§2. The Nelson-Oppen Method Purification : an equi-satisfiable set of pure constraints Γ 1 ∪ Γ 2 is produced (this is achieved by Purification Rule, replacing a subterm t by a fresh variables x - the equation x = t is also added to the current set of constraints); Propagation : the T 1 -constraint solving procedure and the T 2 -constraint solving procedure fairly exchange information concerning unsatisfiability of Σ 0 -constraints; A Comprehensive Framework for Combined Decision Procedures – p. 7/4
§2. The Nelson-Oppen Method Purification : an equi-satisfiable set of pure constraints Γ 1 ∪ Γ 2 is produced (this is achieved by Purification Rule, replacing a subterm t by a fresh variables x - the equation x = t is also added to the current set of constraints); Propagation : the T 1 -constraint solving procedure and the T 2 -constraint solving procedure fairly exchange information concerning unsatisfiability of Σ 0 -constraints; Until : an inconsistency is detected or a saturation state is reached. A Comprehensive Framework for Combined Decision Procedures – p. 7/4
§2. The Nelson-Oppen Method To make the above schema more precise, we need to solve some problems: A Comprehensive Framework for Combined Decision Procedures – p. 8/4
§2. The Nelson-Oppen Method To make the above schema more precise, we need to solve some problems: About Purification : here the problem is how to implement efficiently the preprocessing purification step (we won’t discuss this matter, see (Baader-Tinelli, 2002) for a thorough discussion); A Comprehensive Framework for Combined Decision Procedures – p. 8/4
§2. The Nelson-Oppen Method To make the above schema more precise, we need to solve some problems: About Purification : here the problem is how to implement efficiently the preprocessing purification step (we won’t discuss this matter, see (Baader-Tinelli, 2002) for a thorough discussion); About Propagation : here it is not clear at all how to implement Propagation (one possible risk is non-termination); A Comprehensive Framework for Combined Decision Procedures – p. 8/4
§2. The Nelson-Oppen Method To make the above schema more precise, we need to solve some problems: About Purification : here the problem is how to implement efficiently the preprocessing purification step (we won’t discuss this matter, see (Baader-Tinelli, 2002) for a thorough discussion); About Propagation : here it is not clear at all how to implement Propagation (one possible risk is non-termination); About the Exit from the Loop : whereas it is evident that the procedure is sound (if an inconsistency is detected the input constraint is unsatisfiable), there is no guarantee at all about completeness , in other words reaching saturation does not imply consistency - there are counterexamples! A Comprehensive Framework for Combined Decision Procedures – p. 8/4
§3. Propagation The most popular method for avoiding the non-termination risk is to assume that T 0 is effectively locally finite : this means that, given a finite set of variables x 0 , there are only finitely many Σ 0 ( x 0 ) -terms up to T 0 -equivalence. Representative terms for each equivalence class should also be computable. A Comprehensive Framework for Combined Decision Procedures – p. 9/4
§3. Propagation The most popular method for avoiding the non-termination risk is to assume that T 0 is effectively locally finite : this means that, given a finite set of variables x 0 , there are only finitely many Σ 0 ( x 0 ) -terms up to T 0 -equivalence. Representative terms for each equivalence class should also be computable. If effective local finiteness of the shared theory T 0 is assumed, the total amount of exchangeable information is finite. Propagation can be implemented e.g. in following two ways: let x 0 be the shared variables after the Purification preprocessing Step. A Comprehensive Framework for Combined Decision Procedures – p. 9/4
§3. Propagation Propagation (Guessing Version) : here we simply make a guess of a Σ 0 ( x 0 ) -arrangement (namely we guess for a maximal set of Σ 0 -literals containing at most the variables x 0 ) and check it for both T 1 ∪ Γ 1 -consistency and T 2 ∪ Γ 2 -consistency (the guess is finite and there are finitely many guesses to try by local finiteness). A Comprehensive Framework for Combined Decision Procedures – p. 10/4
§3. Propagation Propagation (Guessing Version) : here we simply make a guess of a Σ 0 ( x 0 ) -arrangement (namely we guess for a maximal set of Σ 0 -literals containing at most the variables x 0 ) and check it for both T 1 ∪ Γ 1 -consistency and T 2 ∪ Γ 2 -consistency (the guess is finite and there are finitely many guesses to try by local finiteness). The alternative implementation schema of Propagation is the following: A Comprehensive Framework for Combined Decision Procedures – p. 10/4
