←→ I ( x ) = I ( y ) \ ( I ( y ) \ I ( z )) ←→ I ( x ) = I ( y ) ∩ I ( z ) . Let (Σ , V, I ) and ( b Σ , V, b I ) be partition assignments, with Σ and b Σ par-titions of the same size and V a finite set of variables. As noted above,the pairs Σ , I and b Σ , b I induce respectively the set assignments M I and b M b I over V . Towards establishing the small model property for BS ⊗ ⊆ (and thenthe small witness-model property for BS ⊗ ), we prove next some results thatcumulatively will provide sufficient conditions in order that( ⋆ ) any BS ⊗ ⊆ -conjunction (resp., BS ⊗ -conjunction) Φ such that Vars(Φ) ⊆ V is satisfiable by b M b I whenever it is satisfied by M I , i.e., M I | = Φ = ⇒ b M b I | = Φ . Lemma 2.
Let Σ and b Σ be partitions and β : Σ → b Σ a bijection. Let I : V → pow(Σ) be a map over a (finite) set of variables V , and let b I : V → pow( b Σ) be the map induced by β and I by letting b I ( x ) = β [ I ( x )] , for any x ∈ V. Then I ( x ) = I ( y ) ∪ I ( z ) ←→ b I ( x ) = b I ( y ) ∪ b I ( z ) I ( x ) = I ( y ) ∩ I ( z ) ←→ b I ( x ) = b I ( y ) ∩ b I ( z ) I ( x ) = I ( y ) \ I ( z ) ←→ b I ( x ) = b I ( y ) \ b I ( z ) , for any x, y, z ∈ V . PRELIMINARIES Proof.
Since β is a bijection, we have I ( x ) = I ( y ) ⋆ I ( z ) ←→ b I ( x ) = β [ I ( x )]= β [ I ( y ) ⋆ I ( z )]= β [ I ( y )] ⋆ β [ I ( z )]= b I ( y ) ⋆ b I ( z ) , for ⋆ ∈ {∪ , ∩ , \} .From Lemmas 1, 2 and Remark 1, we have at once property ( ⋆ ), butlimited to Boolean set literals of types x = y ⋆ z , with ⋆ ∈ {∪ , ∩ , \} over V , for any two partition assignments (Σ , V, I ) and ( b Σ , V, b I ) related by abijection β : Σ → b Σ.We shall express the conditions that take also care of literals in BS ⊗ ⊆ ofthe form x ⊆ y ⊗ z by means of some useful variants of the power set operator.They are variations of the intersecting power set operator pow ∗ , introducedin [Can91] in connection with the solution of the satisfiability problem for afragment of set theory involving the power set and the singleton operators.Specifically, for any set S , we putpow ∗ ( S ) := n t ⊆ [ S | t ∩ s = ∅ , for every s ∈ S o , pow ∗ , ( S ) := n t ∈ pow ∗ ( S ) | | t | o , pow ∗ > ( S ) := n t ∈ pow ∗ ( S ) | | t | > o . Thus,- pow ∗ ( S ) is the collection of all subsets of S S that have nonemptyintersection with all the members of S ;- pow ∗ , ( S ) is the set of all members of pow ∗ ( S ) of cardinality 1 or 2;- pow ∗ > ( S ) is the set of all members of pow ∗ ( S ) of cardinality strictlygreater than 2.Further properties of pow ∗ are listed in [CU18, pp. 16–20].Some useful properties of the operators pow ∗ , , pow ∗ > , and ⊗ are con-tained in the following lemmas.The pow ∗ , operator is strictly connected with the unordered Cartesianoperator ⊗ , as shown next. PRELIMINARIES Lemma 3.
For all sets s and t (not necessarily distinct), we have pow ∗ , ( { s, t } ) = s ⊗ t. Proof.
