A Comprehensive Framework for Combined Decision Procedures Silvio G - - PowerPoint PPT Presentation
A Comprehensive Framework for Combined Decision Procedures Silvio G HILARDI 1 Enrica N ICOLINI 2 Daniele Z UCCHELLI 1 {ghilardi, zucchelli}@dsi.unimi.it, nicolini@mat.unimi.it 1 Dipartimento di Scienze dellInformazione 2 Dipartimento di
Silvio GHILARDI1 Enrica NICOLINI2 Daniele ZUCCHELLI1
{ghilardi, zucchelli}@dsi.unimi.it, nicolini@mat.unimi.it
1Dipartimento di Scienze dell’Informazione 2Dipartimento di Matematica
Università degli Studi di Milano - Italy
A Comprehensive Framework for Combined Decision Procedures – p. 1/4
Our aim is that of providing a unifying approach to combination methodologies for deciding satisfiability of various kinds of constraints. Part I : here we analyze Nelson-Oppen combination method within first-order theories, making special emphasis on the case of non-disjoint signatures; Part II : here we show how it is possible to reduce to the Nelson-Oppen approach various kinds of semantic decision problems involving intensional logics; Part III : taking inspiration from the already analyzed examples, we make our proposal of a comprehensive Nelson-Oppen combination schema.
A Comprehensive Framework for Combined Decision Procedures – p. 2/4
Our aim is that of providing a unifying approach to combination methodologies for deciding satisfiability of various kinds of constraints. Part I : here we analyze Nelson-Oppen combination method within first-order theories, making special emphasis on the case of non-disjoint signatures; Part II : here we show how it is possible to reduce to the Nelson-Oppen approach various kinds of semantic decision problems involving intensional logics; Part III : taking inspiration from the already analyzed examples, we make our proposal of a comprehensive Nelson-Oppen combination schema.
A Comprehensive Framework for Combined Decision Procedures – p. 2/4
Our aim is that of providing a unifying approach to combination methodologies for deciding satisfiability of various kinds of constraints. Part I : here we analyze Nelson-Oppen combination method within first-order theories, making special emphasis on the case of non-disjoint signatures; Part II : here we show how it is possible to reduce to the Nelson-Oppen approach various kinds of semantic decision problems involving intensional logics; Part III : taking inspiration from the already analyzed examples, we make our proposal of a comprehensive Nelson-Oppen combination schema.
A Comprehensive Framework for Combined Decision Procedures – p. 2/4
Our aim is that of providing a unifying approach to combination methodologies for deciding satisfiability of various kinds of constraints. Part I : here we analyze Nelson-Oppen combination method within first-order theories, making special emphasis on the case of non-disjoint signatures; Part II : here we show how it is possible to reduce to the Nelson-Oppen approach various kinds of semantic decision problems involving intensional logics; Part III : taking inspiration from the already analyzed examples, we make our proposal of a comprehensive Nelson-Oppen combination schema.
A Comprehensive Framework for Combined Decision Procedures – p. 2/4
§ 1. Constraint Satisfiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 4 § 2. The Nelson-Oppen Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 6 § 3. Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 9 § 4. Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p.13
A Comprehensive Framework for Combined Decision Procedures – p. 3/4
We fix a first-order theory T (in a first-order signature with equality Σ) as well as a finite set of literals Γ. We ask whether there is a model of T satisfying Γ. There are many examples of theories in which this problem is solvable:
A Comprehensive Framework for Combined Decision Procedures – p. 4/4
We fix a first-order theory T (in a first-order signature with equality Σ) as well as a finite set of literals Γ. We ask whether there is a model of T satisfying Γ. There are many examples of theories in which this problem is solvable: the empty theory;
A Comprehensive Framework for Combined Decision Procedures – p. 4/4
We fix a first-order theory T (in a first-order signature with equality Σ) as well as a finite set of literals Γ. We ask whether there is a model of T satisfying Γ. There are many examples of theories in which this problem is solvable: the empty theory; linear (rational or integer) arithmetic;
A Comprehensive Framework for Combined Decision Procedures – p. 4/4
We fix a first-order theory T (in a first-order signature with equality Σ) as well as a finite set of literals Γ. We ask whether there is a model of T satisfying Γ. There are many examples of theories in which this problem is solvable: the empty theory; linear (rational or integer) arithmetic; theories axiomatizing common datatypes (lists, arrays, ...);
A Comprehensive Framework for Combined Decision Procedures – p. 4/4
We fix a first-order theory T (in a first-order signature with equality Σ) as well as a finite set of literals Γ. We ask whether there is a model of T satisfying Γ. There are many examples of theories in which this problem is solvable: the empty theory; linear (rational or integer) arithmetic; theories axiomatizing common datatypes (lists, arrays, ...); theories coming from computer algebra (K-algebras, ...);
A Comprehensive Framework for Combined Decision Procedures – p. 4/4
We fix a first-order theory T (in a first-order signature with equality Σ) as well as a finite set of literals Γ. We ask whether there is a model of T satisfying Γ. There are many examples of theories in which this problem is solvable: the empty theory; linear (rational or integer) arithmetic; theories axiomatizing common datatypes (lists, arrays, ...); theories coming from computer algebra (K-algebras, ...); algebraic counterparts of modal logics (i.e. theories axiomatizing boolean algebras with operators).