§3. Propagation Propagation (Guessing Version) : here we simply make a guess of a Σ 0 ( x 0 ) -arrangement (namely we guess for a maximal set of Σ 0 -literals containing at most the variables x 0 ) and check it for both T 1 ∪ Γ 1 -consistency and T 2 ∪ Γ 2 -consistency (the guess is finite and there are finitely many guesses to try by local finiteness). The alternative implementation schema of Propagation is the following: Propagation (Backtracking Version) : identify a disjunction of x 0 -atoms A 1 ∨ · · · ∨ A n which is entailed by T i ∪ Γ i ( i = 1 or 2 ) and make case splitting by adding some A j to both Γ 1 , Γ 2 (if none of the A 1 , . . . , A n is already there). Repeat until possible. A Comprehensive Framework for Combined Decision Procedures – p. 10/4
§3. Propagation An advantage of the first option is that whenever constraints are represented not as sets of literals, but as boolean combinations of atoms, one may combine heuristics of powerful DPLL-based SAT-solvers with specific features of the theories to be combined in order to produce efficiently the right arrangement. A Comprehensive Framework for Combined Decision Procedures – p. 11/4
§3. Propagation An advantage of the first option is that whenever constraints are represented not as sets of literals, but as boolean combinations of atoms, one may combine heuristics of powerful DPLL-based SAT-solvers with specific features of the theories to be combined in order to produce efficiently the right arrangement. This idea has been recently developed in the case of disjoint signatures by (Bozzano et al., in print), where positive experimental are obtained. A Comprehensive Framework for Combined Decision Procedures – p. 11/4
§3. Propagation An advantage of the second option is that it works under hypotheses which are weaker than local finiteness: it is sufficient to assume (a) noetherianity (namely that there are no infinite properly ascending chains of finite sets of T 0 -atoms) and (b) that suitable positive residue enumerators are available (see the FroCoS ’05 Proceedings or our extended Technical Report for details). A Comprehensive Framework for Combined Decision Procedures – p. 12/4
§3. Propagation An advantage of the second option is that it works under hypotheses which are weaker than local finiteness: it is sufficient to assume (a) noetherianity (namely that there are no infinite properly ascending chains of finite sets of T 0 -atoms) and (b) that suitable positive residue enumerators are available (see the FroCoS ’05 Proceedings or our extended Technical Report for details). Another advantage of the second method is that the procedure can be made really deterministic in case the T i are both Σ 0 -convex ( T i is said to be Σ 0 -convex iff whenever T i ∪ Γ i entails a disjunction of n > 1 Σ 0 -atoms, then it entails one of them). A Comprehensive Framework for Combined Decision Procedures – p. 12/4
§4. Completeness In the following we always assume that T 0 is effectively locally finite (for simplicity). Hence we know how to implement propagation. A Comprehensive Framework for Combined Decision Procedures – p. 13/4
§4. Completeness In the following we always assume that T 0 is effectively locally finite (for simplicity). Hence we know how to implement propagation. Completeness is still a problem , however. This is because whenever a saturation state is reached, we can just build two models: A Comprehensive Framework for Combined Decision Procedures – p. 13/4
§4. Completeness In the following we always assume that T 0 is effectively locally finite (for simplicity). Hence we know how to implement propagation. Completeness is still a problem , however. This is because whenever a saturation state is reached, we can just build two models: a model M 1 of T 1 ∪ Γ 1 ; A Comprehensive Framework for Combined Decision Procedures – p. 13/4
§4. Completeness In the following we always assume that T 0 is effectively locally finite (for simplicity). Hence we know how to implement propagation. Completeness is still a problem , however. This is because whenever a saturation state is reached, we can just build two models: a model M 1 of T 1 ∪ Γ 1 ; a model M 2 of T 2 ∪ Γ 2 ; A Comprehensive Framework for Combined Decision Procedures – p. 13/4
§4. Completeness In the following we always assume that T 0 is effectively locally finite (for simplicity). Hence we know how to implement propagation. Completeness is still a problem , however. This is because whenever a saturation state is reached, we can just build two models: a model M 1 of T 1 ∪ Γ 1 ; a model M 2 of T 2 ∪ Γ 2 ; M 1 and M 2 (under suitable assignments) are equivalent w.r.t. the truth Σ 0 ( x 0 ) -atoms only. A Comprehensive Framework for Combined Decision Procedures – p. 13/4