Plainly, s ⊗ t ⊆ pow ∗ , ( { s, t } ). Indeed, if u ∈ s ⊗ t , then1 | u | , u ⊆ s ∪ t, and u ∩ s = ∅ 6 = u ∩ t, so that u ∈ pow ∗ , ( { s, t } ).Conversely, let { u, v } ∈ pow ∗ , ( { s, t } ). Then { u, v } ⊆ s ∪ t and { u, v } ∩ s = ∅ 6 = { u, v } ∩ t. Without loss of generality, let us assume that u ∈ s . If v ∈ t , we are done.Otherwise, if v / ∈ t , then v ∈ s (since { u, v } ⊆ s ∪ t ) and u ∈ t (since { u, v } ∩ t = ∅ ). Hence, { u, v } ∈ s ⊗ t , proving that also the inverse inclusionpow ∗ , ( { s, t } ) ⊆ s ⊗ t holds.The following is a simple yet useful property of the unordered Cartesianoperator ⊗ . Lemma 4.
For any sets s , s , t , t , ( s ⊗ s ) ∩ ( t ⊗ t ) = ∅ −→ ( t ∩ s i = ∅ ∧ t ∩ s − i = ∅ ) , for some i ∈ { , } .Proof. Preliminarily, we observe that ( s ⊗ s ) ∩ ( t ⊗ t ) = ∅ plainly impliesthe following inequalities: s ∩ ( t ∪ t ) = ∅ , s ∩ ( t ∪ t ) = ∅ ,t ∩ ( s ∪ s ) = ∅ , t ∩ ( s ∪ s ) = ∅ . Thus, if s i ∩ t j = ∅ for some i, j ∈ { , } , then s i ∩ t − j = ∅ and s − i ∩ t j = ∅ ,and we are done. On the other hand, if s i ∩ t j = ∅ for all i, j ∈ { , } , weare immediately done.Then, in order to get property ( ⋆ ) also for literals of type x ⊆ y ⊗ z , it isenough to require that the bijection β : Σ → b Σ relating two given partitionassignments (Σ , V, I ) and ( b Σ , V, b I ) satisfies the following conditions, for every X ⊆ Σ and σ ∈ Σ:(C ) pow ∗ , ( X ) ∩ σ = ∅ −→ pow ∗ , ( β [ X ]) ∩ β ( σ ) = ∅ , PRELIMINARIES ) pow ∗ > ( X ) ∩ σ = ∅ −→ pow ∗ > ( β [ X ]) ∩ β ( σ ) = ∅ ,and that the partition b Σ is weakly ⊗ -transitive , i.e., for all ⊗ place p , S b p ⊆ b Σ. A place p such that p ⊆ S A ⊆ Σ pow ∗ , ( A ) is an ⊗ place.For a given partition Σ an ⊗ -place q is such that q ⊆ [ B ⊆P pow ∗ , ( B )This defines a labelling ⊗ on the set of P . The collection of all ⊗ placesare denoted by ⊗P . Lemma 5.
Let Σ and b Σ be partitions related by a bijection β : Σ → b Σ suchthat b Σ is weakly ⊗ -transitive and V a set of variables. In addition, let usassume that, for all σ ∈ Σ and X ⊆ Σ , conditions (C ) and (C ) hold.Then, for all X, Y, Z ⊆ Σ and every σ ∈ Σ ,(a) σ ⊆ S A ⊆ Σ pow ∗ , ( A ) −→ β ( σ ) ⊆ S A ⊆ Σ pow ∗ , ( β [ A ]) ,(b) S X ⊆ S Y ⊗ S Z −→ S β [ X ] ⊆ S β [ Y ] ⊗ S β [ Z ] ,(c) Let I : V → pow(Σ) and let b I : V → pow( b Σ) be the map induced by I and β . Also, let M I and b M b I be the set assignments over V induced by (Σ , V, I ) and ( b Σ , V, b I ) respectively, then for all x, y, z ∈ V we have: M I | = x ⊆ y ⊗ z = ⇒ b M b I | = x ⊆ y ⊗ z. Proof.