A Comprehensive Framework for Combined Decision Procedures – p. 4/4
Quite often data appearing in concrete problems are heterogeneous. In software verification one has to deal simultaneously with congruence closure for free symbols, numerical constraints, constraint satisfiability for lists, arrays, booleans, sets and so on. The same applies in other areas: in knowledge representation areas, rich logics need to be considered in which temporal, epistemic, descriptive aspects are present.
A Comprehensive Framework for Combined Decision Procedures – p. 5/4
Quite often data appearing in concrete problems are heterogeneous. In software verification one has to deal simultaneously with congruence closure for free symbols, numerical constraints, constraint satisfiability for lists, arrays, booleans, sets and so on. The same applies in other areas: in knowledge representation areas, rich logics need to be considered in which temporal, epistemic, descriptive aspects are present. The Nelson-Oppen method (Nelson-Oppen, 1979) is the most simple method for combining decision procedures for constraint satisfiability.
A Comprehensive Framework for Combined Decision Procedures – p. 5/4
Nelson-Oppen method was originally proposed for disjoint (first-order)
this reason, we summarize here the essence of Nelson-Oppen method from an intuitive point of view.
A Comprehensive Framework for Combined Decision Procedures – p. 6/4
Nelson-Oppen method was originally proposed for disjoint (first-order)
this reason, we summarize here the essence of Nelson-Oppen method from an intuitive point of view. Let us fix two first-order theories T1, T2 in finite signatures Σ1, Σ2 (we let Σ0 be Σ1 ∩ Σ2 and T0 be a Σ0-theory contained in both T1, T2). Let Γ be a finite set of Σ1 ∪ Σ2-literals.
A Comprehensive Framework for Combined Decision Procedures – p. 6/4
Nelson-Oppen method was originally proposed for disjoint (first-order)
this reason, we summarize here the essence of Nelson-Oppen method from an intuitive point of view. Let us fix two first-order theories T1, T2 in finite signatures Σ1, Σ2 (we let Σ0 be Σ1 ∩ Σ2 and T0 be a Σ0-theory contained in both T1, T2). Let Γ be a finite set of Σ1 ∪ Σ2-literals. Checking satisfiability of T1 ∪ T2 ∪ Γ by Nelson-Oppen requires the following phases:
A Comprehensive Framework for Combined Decision Procedures – p. 6/4
Purification : an equi-satisfiable set of pure constraints Γ1 ∪ Γ2 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);
A Comprehensive Framework for Combined Decision Procedures – p. 7/4
Purification : an equi-satisfiable set of pure constraints Γ1 ∪ Γ2 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 T1-constraint solving procedure and the T2-constraint solving procedure fairly exchange information concerning unsatisfiability of Σ0-constraints;
A Comprehensive Framework for Combined Decision Procedures – p. 7/4
Purification : an equi-satisfiable set of pure constraints Γ1 ∪ Γ2 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 T1-constraint solving procedure and the T2-constraint solving procedure fairly exchange information concerning unsatisfiability of Σ0-constraints; Until : an inconsistency is detected or a saturation state is reached.
A Comprehensive Framework for Combined Decision Procedures – p. 7/4
To make the above schema more precise, we need to solve some problems:
A Comprehensive Framework for Combined Decision Procedures – p. 8/4
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
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
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
The most popular method for avoiding the non-termination risk is to assume that T0 is effectively locally finite: this means that, given a finite set of variables x0, there are only finitely many Σ0(x0)-terms up to T0-equivalence. Representative terms for each equivalence class should also be computable.
