D. C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12(3):291--302, Nov. 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....U (E) For most 0 order calculi it turns out that such an entailment can hold only if one of the disjuncts on the r.h.s. is itself entailed by the equations on the l.h.s. The property of a theory logic where disjunctions are only entailed if at least one disjunct is entailed is known as convexity [18]. If this convexity is established then one can check the entailment by checking whether any of the entailment constraints is individually entailed by the model constraints; and if the semantics is faithful to the logic L 0 then the correspondence CE enables this entailment to be determined by ....

D.C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, (12):291--302, 1980.

....problems of S and of T are decidable, then the quantifier free satisfiability problem of S T is also decidable. 6 The theory of equality It is known that the theory of equality (over an arbitrary signature) is stably infinite and has a decidable quantifier free satisfiability problem [10]. We show here that it is also shiny. We will use the following basic lemma of model theory [5] Lemma 6.1 Let A, B be two interpretations such that there is an embedding B, and let # be a quantifier free formula. Then # is satisfied by if and only if it is satisfied by B. Proposition ....

Oppen, D. C., Complexity, convexity and combination of theories, Theoretical Computer Science 12 (1980), pp. 291--302.

....a specialization of the one presented here. One essential di#erence is that the signatures of the two background theories share at most the equality and the constant symbols, whereas in our case they must share all function symbols. Another is that the two theories are stably infinite (see, e.g. [6]) which is not required in our case. The net e#ect of these di#erences, leading to stronger decidability results, is that for each key set # it is enough to consider only the finitely many residues of the form ## where # is the empty substitution and # a disjunction of equations between the ....

D. C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:245--257, 1980.

....In Nelson and Oppen s method, such communication is achieved by propagating from one procedure to the other any implied equalities between the variables of #. The correctness of this approach is ensured by Theorem 1 below [TH96b] provided that the component theories are stably infinite [Opp80] Definition 2 A consistent, universal theory of signature # is cal led stably infinite if every quantifier free # formula satisfiable in is satisfiable in an infinite model of . Definition 3 If P is any partition on a set of variables V , and R is the corresponding equivalence ....

Derek C. Oppen. Complexity, convexity, and combinations of theories. Theoretical Computer Science, 12, 1980.

....resolution [B 94] It is also enough in [KZ90] because in the used tableau calculus # formulas are expanded into their ground instances which makes the calculus very ine#cient though. Another major di#erence with our approach is that the two background theories are stably infinite (see, e.g. Opp80] and share at most the equality and the constant symbols, whereas in our case the theories, although possibly not stably infinite, share all function symbols. The net e#ect of these di#erences leading to more restricted but stronger computational results than ours is that for each key set # ....

Derek C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:291--302, 1980.

....a specialization of the one presented here. One essential di erence is that the signatures of the two background theories share at most the equality and the constant symbols, whereas in our case they must share all function symbols. Another is that the two theories are stably in nite (see, e.g. [6]) which is not required in our case. The net e ect of these di erences, leading to stronger decidability results, is that for each key set it is enough to consider only the nitely many residues of the form h ; i where is the empty substitution and a disjunction of equations between the ....

D. C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:245-257, 1980.

....94] It is also enough in [KZ90] because in the used tableau calculus formulas are expanded into their ground instances which makes the calculus very inecient though. Another major di erence with our approach is that the signatures of the two background theories are stably in nite (see, e.g. Opp80] and share at most the equality and the constant symbols, whereas in our case the theories, although possibly not stably in nite, share all function symbols. The net e ect of these di erences leading to more restricted but stronger computational results is that for each key set it is enough to ....

Derek C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:291-302, 1980.

....the solvers. Since there is a great variety of constraint systems and of approaches for combining them, we will start with a short classi cation of the di erent approaches. Subsequently, we will describe two of the most prominent combination approaches in this area: the Nelson Oppen scheme [49, 54] for combining decision procedures for the validity of quanti er free formulae in rst order theories, which was originally motivated by program veri cation; methods for combining E uni cation algorithms [60, 15, 6] and more general constraint solvers [7] which are of interest in theorem ....

....A (a theory T ) the positive theory of A (T ) consists of the set of all positive sentences valid in A (T ) The positive existential, positive AE, universal, and conjunctive universal theories of A (T ) are de ned analogously. Quanti er free formulae The Nelson Oppen combination method [49, 54] applies to constraint systems of the following form: For a given signature , the constraint language consists of all quanti er free formulae, i.e. all matrices. The semantics is de ned by an arbitrary theory T . One is interested in satis ability in some model of T . Thus, the ....