Concerning (a), let σ ⊆ S A ⊆ Σ pow ∗ , ( A ), so that σ ∩ [ A ⊆ Σ pow ∗ > ( A ) = ∅ . (3)Being σ an ⊗ place, from weak ⊗ -transitive of b Σ and the bijectivity of β , itfollows that β ( σ ) ⊆ b Σ ⊆ pow( b Σ) = [ b A ⊆ b Σ pow ∗ ( b A ) = [ A ⊆ Σ pow ∗ ( β [ A ]) . (4)From (3) and (C ), we have β ( σ ) ∩ [ A ⊆ Σ pow ∗ > ( β [ A ]) = ∅ , PRELIMINARIES β ( σ ) ⊆ [ A ⊆ Σ pow ∗ , ( β [ A ]) , and therefore σ ⊆ [ A ⊆ Σ pow ∗ , ( A ) −→ β ( σ ) ⊆ [ A ⊆ Σ pow ∗ , ( β [ A ]) . Concerning (b), let [ X ⊆ [ Y ⊗ [ Z, (5)for some X, Y, Z ⊆ Σ, and let t ∈ S β [ X ]. Hence, t ∈ β ( σ ), for some σ ∈ X . By (5) and Lemma 3, we have σ ⊆ S A ⊆ Σ pow ∗ , ( A ), so that, by(a), β ( σ ) ⊆ S A ⊆ Σ pow ∗ , ( β [ A ]). Hence, t = { a, b } , for some sets a and b not necessarily distinct. Since b Σ is weakly ⊗ transitive, t ⊆ b Σ. Thus, t ⊆ β ( σ ) ∪ β ( σ ), for some σ , σ ∈ Σ not necessarily distinct such that t ∩ β ( σ ) = ∅ and t ∩ β ( σ ) = ∅ . Plainly, t ∈ pow ∗ , ( { β ( σ ) , β ( σ ) } ), and sopow ∗ , ( { β ( σ ) , β ( σ ) } ) ∩ β ( σ ) = ∅ . Therefore, by (C ), pow ∗ , ( { σ , σ } ) ∩ σ = ∅ and, by Lemma 3, ( σ ⊗ σ ) ∩ σ = ∅ , so that a fortiori ( σ ⊗ σ ) ∩ ( S Y ⊗ S Z ) = ∅ . Thus, by Lemma 4, S Y ∩ σ i = ∅ and S Z ∩ σ − i = ∅ ,for some i ∈ { , } . But then, σ i ∈ Y and σ − i ∈ Z , so that β ( σ i ) ⊆ S β [ Y ]and β ( σ − i ) ⊆ S β [ Z ]. Hence, β ( σ ) ⊗ β ( σ ) ⊆ S β [ Y ] ⊗ S β [ Z ] and so t ∈ S β [ Y ] ⊗ S β [ Z ]. By the arbitrariness of t ∈ S β [ X ], it follows that [ β [ X ] ⊆ [ β [ Y ] ⊗ [ β [ Z ] , and therefore [ X ⊆ [ Y ⊗ [ Z −→ [ β [ X ] ⊆ [ β [ Y ] ⊗ [ β [ Z ]holds. PRELIMINARIES M I | = x ⊆ y ⊗ z = ⇒ M I x ⊆ M I y ⊗ M I z = ⇒ [ I ( x ) ⊆ [ I ( y ) ⊗ [ I ( y )= ⇒ [ β [ I ( x )] ⊆ [ β [ I ( y )] ⊗ [ β [ I ( y )] (by (b))= ⇒ [ b I ( x ) ⊆ [ b I ( y ) ⊗ [ b I ( y )= ⇒ b M b I x ⊆ b M b I y ⊗ b M b I z = ⇒ b M b I | = x ⊆ y ⊗ z. The following definition and theorem summarize the above considera-tions.