§4. Completeness In the following we always assume that T 0 is effectively locally finite (for simplicity). Hence we know how to implement propagation. Completeness is still a problem , however. This is because whenever a saturation state is reached, we can just build two models: a model M 1 of T 1 ∪ Γ 1 ; a model M 2 of T 2 ∪ Γ 2 ; M 1 and M 2 (under suitable assignments) are equivalent w.r.t. the truth Σ 0 ( x 0 ) -atoms only. If M 1 and M 2 were Σ 0 ( x 0 ) -isomorphic, then we could easily build the desired model of T 1 ∪ T 2 ∪ Γ 1 ∪ Γ 2 . A Comprehensive Framework for Combined Decision Procedures – p. 13/4
§4. Completeness The reason why Nelson-Oppen may happen to work is because (in some situations, to be identified) we can jump from equivalence w.r.t. a certain fragment of Σ 0 ( x 0 ) -language to Σ 0 ( x 0 ) -isomorphism . A Comprehensive Framework for Combined Decision Procedures – p. 14/4
§4. Completeness The reason why Nelson-Oppen may happen to work is because (in some situations, to be identified) we can jump from equivalence w.r.t. a certain fragment of Σ 0 ( x 0 ) -language to Σ 0 ( x 0 ) -isomorphism . There are devices (which we shall call isomorphism theorems ) that guarantee the safety of this big jump. A Comprehensive Framework for Combined Decision Procedures – p. 14/4
§4. Completeness The reason why Nelson-Oppen may happen to work is because (in some situations, to be identified) we can jump from equivalence w.r.t. a certain fragment of Σ 0 ( x 0 ) -language to Σ 0 ( x 0 ) -isomorphism . There are devices (which we shall call isomorphism theorems ) that guarantee the safety of this big jump. Our claim is that we can extend Nelson-Oppen method quite far (even to higher-order contexts), provided there are such devices at hand. A Comprehensive Framework for Combined Decision Procedures – p. 14/4
§4. Completeness The reason why Nelson-Oppen may happen to work is because (in some situations, to be identified) we can jump from equivalence w.r.t. a certain fragment of Σ 0 ( x 0 ) -language to Σ 0 ( x 0 ) -isomorphism . There are devices (which we shall call isomorphism theorems ) that guarantee the safety of this big jump. Our claim is that we can extend Nelson-Oppen method quite far (even to higher-order contexts), provided there are such devices at hand. To explain our ideas, let us turn to the case of first-order theories T 1 , T 2 sharing the theory T 0 in the common subsignature. A Comprehensive Framework for Combined Decision Procedures – p. 14/4
§4. Completeness Let’s even assume for a while that the signatures Σ 1 , Σ 2 are disjoint and let T 0 be the empty theory. We call T ∗ 0 the theory saying that the domain is infinite: that is, to get T ∗ 0 we need the infinite axioms ∀ x 1 · · · ∀ x n ∃ y ( x 1 � = y ∧ · · · ∧ x n � = y ) . A Comprehensive Framework for Combined Decision Procedures – p. 15/4
§4. Completeness Let’s even assume for a while that the signatures Σ 1 , Σ 2 are disjoint and let T 0 be the empty theory. We call T ∗ 0 the theory saying that the domain is infinite: that is, to get T ∗ 0 we need the infinite axioms ∀ x 1 · · · ∀ x n ∃ y ( x 1 � = y ∧ · · · ∧ x n � = y ) . We assume that a T i -satisfiable constraint is satisfiable in a model of T i ∪ T ∗ 0 (this means that T i - for i = 1 , 2 - is stably infinite ). A Comprehensive Framework for Combined Decision Procedures – p. 15/4
§4. Completeness Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗ 0 admits quantifier elimination, we get A Comprehensive Framework for Combined Decision Procedures – p. 16/4
§4. Completeness Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗ 0 admits quantifier elimination, we get a model M 1 of T 1 ∪ Γ 1 ; A Comprehensive Framework for Combined Decision Procedures – p. 16/4
§4. Completeness Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗ 0 admits quantifier elimination, we get a model M 1 of T 1 ∪ Γ 1 ; a model M 2 of T 2 ∪ Γ 2 ; A Comprehensive Framework for Combined Decision Procedures – p. 16/4
§4. Completeness Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗ 0 admits quantifier elimination, we get a model M 1 of T 1 ∪ Γ 1 ; a model M 2 of T 2 ∪ Γ 2 ; M 1 and M 2 (under suitable assignments) are equivalent w.r.t. the truth of Σ 0 ( x 0 ) - first-order formulae (not just of Σ 0 ( x 0 ) -atoms). A Comprehensive Framework for Combined Decision Procedures – p. 16/4