A Comprehensive Framework for Combined Decision Procedures – p. 9/4
The most popular method for avoiding the non-termination risk is to assume that T0 is effectively locally finite: this means that, given a finite set of variables x0, there are only finitely many Σ0(x0)-terms up to T0-equivalence. Representative terms for each equivalence class should also be computable. If effective local finiteness of the shared theory T0 is assumed, the total amount of exchangeable information is finite. Propagation can be implemented e.g. in following two ways: let x0 be the shared variables after the Purification preprocessing Step.
A Comprehensive Framework for Combined Decision Procedures – p. 9/4
Propagation (Guessing Version) : here we simply make a guess of a Σ0(x0)-arrangement (namely we guess for a maximal set of Σ0-literals containing at most the variables x0) and check it for both T1 ∪ Γ1-consistency and T2 ∪ Γ2-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
Propagation (Guessing Version) : here we simply make a guess of a Σ0(x0)-arrangement (namely we guess for a maximal set of Σ0-literals containing at most the variables x0) and check it for both T1 ∪ Γ1-consistency and T2 ∪ Γ2-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
Propagation (Guessing Version) : here we simply make a guess of a Σ0(x0)-arrangement (namely we guess for a maximal set of Σ0-literals containing at most the variables x0) and check it for both T1 ∪ Γ1-consistency and T2 ∪ Γ2-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 x0-atoms A1 ∨ · · · ∨ An which is entailed by Ti ∪ Γi (i = 1 or 2) and make case splitting by adding some Aj to both Γ1, Γ2 (if none of the A1, . . . , An is already there). Repeat until possible.
A Comprehensive Framework for Combined Decision Procedures – p. 10/4
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
A Comprehensive Framework for Combined Decision Procedures – p. 11/4
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
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
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 T0-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
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 T0-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 Ti are both Σ0-convex (Ti is said to be Σ0-convex iff whenever Ti ∪ Γi entails a disjunction of n > 1 Σ0-atoms, then it entails one of them).
A Comprehensive Framework for Combined Decision Procedures – p. 12/4
In the following we always assume that T0 is effectively locally finite (for simplicity). Hence we know how to implement propagation.
A Comprehensive Framework for Combined Decision Procedures – p. 13/4
In the following we always assume that T0 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
In the following we always assume that T0 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 M1 of T1 ∪ Γ1;
A Comprehensive Framework for Combined Decision Procedures – p. 13/4
In the following we always assume that T0 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 M1 of T1 ∪ Γ1; a model M2 of T2 ∪ Γ2;
A Comprehensive Framework for Combined Decision Procedures – p. 13/4
In the following we always assume that T0 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 M1 of T1 ∪ Γ1; a model M2 of T2 ∪ Γ2; M1 and M2 (under suitable assignments) are equivalent w.r.t. the truth Σ0(x0)-atoms only.
A Comprehensive Framework for Combined Decision Procedures – p. 13/4
In the following we always assume that T0 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 M1 of T1 ∪ Γ1; a model M2 of T2 ∪ Γ2; M1 and M2 (under suitable assignments) are equivalent w.r.t. the truth Σ0(x0)-atoms only. If M1 and M2 were Σ0(x0)-isomorphic, then we could easily build the desired model of T1 ∪ T2 ∪ Γ1 ∪ Γ2.
A Comprehensive Framework for Combined Decision Procedures – p. 13/4
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 Σ0(x0)-language to Σ0(x0)-isomorphism.
A Comprehensive Framework for Combined Decision Procedures – p. 14/4
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 Σ0(x0)-language to Σ0(x0)-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
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 Σ0(x0)-language to Σ0(x0)-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
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 Σ0(x0)-language to Σ0(x0)-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 T1, T2 sharing the theory T0 in the common subsignature.
A Comprehensive Framework for Combined Decision Procedures – p. 14/4
Let’s even assume for a while that the signatures Σ1, Σ2 are disjoint and let T0 be the empty theory. We call T ∗
0 the theory saying that the domain is infinite: that is, to get T ∗
we need the infinite axioms ∀x1 · · · ∀xn∃y (x1 = y ∧ · · · ∧ xn = y).