Definition 4 ( BS ⊗ -imitation) . A weakly ⊗ -transitive partition b Σ is said to weakly BS ⊗ -imitates another partition Σ, when there exists a bijection β : Σ → b Σ such that, for all X ⊆ Σ and σ ∈ Σ,(C ) pow ∗ , ( X ) ∩ σ = ∅ −→ pow ∗ , ( β [ X ]) ∩ β ( σ ) = ∅ ,(C ) pow ∗ > ( X ) ∩ σ = ∅ −→ pow ∗ > ( β [ X ]) ∩ β ( σ ) = ∅ .Theorem 1 contains sufficient conditions to achieve property ( ⋆ ) above(just before Lemma 2) for BS ⊗ ⊆ -conjunctions. Theorem 1.
Let Σ and b Σ be partitions such that b Σ is weakly ⊗ -transitiveand weakly BS ⊗ -imitates Σ via a bijection β : Σ → b Σ . Also, let I : V → pow(Σ) be any map over a given finite collection V of variables, and let b I be the map over V induced by I and β . Then, for every BS ⊗ ⊆ -conjunction Φ such that Vars(Φ) ⊆ V , we have M I | = Φ = ⇒ b M b I | = Φ , where M I and b M b I are the set assignments over V induced by the partitionassignments (Σ , V, I ) and ( b Σ , V, b I ) , respectively. BS ⊗ -imitation of a partitionDefinition 5 ((Strong) BS ⊗ -imitation) . A weakly ⊗ -transitive b Σ is said to (strong) BS ⊗ -imitates another partition Σ, when it weakly BS ⊗ -imitatesΣ via a bijection β and the following additional ⊗ -saturatedness conditionholds, for every X ⊆ Σ: PRELIMINARIES ) pow ∗ , ( X ) ⊆ S Σ −→ pow ∗ , ( β [ X ]) ⊆ S b
Σ.In general, a node A is said to be saturated if pow ∗ , ( A ) ⊆ S Σ holds.
Theorem 2.
Let Σ and b Σ be partitions such that b Σ is weakly ⊗ -transitiveand BS ⊗ -imitates Σ via a bijection β : Σ → b Σ . Also, let I : V → pow(Σ) beany map over a given finite collection V of variables, and let b I be the mapover V induced by I and β . Then, for every BS ⊗ -conjunction Φ such that Vars(Φ) ⊆ V , we have M I | = Φ = ⇒ b M b I | = Φ , where M I and b M b I are the set assignments over V induced by the partitionassignments (Σ , V, I ) and ( b Σ , V, b I ) , respectively.Proof. In view of Theorem 1, it is enough to prove that for every literal ofthe form y ⊗ z ⊆ x , with x, y, z ∈ V , we have M I | = y ⊗ z ⊆ x = ⇒ b M b I | = y ⊗ z ⊆ x. (6)Thus, let us assume that M I | = y ⊗ z ⊆ x , so that [ I ( y ) ⊗ [ I ( z ) ⊆ [ I ( x ) , (7)and let t ∈ S b I ( y ) ⊗ S b I ( z ) = S β [ I ( y )] ⊗ S β [ I ( z )]. Hence, t ∈ β ( σ ) ⊗ β ( σ ),for some σ ∈ I ( y ) and σ ∈ I ( z ). By (7) and Lemma 3, we havepow ∗ , ( { σ , σ } ) = σ ⊗ σ ⊆ [ I ( y ) ⊗ [ I ( z ) ⊆ [ I ( x ) ⊆ [ Σ . (8)Hence, by condition (7) and Lemma 3 again, we have β ( σ ) ⊗ β ( σ ) = pow ∗ , ( { β ( σ ) , β ( σ ) } ) ⊆ [ b Σ , so that t ∈ S b
Σ.Let γ ∈ Σ be such that t ∈ β ( γ ). Since pow ∗ , ( { β ( σ ) , β ( σ ) } ) ∩ β ( γ ) = ∅ ,from condition (C ) of Definition 4 it follows that pow ∗ , ( { σ , σ } ) ∩ γ = ∅ .Hence, by (8), we have γ ∩ S I ( x ) = ∅ , and therefore γ ∈ I ( x ), so that t ∈ β ( γ ) ⊆ S β [ I ( x )], which in turn implies t ∈ S β [ I ( x )] = b I ( x ).By the arbitrariness of t ∈ S b I ( y ) ⊗ S b I ( z ), it follows that S b I ( y ) ⊗ S b I ( z ) ⊆ S b I ( z ), namely b M b I y ⊗ b M b I z ⊆ b M b I x , and therefore b M b I | = y ⊗ z ⊆ x .This completes the proof of (6), and in turn of the theorem. PRELIMINARIES ⊗ -graphs By relying on the results in Theorem 1, our next task will be to address theproblem of how to generate a transitive partition b Σ of bounded rank thatweakly BS ⊗ -imitates a given finite partition Σ.The basic idea consists in a progressive copying of the structure of thegraph linked to the partition. The basic idea behind the generation of asuitable imitating partition b Σ of a given partition Σ is to single out a rep-resentation of the structure of a partition through a graph structure. Sincethe only operator we take in account is the unordered cartesian product ourgraph structure will do the same.The reason to move from partitions towards graphs lies in a greaterflexibility of this last representation in building a new model.The path to reach a decidability test for languages which involve un-ordered cartesian product operator, requires a construction procedure of Σconveniently modified so as to obtain another transitive b Σ that imitates Σ.We describe in which way to create a graph in order to take into accountunordered cartesian operator. We call such a graph related to a partition as ⊗ -graph.The idea is to associate with every transitive partition Σ a bipartitegraph with two types of vertices, places and nodes .We consider a non-empty finite set P , whose elements are called places (or syntactical Venn regions ) and whose subsets are called nodes . Wewe denote by N the collection of nodes and assume that P ∩ N = ∅ , so thatno node is a place, and vice versa. We shall use these places and nodes as thevertices of a directed bipartite graph G of a special kind, called ⊗ -graph .The edges issuing from each place q are exactly all pairs h q, B i such that q ∈ B ⊆ P : these are called membership edges . The remaining edges of G , called distribution edges , go from nodes to places; hence, G is fullycharacterized by the function T ∈ pow( P ) pow( P ) associating with each node B the set of all places t such that h B, t i is anedge of G . The elements of T ( B ) are the targets of B , and T is the ⊗ target function of G . Thus, we usually represent G by T .Edges B → ⊗ q of a P -graph, where q is a ⊗ -place, will be referred to as ⊗ -edges .When B is a subset of P we denote by G ⇂ B the subgraph restricted to PRELIMINARIES B (and obviously the corresponding nodes). Definition 6 (Compliance with a ⊗ -graph) . Given a ⊗ -graph G , a transitivepartition Σ and a , Σ is said to comply with G (and, symmetrically, G issaid to be induced by Σ) via the map q q ( • ) , where | Σ | = |P| and q q ( • ) belongs to Σ P , if(a) the map q q ( • ) is bijective,(b) the target function T of G satisfies T ( B ) = { q ∈ ⊗P | q ( • ) ∩ pow ∗ , ( B ( • ) ) = ∅} for every B ⊆ P , and(c) for every ⊗ -place q ∈ P , the set q ( • ) may contain only singletons anddoubletons, hence: q ( • ) ⊆ S (cid:8) pow ∗ , ( B ) | q ∈ T ( B ) (cid:9) .A ⊗ -graph is realizable if it is induced by some partition. Definition 7.