§4. Completeness Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗ 0 admits quantifier elimination, we get a model M 1 of T 1 ∪ Γ 1 ; a model M 2 of T 2 ∪ Γ 2 ; M 1 and M 2 (under suitable assignments) are equivalent w.r.t. the truth of Σ 0 ( x 0 ) - first-order formulae (not just of Σ 0 ( x 0 ) -atoms). Our ‘magic’ device ( Keisler-Shelah Isomorphism Theorem ) says that two elementarily equivalent structures become isomorphic if a suitable ultrapower is applied. About ultrapowers we only need to know that they are operations on semantic structures that preserve truth of elementary formulae (hence, a fortiori, of our constraints). A Comprehensive Framework for Combined Decision Procedures – p. 16/4
§4. Completeness Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗ 0 admits quantifier elimination, we get a model M 1 of T 1 ∪ Γ 1 ; a model M 2 of T 2 ∪ Γ 2 ; M 1 and M 2 (under suitable assignments) are equivalent w.r.t. the truth of Σ 0 ( x 0 ) - first-order formulae (not just of Σ 0 ( x 0 ) -atoms). Our ‘magic’ device ( Keisler-Shelah Isomorphism Theorem ) says that two elementarily equivalent structures become isomorphic if a suitable ultrapower is applied. About ultrapowers we only need to know that they are operations on semantic structures that preserve truth of elementary formulae (hence, a fortiori, of our constraints). The above argument proves completeness of Nelson-Oppen procedure. A Comprehensive Framework for Combined Decision Procedures – p. 16/4
§4. Completeness It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: A Comprehensive Framework for Combined Decision Procedures – p. 17/4
§4. Completeness It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: 0 ⊇ T 0 that eliminates quantifiers such that (1) there is a Σ 0 -theory T ∗ A Comprehensive Framework for Combined Decision Procedures – p. 17/4
§4. Completeness It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: 0 ⊇ T 0 that eliminates quantifiers such that (1) there is a Σ 0 -theory T ∗ (2) for i = 0 , 1 , 2 , a T i -satisfiable constraint is satisfiable in a model of T i ∪ T ∗ 0 . A Comprehensive Framework for Combined Decision Procedures – p. 17/4
§4. Completeness It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: 0 ⊇ T 0 that eliminates quantifiers such that (1) there is a Σ 0 -theory T ∗ (2) for i = 0 , 1 , 2 , a T i -satisfiable constraint is satisfiable in a model of T i ∪ T ∗ 0 . Condition (2) can be equivalently reformulated by saying that every model of T i embeds into a model of T i ∪ T ∗ 0 . A Comprehensive Framework for Combined Decision Procedures – p. 17/4
§4. Completeness It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: 0 ⊇ T 0 that eliminates quantifiers such that (1) there is a Σ 0 -theory T ∗ (2) for i = 0 , 1 , 2 , a T i -satisfiable constraint is satisfiable in a model of T i ∪ T ∗ 0 . Condition (2) can be equivalently reformulated by saying that every model of T i embeds into a model of T i ∪ T ∗ 0 . Theorem 1. Under the above hypotheses (1)-(2) and under effective local finiteness of T 0 , Nelson-Oppen procedure transfers decidability of constraint satisfiability problems from T 1 and T 2 to T 1 ∪ T 2 . A Comprehensive Framework for Combined Decision Procedures – p. 17/4
§4. Completeness The above Theorem is equivalent to the main combination result in (Ghilardi, 2003) [FTP’03 Conference], (Ghilardi, 2005). For a many-sorted version, see (Ganesh-Berezin-Tinelli-Dill, 2004). For an enrichment (taking into consideration modules integration, refinements, etc.), see (Ganzinger-Ruess-Shankar, 2004) [invited talk at PDPAR ’05]. A Comprehensive Framework for Combined Decision Procedures – p. 18/4
Part II Modal-like Constraints § 5. Algebraic Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 20 § 6. Semantic Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 23 § 7. Combined Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 29 A Comprehensive Framework for Combined Decision Procedures – p. 19/4
§5. Algebraic Translations It is well-known that there is a one-to-one correspondence between classical modal propositional logics and equational theories axiomatizing varieties of Boolean algebras with operators (union of such equational theories is called fusion in modal terminology). A Comprehensive Framework for Combined Decision Procedures – p. 20/4
§5. Algebraic Translations It is well-known that there is a one-to-one correspondence between classical modal propositional logics and equational theories axiomatizing varieties of Boolean algebras with operators (union of such equational theories is called fusion in modal terminology). Decidability of constraint satisfiability problems (in our sense) is mapped to decidability of global consequence relation under the above correspondence. A Comprehensive Framework for Combined Decision Procedures – p. 20/4