A Comprehensive Framework for Combined Decision Procedures – p. 15/4
Let’s even assume for a while that the signatures Σ1, Σ2 are disjoint and let T0 be the empty theory. We call T ∗
0 the theory saying that the domain is infinite: that is, to get T ∗
we need the infinite axioms ∀x1 · · · ∀xn∃y (x1 = y ∧ · · · ∧ xn = y). We assume that a Ti-satisfiable constraint is satisfiable in a model of Ti ∪ T ∗
0 (this means that Ti - for i = 1, 2 - is stably infinite).
A Comprehensive Framework for Combined Decision Procedures – p. 15/4
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
Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗
0 admits quantifier elimination, we get
a model M1 of T1 ∪ Γ1;
A Comprehensive Framework for Combined Decision Procedures – p. 16/4
Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗
0 admits quantifier elimination, we get
a model M1 of T1 ∪ Γ1; a model M2 of T2 ∪ Γ2;
A Comprehensive Framework for Combined Decision Procedures – p. 16/4
Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗
0 admits quantifier elimination, we get
a model M1 of T1 ∪ Γ1; a model M2 of T2 ∪ Γ2; M1 and M2 (under suitable assignments) are equivalent w.r.t. the truth of Σ0(x0)-first-order formulae (not just of Σ0(x0)-atoms).
A Comprehensive Framework for Combined Decision Procedures – p. 16/4
Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗
0 admits quantifier elimination, we get
a model M1 of T1 ∪ Γ1; a model M2 of T2 ∪ Γ2; M1 and M2 (under suitable assignments) are equivalent w.r.t. the truth of Σ0(x0)-first-order formulae (not just of Σ0(x0)-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
Suppose now that we reach a saturation state during the execution of Nelson-Oppen; since T ∗
0 admits quantifier elimination, we get
a model M1 of T1 ∪ Γ1; a model M2 of T2 ∪ Γ2; M1 and M2 (under suitable assignments) are equivalent w.r.t. the truth of Σ0(x0)-first-order formulae (not just of Σ0(x0)-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
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
It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: (1) there is a Σ0-theory T ∗
0 ⊇ T0 that eliminates quantifiers such that
A Comprehensive Framework for Combined Decision Procedures – p. 17/4
It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: (1) there is a Σ0-theory T ∗
0 ⊇ T0 that eliminates quantifiers such that
(2) for i = 0, 1, 2, a Ti-satisfiable constraint is satisfiable in a model of Ti ∪ T ∗
0 .
A Comprehensive Framework for Combined Decision Procedures – p. 17/4
It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: (1) there is a Σ0-theory T ∗
0 ⊇ T0 that eliminates quantifiers such that
(2) for i = 0, 1, 2, a Ti-satisfiable constraint is satisfiable in a model of Ti ∪ T ∗
0 .
Condition (2) can be equivalently reformulated by saying that every model
0 .
A Comprehensive Framework for Combined Decision Procedures – p. 17/4
It is now evident under which hypotheses we can extend completeness of Nelson-Oppen to non disjoint signatures case. We need to assume that: (1) there is a Σ0-theory T ∗
0 ⊇ T0 that eliminates quantifiers such that
(2) for i = 0, 1, 2, a Ti-satisfiable constraint is satisfiable in a model of Ti ∪ T ∗
0 .
Condition (2) can be equivalently reformulated by saying that every model
0 .
Theorem 1. Under the above hypotheses (1)-(2) and under effective local finiteness of T0, Nelson-Oppen procedure transfers decidability of constraint satisfiability problems from T1 and T2 to T1 ∪ T2.
A Comprehensive Framework for Combined Decision Procedures – p. 17/4
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
§ 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
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
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
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 T1, T2 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
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 T1, T2 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
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
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
Solving combined modal constraints by translating them into the formalism
A Comprehensive Framework for Combined Decision Procedures – p. 22/4
Solving combined modal constraints by translating them into the formalism
is elegant;
A Comprehensive Framework for Combined Decision Procedures – p. 22/4
Solving combined modal constraints by translating them into the formalism
is elegant; produces simple and easy to understand completeness proofs;
A Comprehensive Framework for Combined Decision Procedures – p. 22/4
Solving combined modal constraints by translating them into the formalism
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
Solving combined modal constraints by translating them into the formalism
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
Solving combined modal constraints by translating them into the formalism
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
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
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
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
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
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
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
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
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(ψ1 ∨ ψ2, w) = ST(ψ1, w) ∨ ST(ψ2, w) ST(ψ1 ∧ ψ2, w) = ST(ψ1, w) ∧ ST(ψ2, w) ST(♦ψ, w) = ∃v(R(w, v) ∧ ST(ψ, v)).