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D.C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:291-302, 1980.

.... that the system G corresponding to PA has also the Lob rule G 2A A 2 We use here the name G, as in Fitting [3] and Goldblatt [6] to denote the modal system obtained by adding to K the Lob s axiom 2(2p p) 2p, this system is also denoted KW by Bull and Segerberg in [1] and by Segerberg in [18], while it is denoted PRL by Smorynski in [21] 4. REITER S DEFAULT LOGIC AND THE PROVABILITY OPERATOR 15 implies G A. As already pointed out this rule gives an inconsistency with the further hypothesis of 3 : more generally the schema of the unprovability as failure to prove yields ....

....U of worlds. By the way, also the logic S4f has a class of models that is closed with respect to insertion: indeed S4f is complete with respect to the class of frames for which the frame hW; R; V i has a unique last cluster V , and such that the frame restricted to the set W V is also a cluster [12, 18]. See Fig. 1) Proposition 4.1 (Insertion Lemma) KD4Z and S4f have a class of frames closed with respect to the insertion of arbitrary sets U of worlds whereas KD4LZ has a class of frames closed with respect to insertion of finite sets U . In the sequel I denotes the set 2W [f2ff 23fi 2fl : ....

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D.C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, (12):291--302, 1980. 8. FURTHER WORK 45

....for constraint programming and automated deduction. The modular construction of decision procedures has been considered for the rst time by Shostak [17, 18] in order to solve heterogeneous formulae involving arithmetic and additional function symbols. Approximately in the same time, Nelson Oppen [12, 13, 11] proposed an algorithm dedicated to the union of theories axiomatizing reals, arrays, lists and additional function symbols. The aim was to build a validity checker for programming languages in which such formulae frequently appear. Formally, the combined decision problem can be stated as ....

....quantier free Sigma 1 [ Sigma 2 formulae is decidable w.r.t. Th 1 [ Th 2 if Gamma Sigma 1 Sigma 2 = Gamma satisability of quantier free Sigma i formulae is decidable w.r.t. Th i , Gamma Th i is a stably innite theory (for i = 1; 2) We thus get the result due to Nelson Oppen [13, 11] with a dioeerent proof. It is important to note that even for the disjoint case, we have (like them) additional assumptions concerning the individual theories. This result holds however for some non universal theories since we can also consider arbitrary theories Th i , provided that each ....

D. C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:291302, 1980.

....constraint programming and automated deduction. The modular construction of decision procedures has been considered for the first time by Shostak [17, 18] in order to solve heterogeneous formulae involving arithmetic and additional function symbols. Approximately in the same time, Nelson Oppen [12, 13, 11] proposed an algorithm dedicated to the union of theories axiomatizing reals, arrays, lists and additional function symbols. The aim was to build a validity checker for programming languages in which such formulae frequently appear. Formally, the combined decision problem can be stated as ....

....of quantifier free Sigma 1 [ Sigma 2 formulae is decidable w.r.t. Th 1 [ Th 2 if ffl Sigma 1 Sigma 2 = ffl satisfiability of quantifier free Sigma i formulae is decidable w.r.t. Th i , ffl Th i is a stably infinite theory (for i = 1; 2) We thus get the result due to Nelson Oppen [13, 11] with a different proof. It is important to note that even for the disjoint case, we have (like them) additional assumptions concerning the individual theories. This result holds however for some non universal theories since we can also consider arbitrary theories Th i , provided each ....

D. C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:291--302, 1980.

....a specialization of the one presented here. One essential di#erence is that the signatures of the two background theories share at most the equality and the constant symbols, whereas in our case they must share all function symbols. Another is that the two theories are stably infinite (see, e.g. [6]) which is not required in our case. The net e#ect of these di#erences, leading to stronger decidability results, is that for each key set # it is enough to consider only the finitely many residues of the form ##, ## where # is the empty substitution and # a disjunction of equations between the ....

D. C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:245--257, 1980.

No context found.

D. Oppen. Complexity, convexity and combination of theories. Theoretical Computer Science, 12:291-302, 1980.

No context found.

Derek C. Oppen. Complexity, convexity and combination of theories. Theoretical Computer Science, 12:291--302, 1980.