§5. Algebraic Translations It is well-known that there is a one-to-one correspondence between classical modal propositional logics and equational theories axiomatizing varieties of Boolean algebras with operators (union of such equational theories is called fusion in modal terminology). Decidability of constraint satisfiability problems (in our sense) is mapped to decidability of global consequence relation under the above correspondence. The hypotheses of Theorem 1 are true whenever T 1 , T 2 are theories axiomatizing equational classes of Boolean algebras with operators (provided only Boolean operators are shared). A Comprehensive Framework for Combined Decision Procedures – p. 20/4
§5. Algebraic Translations It is well-known that there is a one-to-one correspondence between classical modal propositional logics and equational theories axiomatizing varieties of Boolean algebras with operators (union of such equational theories is called fusion in modal terminology). Decidability of constraint satisfiability problems (in our sense) is mapped to decidability of global consequence relation under the above correspondence. The hypotheses of Theorem 1 are true whenever T 1 , T 2 are theories axiomatizing equational classes of Boolean algebras with operators (provided only Boolean operators are shared). It follows that Theorem 1 implies transfer of decidability of global consequence relation to fusions of classical modal logics (Wolter, 1999). See also (Ghilardi-Santocanale, 2003) for further applications of Theorem 1 to modal logic. A Comprehensive Framework for Combined Decision Procedures – p. 20/4
§5. Algebraic Translations Much less immediate (more work is required, both from the technical and the conceptual side, as well as integration with methods coming from difference source) are transfer results for (standard) local consequence relation [that corresponds to word problem ], see (Baader-Ghilardi-Tinelli, 2004). A Comprehensive Framework for Combined Decision Procedures – p. 21/4
§5. Algebraic Translations Much less immediate (more work is required, both from the technical and the conceptual side, as well as integration with methods coming from difference source) are transfer results for (standard) local consequence relation [that corresponds to word problem ], see (Baader-Ghilardi-Tinelli, 2004). Recent extensions cover also the case of E -connections (Baader-Ghilardi, 2005a-b). Techniques are specific for the E -connection case, but the combined decision algorithms are still Nelson-Oppen like. A Comprehensive Framework for Combined Decision Procedures – p. 21/4
§5. Algebraic Translations Solving combined modal constraints by translating them into the formalism of Boolean algebras with operators and by applying Nelson-Oppen A Comprehensive Framework for Combined Decision Procedures – p. 22/4
§5. Algebraic Translations Solving combined modal constraints by translating them into the formalism of Boolean algebras with operators and by applying Nelson-Oppen is elegant; A Comprehensive Framework for Combined Decision Procedures – p. 22/4
§5. Algebraic Translations Solving combined modal constraints by translating them into the formalism of Boolean algebras with operators and by applying Nelson-Oppen is elegant; produces simple and easy to understand completeness proofs; A Comprehensive Framework for Combined Decision Procedures – p. 22/4
§5. Algebraic Translations Solving combined modal constraints by translating them into the formalism of Boolean algebras with operators and by applying Nelson-Oppen is elegant; produces simple and easy to understand completeness proofs; does not require specific implementation (just standard Nelson-Oppen interface); A Comprehensive Framework for Combined Decision Procedures – p. 22/4
§5. Algebraic Translations Solving combined modal constraints by translating them into the formalism of Boolean algebras with operators and by applying Nelson-Oppen is elegant; produces simple and easy to understand completeness proofs; does not require specific implementation (just standard Nelson-Oppen interface); it is (at least) not worse than other methods from the complexity viewpoint. A Comprehensive Framework for Combined Decision Procedures – p. 22/4
§5. Algebraic Translations Solving combined modal constraints by translating them into the formalism of Boolean algebras with operators and by applying Nelson-Oppen is elegant; produces simple and easy to understand completeness proofs; does not require specific implementation (just standard Nelson-Oppen interface); it is (at least) not worse than other methods from the complexity viewpoint. However, it does not seem to be always possible. There are modal-like formalisms (some description logics constructors, first order fragments like the guarded ones...) that resist from attempts to encapsulate them into the formalism of Boolean algebras with operators. A Comprehensive Framework for Combined Decision Procedures – p. 22/4