A Comprehensive Framework for Combined Decision Procedures – p. 24/4
Notice that the signature ΣR 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
Notice that the signature ΣR 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 ΣR-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
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
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(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, w)} = {w | ST(φ′ m, w)}.
A Comprehensive Framework for Combined Decision Procedures – p. 26/4
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(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, 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
The satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, w)} = {w | ST(φ′ m, w)}.
A Comprehensive Framework for Combined Decision Procedures – p. 27/4
The satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, w)} = {w | ST(φ′ m, w)}.
contains bounded variables w, v, . . . of type W (let W be the name of the unique sort of ΣR);
A Comprehensive Framework for Combined Decision Procedures – p. 27/4
The satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, w)} = {w | ST(φ′ m, w)}.
contains bounded variables w, v, . . . of type W (let W be the name of the unique sort of ΣR); 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
The satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, w)} = {w | ST(φ′ m, w)}.
contains bounded variables w, v, . . . of type W (let W be the name of the unique sort of ΣR); 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
The satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, w)} = {w | ST(φ′ m, w)}.
contains bounded variables w, v, . . . of type W (let W be the name of the unique sort of ΣR); 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
The (Kripke semantics) meaning of the satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, w)} = {w | ST(φ′ m, w)}.
is that we must find
A Comprehensive Framework for Combined Decision Procedures – p. 28/4
The (Kripke semantics) meaning of the satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, 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
The (Kripke semantics) meaning of the satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, 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 ψi ↔ ψ′
i hold globally (that is,
in every state);
A Comprehensive Framework for Combined Decision Procedures – p. 28/4
The (Kripke semantics) meaning of the satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, 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 ψi ↔ ψ′
i hold globally (that is,
in every state); and in such a way that the modal formulae ¬(φj ↔ φ′
j) hold locally
(that is, in some state).
A Comprehensive Framework for Combined Decision Procedures – p. 28/4
The (Kripke semantics) meaning of the satisfiability problem {w | ST(ψ1, w)} = {w | ST(ψ′
1, w)}, . . . , {w | ST(ψn, w)} = {w | ST(ψ′ n, w)},
{w | ST(φ1, w)} = {w | ST(φ′
1, w)}, . . . , {w | ST(φm, 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 ψi ↔ ψ′
i hold globally (that is,
in every state); and in such a way that the modal formulae ¬(φj ↔ 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
Combined problems of the above kind are defined in the obvious way: we consider two modal operators ♦1 and ♦2, the signature ΣR1 ∪ ΣR1, two frame classes F1 and F2, and we ask for satisfiability in the frame class F1 ⊕ F2 = {(W, R1, R2) | (W, R1) ∈ F1 & (W, R2) ∈ F2}.
A Comprehensive Framework for Combined Decision Procedures – p. 29/4
Combined problems of the above kind are defined in the obvious way: we consider two modal operators ♦1 and ♦2, the signature ΣR1 ∪ ΣR1, two frame classes F1 and F2, and we ask for satisfiability in the frame class F1 ⊕ F2 = {(W, R1, R2) | (W, R1) ∈ F1 & (W, R2) ∈ F2}. Let’s analyze why Nelson-Oppen method applies through an example.
A Comprehensive Framework for Combined Decision Procedures – p. 29/4
Combined problems of the above kind are defined in the obvious way: we consider two modal operators ♦1 and ♦2, the signature ΣR1 ∪ ΣR1, two frame classes F1 and F2, and we ask for satisfiability in the frame class F1 ⊕ F2 = {(W, R1, R2) | (W, R1) ∈ F1 & (W, R2) ∈ F2}. 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(R1(w, v) ∧ ∀z(R2(v, z) → X(z)))} = ∅.
A Comprehensive Framework for Combined Decision Procedures – p. 29/4
Purification : the purified constraint is {w | ∃v(R1(w, v)∧Y (v)} = ∅, {w | Y (w)} = {w | ∀z(R2(w, z) → X(z)))}. Notice that:
A Comprehensive Framework for Combined Decision Procedures – p. 30/4
Purification : the purified constraint is {w | ∃v(R1(w, v)∧Y (v)} = ∅, {w | Y (w)} = {w | ∀z(R2(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
Purification : the purified constraint is {w | ∃v(R1(w, v)∧Y (v)} = ∅, {w | Y (w)} = {w | ∀z(R2(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
Purification : the purified constraint is {w | ∃v(R1(w, v)∧Y (v)} = ∅, {w | Y (w)} = {w | ∀z(R2(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
Propagation : information that can be exchanged consists of equations and inequations among terms of the kind {w | ST(ψ, w)}, where ψ does not contain modal operators.