Let Σ , b Σ two partitions such that b Σ is weakly ⊗ -transitiveand G , b G the induced ⊗ -graphs. An bijective map q b q naturally extends tothe nodes B b B = { b q | q ∈ B } and obviously to the ⊗ -graphs. We definea map β : G → b G a weak isomorphism between ⊗ -graphs when(C ) v → ⊗ w ↔ β ( v ) → ⊗ β ( w ) ,and we denote this relation in the following way G ≃ − b G and we say thatthe ⊗ -graphs are weakly isomorphic.Moreover, if the following statement(I ) pow ∗ , ( X ) ⊆ S Σ ↔ pow ∗ , ( β [ X ]) ⊆ S b Σ .is fulfilled then we say that the map β : G → b G is an isomorphism between ⊗ -graphs, and we write G ≃ b G . Obviously the following holds
Theorem 3.
Let Σ and b Σ be partitions such that G and b G are (weak) iso-morphic then b Σ and Σ (weakly) BS ⊗ -imitates each other. Intuitively speaking, only elements in pow ∗ , ( B ( • ) ) can flow from node B to a place q along any ⊗ -edge B → ⊗ q (see Definition 6). THE DECIDABILITY OF BS ⊗ BS ⊗ Consider a BS ⊗ ⊆ -conjunction Φ satisfied by a partition Σ with ⊗ -graph G .Assume that the longest path without repetitions through ⊗ -arrows is k and |P| = P . Theorem 4. BS ⊗ ⊆ is decidable.Proof. Using the procedure
BuilderP artitionM LImitate , we create a rankbounded partition that(weakly) BS ⊗ -imitates Σ.Theorem 1 and Assert A imply the small model property for BS ⊗ ⊆ and,by-product, our result.In the prosecution we denote by p a non ⊗ -place, by q an ⊗ -place. Ifthe procedure has passed through a path of length h starting from a place p until a node A we write p h A .We define minrank ( q ) = min { α | t ∈ q, rank ( t ) = α } .Looking the status of Assert A at the end of execution of procedure BuilderP artitionM LImitate , it results b G ≃ − G . Then, by Theorem 3, b Σ(weakly)- BS ⊗ -imitates Σ.We are left to prove inductively the asserts A − A . THE DECIDABILITY OF BS ⊗ procedure BuilderPartitionMLImitate (Σ, ordered sequence of minimal ranks i , . . . , i ℓ ); - We denote by p places not in ⊗P , for each p charge b p with P k elements of rank H , - Label signed node A such that p ∈ A - Pick i j and all signed nodes A not distributed such that for each q ∈ A minrank ( q ) i j - let q
7→ ∇ ( b q ) be a set-valued map over ⊗P such that- (a) {∇ ( b q [ i ] ) | q ∈ ⊗P} \ {∅} is a partition of a non-null subset of- pow ∗ , ( (cid:2) b A (cid:3) ) \ b P ,- (b) |∇ ( b q ) | > P k − j − (using Assert A )- (c) b q = b q ∪ ∇ ( b q ) - Label as distributed node A . - Label as signed all nodes B such that B ∩ T ( A ) = ∅ - Assert A1 : for each q ∈ ⊗P and minrank ( q ) i j +1 , b q = ∅ - Assert A2 : if p j A A signed then (cid:12)(cid:12)(cid:12) pow ∗ , ( b A ) \ b P (cid:12)(cid:12)(cid:12) > P k − j - Assert A3 : b Σ is a partition,-
Assert A4 : The rank of b Σ does not exceed H + j - Assert A5 : b G ⇂ p,q,minrank ( q ) i j ≃ − G ⇂ p,q,minrank ( q ) i j , end procedure ; Table 1: A procedure to create a partition of bounded rank which weakly BS ⊗ -imitates a given partition Σ.[Base Step 0][Proof-Assert A ] At the beginning of procedure { b p } p ∈⊗P is a partition, byconstruction { b p, b q } p ∈⊗P ,minrank ( q )= i is a partition.[Proof-Assert A ] At the beginning of procedure both b G and G ⇂ p ...p r hasno arrows. After the first execution of the procedures all outgoing arrowsstarting from nodes composed only by p type of places have been activated.Since for any place q with minrank ( q ) = i there must be an incomingarrow from nodes composed only by p type of places, A holds.