§6. Semantic Translations Usually papers oriented to computer science applications present decision problems in modal/temporal/description logics as semantic decision problems. A Comprehensive Framework for Combined Decision Procedures – p. 23/4
§6. Semantic Translations Usually papers oriented to computer science applications present decision problems in modal/temporal/description logics as semantic decision problems. For instance, (Baader-Lutz-Sturm-Wolter, 2002) prove semantically general decidability transfer results for relativized (and also un-relativized) satisfiability in so-called local abstract description systems . A Comprehensive Framework for Combined Decision Procedures – p. 23/4
§6. Semantic Translations Usually papers oriented to computer science applications present decision problems in modal/temporal/description logics as semantic decision problems. For instance, (Baader-Lutz-Sturm-Wolter, 2002) prove semantically general decidability transfer results for relativized (and also un-relativized) satisfiability in so-called local abstract description systems . As another example, semantic decidability results on monodic modal and temporal fragments (Wolter-Zakharyaschev, 2001) look like genuine combination results. A Comprehensive Framework for Combined Decision Procedures – p. 23/4
§6. Semantic Translations Usually papers oriented to computer science applications present decision problems in modal/temporal/description logics as semantic decision problems. For instance, (Baader-Lutz-Sturm-Wolter, 2002) prove semantically general decidability transfer results for relativized (and also un-relativized) satisfiability in so-called local abstract description systems . As another example, semantic decidability results on monodic modal and temporal fragments (Wolter-Zakharyaschev, 2001) look like genuine combination results. In all these cases, algorithms are rather similar to Nelson-Oppen style algorithms (with Propagation implemented by guessing). A Comprehensive Framework for Combined Decision Procedures – p. 23/4
§6. Semantic Translations Usually papers oriented to computer science applications present decision problems in modal/temporal/description logics as semantic decision problems. For instance, (Baader-Lutz-Sturm-Wolter, 2002) prove semantically general decidability transfer results for relativized (and also un-relativized) satisfiability in so-called local abstract description systems . As another example, semantic decidability results on monodic modal and temporal fragments (Wolter-Zakharyaschev, 2001) look like genuine combination results. In all these cases, algorithms are rather similar to Nelson-Oppen style algorithms (with Propagation implemented by guessing). Proofs however have nothing to do with the above proof of Theorem 1... A Comprehensive Framework for Combined Decision Procedures – p. 23/4
§6. Semantic Translations Usually papers oriented to computer science applications present decision problems in modal/temporal/description logics as semantic decision problems. For instance, (Baader-Lutz-Sturm-Wolter, 2002) prove semantically general decidability transfer results for relativized (and also un-relativized) satisfiability in so-called local abstract description systems . As another example, semantic decidability results on monodic modal and temporal fragments (Wolter-Zakharyaschev, 2001) look like genuine combination results. In all these cases, algorithms are rather similar to Nelson-Oppen style algorithms (with Propagation implemented by guessing). Proofs however have nothing to do with the above proof of Theorem 1... ...apparently... A Comprehensive Framework for Combined Decision Procedures – p. 23/4
§6. Semantic Translations Let’s analyze the most simple case of a modal (global) constraint, namely relativized satisfiability . Propositional modal formulae are built up by using propositional variables, Boolean connectives and ♦ ( � is defined as ¬ ♦ ¬ ). A Comprehensive Framework for Combined Decision Procedures – p. 24/4