No context found.

Derek C. Oppen. Complexity, convexity and combination of theories. Theoretical Computer Science, 12:291--302, 1980.

....(see Example 2.3 below) One must assume that each Fi is stably infinite, that is, such that a quantifier free formula qoi is satisfiable in Fi iff it is satisfiable in an infinite model of Fi. This restriction was not mentioned in Nelson and Oppen s original article [7] it was introduced in [10]. Two new and simple proofs of soundness and completeness of the procedure are given in [11, 15] In essence, both depend on the following proposition (see [15] for a proof) For a finite set of variables X, let A(X) denote the conjunction of all disequations x y for x, y C X, x y. Proposition ....

Derek C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12, 1980.

....to assessing the time and space complexity of a combination procedure based on the above method because of the case reasoning required with non convex theories. Computational complexities issues related to the implementation of the Nelson Oppen method have been extensively investigated in [18] and we refer the reader to that work. We will ignore those issues here by considering a nondeterministic combination procedure that we have adapted from those in [18] itself and [2] and which applies to convex as well as non convex theories. Essentially, instead of propagating variable equalities ....

....Computational complexities issues related to the implementation of the Nelson Oppen method have been extensively investigated in [18] and we refer the reader to that work. We will ignore those issues here by considering a nondeterministic combination procedure that we have adapted from those in [18] itself and [2] and which applies to convex as well as non convex theories. Essentially, instead of propagating variable equalities back and forth and having to reason by cases when splits are implied, the non deterministic procedure guesses in advance all the equalities that hold between the ....

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Derek C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12, 1980.

....sound (see Example 2. 2 below) One must assume that each i is stably infinite, that is, such that a quantifier free formula i is satisfiable in i iff it is satisfiable in an infinite model of i This restriction was not mentioned in Nelson and Oppen s original article [8] it was introduced in [11]. Two new and simple proofs of soundness and completeness of the procedure are given in [12, 16] In essence, both depend on the following proposition (see [16] for a proof) For a finite set of variables X, let A(X) denote the conjunction of all disequations y for z,y X,zy. Proposition 2.1 Let 1 ....

D.C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:291-302, 1980.

....combination results apply: the theory of equality, the theory of partial orders, and the theory of total orders. 6. 1 The theory of equality It is well known that the theory of equality (over an arbitrary signature) is stably infinite and has a decidable quantifier free satisfiability problem [Opp80] 11 We show here that the theory of equality is also shiny. To do that we will use the following basic lemma of model theory, adapted from page 44 of Hodges s book [Hod97] Lemma 14. Let A, B be two interpretations such that there is an embedding of B, and let # be a quantifier free ....

Derek C. Oppen. Complexity, convexity and combination of theories. Theoretical Computer Science, 12:291--302, 1980.

....P i that decides the satis ability of quanti er free formulae in a universal theory T i , their method yields a procedure that decides the satis ability of quanti er free formulae in the theory T 1 [ T n . A declarative and non deterministic view of the procedure was suggested by Oppen in [Opp80] In [TH96] C. Tinelli (the rst of us) and M. Harandi followed up on this suggestion describing a non deterministic version of the Nelson Oppen combination procedure and providing a simpler correctness proof. A similar approach had also been followed by C. Ringeissen (the second of us) in ....

....If S1 = and S2 = then t is pure and so it has no aliens subterms. 32 6.1 Disjoint Signatures A sucient, and local, condition for the N O combinability of two signature disjoint theories over the language of quanti er free formulae has been known for quite some time. It was introduced in [Opp80] to justify the correctness of the Nelson Oppen combination method. There, each theory T i is required to be stably in nite, that is, universal and such that every quanti er free formula satis able in T i is satis able in an in nite model of T i . In the following, we show that the notion of ....

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Derek C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12, 1980.

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D. C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12(3):291--302, Nov. 1980.

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Oppen D., Complexity, Convexity and Combination of Theories, Theoretical Computer Science, 12, pp. 291-302 (1980).

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Oppen D., Complexity, Convexity and Combination of Theories, Theoretical Computer Science, 12, pp.291-302, (1980a).

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D. C. Oppen. Complexity, convexity and combinations of theories. Theoretical Computer Science, 12:291-302, 1980.

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Derek C. Oppen. Complexity, convexity and combination of theories. Theoretical Computer Science, 12:291--302, 1980.

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