[Proof-Assert A ] A straightforward consequence of A .[Proof-Assert A ] The only case is p A . By construction of p they haveall P k elements of rank H . A node composed only by p type of places haveat least P k pairs, that necessarily have rank H + 1, therefore they haveintersection null with all b p , hence (cid:12)(cid:12)(cid:12) pow ∗ , ( b A ) \ b P (cid:12)(cid:12)(cid:12) > P k and A holds.[Proof-Assert A ] As observed above, the rank of places of type p is H .[Inductive Step j + 1] THE DECIDABILITY OF BS ⊗ A ] It is true by inductive hypothesis and the construction of ∇ .[Proof-Assert A ] We have to prove that for all q such that minrank ( q ) = i j +1 if A = { q , q } , minrank ( q ) , minrank ( q ) i j and A → ⊗ q then b A → ⊗ b q . By construction all these nodes are distributed then they haveactivated all their outgoing ⊗ -arrows.[Proof-Assert A ] If minrank ( q ) = i j +1 there exists a node { q , q } → ⊗ q such that minrank ( q ) , minrank ( q ) < i j +1 . By Assert A this arrow existsin b G , hence b q = ∅ [Proof-Assert A ] By inductive hypothesis and construction for all q ∈ T ( A ),pow ∗ , ( b A ) \ b P ⊇ ∇ ( b q ) and |∇ ( b q ) | > P k − j − . Let B = { q, m } , obviously whenit is called by the procedure p j +1 B . Observe that for a given partitiondifferent nodes generates disjoint sets of pairs. Since {∇ ( b q ) , b m } 6 = b X for anynode X of Σ, this implies pow ∗ , ( ∇ ( b q ) , b m ) ∩ pow ∗ , ( b X ) = ∅ and, in particularfor any place σ , pow ∗ , ( ∇ ( b q ) , b m ) ∩ b σ = ∅ . Therefore (cid:12)(cid:12) pow ∗ , ( b q ∪ ∇ ( b q ) , b m ) (cid:12)(cid:12) > P k − j − [Proof-Assert A ] By inductive hypothesis the new pairs are composed ofelements of rank at most H + j then the rank of all elements in b Σ cannotexceed H + j + 1.Observe that if minrank ( q ) = m there exists a node { q , q } → ⊗ q suchthat minrank ( q ) , minrank ( q ) < minrank ( q ).Moreover, for each ⊗ -place b q , since at the beginning of procedure thesekind of places are empty, S b q ⊆ b Σ. This last fact, together with Theorem 3,Theorem 1 and Assert A , imply our result.The above construction implies the following: Theorem 5. BS ⊗ ⊆ is NP-complete.Proof. Since BS is NP-complete you can verify in polynomial time if anassignment, that makes non-empty all places, is a model or not. On the otherhand, by procedure BuilderP artitionM LImitate , in order to verify thatthis model satisfies ⊗ literals, it is sufficient to check whether each ⊗ -place q is reachable from a non ⊗ -place, which is a polynomial time research.In order to solve decidability problem for BS ⊗ we have to fulfill propertypow ∗ , ( X ) ⊆ [ Σ −→ pow ∗ , ( β [ X ]) ⊆ [ b ΣFor this purpose we introduce the procedure
CartSaturateP artition . Ifthis procedure does not terminate this implies that there is at least one cycle
THE DECIDABILITY OF BS ⊗ procedure CartSaturatePartition (Σ, Stack S of not cart-saturated nodes); - P op (S) = A , - let q