§6. Semantic Translations Let’s analyze the most simple case of a modal (global) constraint, namely relativized satisfiability . Propositional modal formulae are built up by using propositional variables, Boolean connectives and ♦ ( � is defined as ¬ ♦ ¬ ). The standard translation ST ( ψ, w ) of the modal formula ψ is so defined: ST ( ⊤ , w ) = ⊤ ST ( ⊥ , w ) = ⊥ ST (x , w ) = X ( w ) ST ( ¬ ψ, w ) = ¬ ST ( ψ, w ) ST ( ψ 1 ∨ ψ 2 , w ) = ST ( ψ 1 , w ) ∨ ST ( ψ 2 , w ) ST ( ψ 1 ∧ ψ 2 , w ) = ST ( ψ 1 , w ) ∧ ST ( ψ 2 , w ) ST ( ♦ ψ, w ) = ∃ v ( R ( w, v ) ∧ ST ( ψ, v )) . A Comprehensive Framework for Combined Decision Procedures – p. 24/4
§6. Semantic Translations Notice that the signature Σ R for the translation is a first-order signature with just a binary relational symbol R . We also used unary predicate variables X in correspondence to propositional variables x . A Comprehensive Framework for Combined Decision Procedures – p. 25/4
§6. Semantic Translations Notice that the signature Σ R for the translation is a first-order signature with just a binary relational symbol R . We also used unary predicate variables X in correspondence to propositional variables x . Let F be a class of Kripke frames, i.e. of Σ R -structures, closed under disjoint unions and isomorphisms. Our global modal F -constraints can be formulated in various equivalent ways (given that we have full Boolean connectives and that F is closed under disjoint unions). We choose a formulation that will help the development of our plans. A Comprehensive Framework for Combined Decision Procedures – p. 25/4
§6. Semantic Translations We call global modal F -constraint the problem of deciding whether there is F ∈ F satisfying a conjunction of equations and inequations of the following kind: A Comprehensive Framework for Combined Decision Procedures – p. 26/4
§6. Semantic Translations We call global modal F -constraint the problem of deciding whether there is F ∈ F satisfying a conjunction of equations and inequations of the following kind: { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . A Comprehensive Framework for Combined Decision Procedures – p. 26/4
§6. Semantic Translations We call global modal F -constraint the problem of deciding whether there is F ∈ F satisfying a conjunction of equations and inequations of the following kind: { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . Notice we used a higher-order formalism (although the problem is not a real higher-order problem); let’s have a better look to it. A Comprehensive Framework for Combined Decision Procedures – p. 26/4
§6. Semantic Translations The satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . A Comprehensive Framework for Combined Decision Procedures – p. 27/4
§6. Semantic Translations The satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . contains bounded variables w, v, . . . of type W (let W be the name of the unique sort of Σ R ); A Comprehensive Framework for Combined Decision Procedures – p. 27/4
§6. Semantic Translations The satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . contains bounded variables w, v, . . . of type W (let W be the name of the unique sort of Σ R ); contains a constant R of type W → ( W → Ω) (here Ω is the truth-value type); A Comprehensive Framework for Combined Decision Procedures – p. 27/4
§6. Semantic Translations The satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . contains bounded variables w, v, . . . of type W (let W be the name of the unique sort of Σ R ); contains a constant R of type W → ( W → Ω) (here Ω is the truth-value type); contains free variables X, Y, . . . for subsets, namely of type W → Ω ; A Comprehensive Framework for Combined Decision Procedures – p. 27/4
§6. Semantic Translations The satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . contains bounded variables w, v, . . . of type W (let W be the name of the unique sort of Σ R ); contains a constant R of type W → ( W → Ω) (here Ω is the truth-value type); contains free variables X, Y, . . . for subsets, namely of type W → Ω ; uses in all (in)equations higher-order terms of type W → Ω (this is the same as the type of the free variables of the constraint ). A Comprehensive Framework for Combined Decision Procedures – p. 27/4
§6. Semantic Translations The (Kripke semantics) meaning of the satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . is that we must find A Comprehensive Framework for Combined Decision Procedures – p. 28/4
§6. Semantic Translations The (Kripke semantics) meaning of the satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . is that we must find a frame F in F and an assignment to the free variables of the problem (namely, a Kripke model on F ); A Comprehensive Framework for Combined Decision Procedures – p. 28/4