7→ ∇ ( b q ) be a set-valued map over ⊗P such that- (a) {∇ ( b q [ i ] ) | q ∈ ⊗P} \ {∅} is a partition of a non-null subset of- pow ∗ , ( (cid:2) b A (cid:3) ) \ b P ,- (c) b q = b q ∪ ∇ ( b q ) - P ush (B) all not cart-saturated B not in S. - If S is empty exit. end procedure ; Table 2: A procedure to cart-saturate a partition Σ.in G . Otherwise there is not any cycle.In both cases the assignment b Σ = { S i ∈ α b q [ i ] | q ∈ Σ } where b q [ i ] is theplace b q at the step i of the procedure CartSaturateP artition satisfiespow ∗ , ( X ) ⊆ [ Σ −→ pow ∗ , ( β [ X ]) ⊆ [ b ΣTherefore, by Theorem 3 and Theorem 2, the partition b Σ BS ⊗ -imitatesΣ. In case the procedure terminates, α ∈ N therefore there are no cycles,you have a finite construction and a small model. Otherwise, α = ω , the firstinfinite ordinal and you have a transfinite construction and the assignmentbuilt at the end of the procedure BuilderP artitionM LImitate witnessesthe existence of a model. This in particular implies NP-completeness of BS ⊗ .Consider BS ⊗ fin , as BS ⊗ restricted to finite models. Since a finite modelcannot have a cycle, this, in particular, implies BS ⊗ fin has the small modelproperty.Resuming, Corollary 1. BS ⊗ is NP-complete and BS ⊗ fin has the small model prop-erty. REMARK ON HTP Consider a ⊗ graph G with a set of cardinal constraints of the type | p | | q | with p, q vertices of G . The following problem Problem 1. G is realizable? is undecidable, since this problem is reducible to HTP.On the other side, we conjecture the following result. Conjecture 1.
For an algorithm A and an input x there exists a ⊗ graph G with cardinal inequalities such that A terminates on input x iff G is realizable. If the conjecture 1 were proved it would lead to a straightforward reduc-tion from the HALTING problem to HTP and, contextually, to a completelyalternative proof to that provided by Matyasevich.
The authors are grateful to Martin Davis who kindly gave his permission tocite his personal communication.
References [Can91] D. Cantone. Decision procedures for elementary sublanguages of settheory. X. Multilevel syllogistic extended by the singleton and powersetoperators.
Journal of Automated Reasoning , 7(2):193–230, 1991.[CCP90] D. Cantone, V. Cutello, and A. Policriti. Set-theoretic reductionsof Hilbert’s tenth problem. In
Proc. of 3rd Workshop “Computer Sci-ence Logic” 1989 , pages 65–75, 1990. Lecture Notes in Computer Sci-ence, 440.[CU14] Domenico Cantone and Pietro Ursino. Formative processes withapplications to the decision problem in set theory: II. Powerset andsingleton operators, finiteness predicate.
Inf. Comput. , 237:215–242,2014.[CU18] Domenico Cantone and Pietro Ursino. An Introduction to the Tech-nique of Formative Processes in Set Theory.
Springer InternationalPublishing , 2018.
EFERENCES
Annals of Mathematics , 74(2):425–436, 1961.[FOS80] A. Ferro, E.G. Omodeo, and J.T. Schwartz. Decision proceduresfor elementary sublanguages of set theory. I: Multilevel syllogistic andsome extensions.
Comm. Pure Appl. Math. , 33:599–608, 1980.[Mat70] Matiyasevich, Yu. V. (1970). Enumerable sets are Diophantine (inRussian). Dokl. AN SSSR, 191(2), 278–282; Translated in: Soviet Math.Doklady, 11(2), 354–358. Correction Ibid 11 (6) (1970), vi. Reprintedon pp. 269–273 in: Mathematical logic in the 20th century, G. E. Sacks,(Ed.), (2003). Singapore University Press and World Scientific Publish-ing Co., Singapore and River Edge, NJ.[Hilbert-02] Hilbert, D. . Mathematical Problems.
Bulletin of the AmericanMathematical Society , 8(10), 437–479, 1902.[Rob] Robinson, R. M. . Arithmetical representation of recursively enumer-able sets.
Journal of Symbolic Logic , 21(2), 162–186, 1956.[Sca] B. Scarpellini. Complexity of subcases of Presburger Arithmetic.