§6. Semantic Translations The (Kripke semantics) meaning of the satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . is that we must find a frame F in F and an assignment to the free variables of the problem (namely, a Kripke model on F ); in such a way that the modal formulae ψ i ↔ ψ ′ i hold globally (that is, in every state); A Comprehensive Framework for Combined Decision Procedures – p. 28/4
§6. Semantic Translations The (Kripke semantics) meaning of the satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . is that we must find a frame F in F and an assignment to the free variables of the problem (namely, a Kripke model on F ); in such a way that the modal formulae ψ i ↔ ψ ′ i hold globally (that is, in every state); and in such a way that the modal formulae ¬ ( φ j ↔ φ ′ j ) hold locally (that is, in some state). A Comprehensive Framework for Combined Decision Procedures – p. 28/4
§6. Semantic Translations The (Kripke semantics) meaning of the satisfiability problem { w | ST ( ψ 1 , w ) } = { w | ST ( ψ ′ 1 , w ) } , . . . , { w | ST ( ψ n , w ) } = { w | ST ( ψ ′ n , w ) } , { w | ST ( φ 1 , w ) } � = { w | ST ( φ ′ 1 , w ) } , . . . , { w | ST ( φ m , w ) } � = { w | ST ( φ ′ m , w ) } . is that we must find a frame F in F and an assignment to the free variables of the problem (namely, a Kripke model on F ); in such a way that the modal formulae ψ i ↔ ψ ′ i hold globally (that is, in every state); and in such a way that the modal formulae ¬ ( φ j ↔ φ ′ j ) hold locally (that is, in some state). Since F is closed under disjoint unions, this is standard relativized satisfiability problem . A Comprehensive Framework for Combined Decision Procedures – p. 28/4
§6. Combined Constraints Combined problems of the above kind are defined in the obvious way: we consider two modal operators ♦ 1 and ♦ 2 , the signature Σ R 1 ∪ Σ R 1 , two frame classes F 1 and F 2 , and we ask for satisfiability in the frame class F 1 ⊕ F 2 = { ( W, R 1 , R 2 ) | ( W, R 1 ) ∈ F 1 & ( W, R 2 ) ∈ F 2 } . A Comprehensive Framework for Combined Decision Procedures – p. 29/4
§6. Combined Constraints Combined problems of the above kind are defined in the obvious way: we consider two modal operators ♦ 1 and ♦ 2 , the signature Σ R 1 ∪ Σ R 1 , two frame classes F 1 and F 2 , and we ask for satisfiability in the frame class F 1 ⊕ F 2 = { ( W, R 1 , R 2 ) | ( W, R 1 ) ∈ F 1 & ( W, R 2 ) ∈ F 2 } . Let’s analyze why Nelson-Oppen method applies through an example. A Comprehensive Framework for Combined Decision Procedures – p. 29/4
§6. Combined Constraints Combined problems of the above kind are defined in the obvious way: we consider two modal operators ♦ 1 and ♦ 2 , the signature Σ R 1 ∪ Σ R 1 , two frame classes F 1 and F 2 , and we ask for satisfiability in the frame class F 1 ⊕ F 2 = { ( W, R 1 , R 2 ) | ( W, R 1 ) ∈ F 1 & ( W, R 2 ) ∈ F 2 } . Let’s analyze why Nelson-Oppen method applies through an example. Consider the following combined constraint: { w | ST ( ♦ 1 � 2 x , w ) } � = { w | ST ( ⊥ , w ) } , that is { w | ∃ v ( R 1 ( w, v ) ∧ ∀ z ( R 2 ( v, z ) → X ( z ))) } � = ∅ . A Comprehensive Framework for Combined Decision Procedures – p. 29/4
§6. Combined Constraints Purification : the purified constraint is { w | ∃ v ( R 1 ( w, v ) ∧ Y ( v ) } � = ∅ , { w | Y ( w ) } = { w | ∀ z ( R 2 ( w, z ) → X ( z ))) } . Notice that: A Comprehensive Framework for Combined Decision Procedures – p. 30/4
§6. Combined Constraints Purification : the purified constraint is { w | ∃ v ( R 1 ( w, v ) ∧ Y ( v ) } � = ∅ , { w | Y ( w ) } = { w | ∀ z ( R 2 ( w, z ) → X ( z ))) } . Notice that: { w | Y ( w ) } is (the long βη -normal form of) a fresh variable; A Comprehensive Framework for Combined Decision Procedures – p. 30/4
§6. Combined Constraints Purification : the purified constraint is { w | ∃ v ( R 1 ( w, v ) ∧ Y ( v ) } � = ∅ , { w | Y ( w ) } = { w | ∀ z ( R 2 ( w, z ) → X ( z ))) } . Notice that: { w | Y ( w ) } is (the long βη -normal form of) a fresh variable; the purification equation in the new constraint, if interpreted as a substitution and applied to the purified in-equation, gives back the old constraint (up to a β -conversion); A Comprehensive Framework for Combined Decision Procedures – p. 30/4
§6. Combined Constraints Purification : the purified constraint is { w | ∃ v ( R 1 ( w, v ) ∧ Y ( v ) } � = ∅ , { w | Y ( w ) } = { w | ∀ z ( R 2 ( w, z ) → X ( z ))) } . Notice that: { w | Y ( w ) } is (the long βη -normal form of) a fresh variable; the purification equation in the new constraint, if interpreted as a substitution and applied to the purified in-equation, gives back the old constraint (up to a β -conversion); after all, this purification is possible because types of variables and of terms used in constraints match! A Comprehensive Framework for Combined Decision Procedures – p. 30/4
Recommend
More recommend