A Comprehensive Framework for Combined Decision Procedures – p. 31/4
Propagation : information that can be exchanged consists of equations and inequations among terms of the kind {w | ST(ψ, w)}, where ψ does not contain modal operators. This is a finite amount of information: we are in a locally finite case and Propagation can be implemented for instance by guessing a suitable arrangement (basically, we have to guess which subsets definable as Boolean combinations of the variables X, Y are empty and which are not).
A Comprehensive Framework for Combined Decision Procedures – p. 31/4
Propagation : information that can be exchanged consists of equations and inequations among terms of the kind {w | ST(ψ, w)}, where ψ does not contain modal operators. This is a finite amount of information: we are in a locally finite case and Propagation can be implemented for instance by guessing a suitable arrangement (basically, we have to guess which subsets definable as Boolean combinations of the variables X, Y are empty and which are not). We try with the arrangement saying that {w | X(w) ∧ ¬Y (w)} is empty, whereas {w | ¬X(w) ∧ ¬Y (w)}, {w | X(w) ∧ ¬Y (w)}, {w | X(w) ∧ Y (w)} are not empty.
A Comprehensive Framework for Combined Decision Procedures – p. 31/4
Propagation : information that can be exchanged consists of equations and inequations among terms of the kind {w | ST(ψ, w)}, where ψ does not contain modal operators. This is a finite amount of information: we are in a locally finite case and Propagation can be implemented for instance by guessing a suitable arrangement (basically, we have to guess which subsets definable as Boolean combinations of the variables X, Y are empty and which are not). We try with the arrangement saying that {w | X(w) ∧ ¬Y (w)} is empty, whereas {w | ¬X(w) ∧ ¬Y (w)}, {w | X(w) ∧ ¬Y (w)}, {w | X(w) ∧ Y (w)} are not empty. Exit : the arrangement is consistent for both input procedures and we exit the Nelson-Oppen loop.
A Comprehensive Framework for Combined Decision Procedures – p. 31/4
Are we justified in claiming for combined satisfiability of the constraint?
A Comprehensive Framework for Combined Decision Procedures – p. 32/4
Are we justified in claiming for combined satisfiability of the constraint? The answer is yes, because taking disjoint copies gives us the necessary isomorphism theorem. In fact, a largely indexed disjoint copy
A Comprehensive Framework for Combined Decision Procedures – p. 32/4
Are we justified in claiming for combined satisfiability of the constraint? The answer is yes, because taking disjoint copies gives us the necessary isomorphism theorem. In fact, a largely indexed disjoint copy makes models which are equivalent w.r.t. satisfiability of shared constraints isomorphic as far as the shared signature is concerned;
A Comprehensive Framework for Combined Decision Procedures – p. 32/4
Are we justified in claiming for combined satisfiability of the constraint? The answer is yes, because taking disjoint copies gives us the necessary isomorphism theorem. In fact, a largely indexed disjoint copy makes models which are equivalent w.r.t. satisfiability of shared constraints isomorphic as far as the shared signature is concerned; does not affect satisfiability of the ‘blue’ and the ‘red’ constraints, because such constraints are preserved under taking disjoint copies and our frame classes are closed under disjoint unions (we shall say later on that the semantic operation of taking disjoint copies is extensible to the input kind of fragments of the language).
A Comprehensive Framework for Combined Decision Procedures – p. 32/4
Are we justified in claiming for combined satisfiability of the constraint? The answer is yes, because taking disjoint copies gives us the necessary isomorphism theorem. In fact, a largely indexed disjoint copy makes models which are equivalent w.r.t. satisfiability of shared constraints isomorphic as far as the shared signature is concerned; does not affect satisfiability of the ‘blue’ and the ‘red’ constraints, because such constraints are preserved under taking disjoint copies and our frame classes are closed under disjoint unions (we shall say later on that the semantic operation of taking disjoint copies is extensible to the input kind of fragments of the language). Next slide tries to illustrate the above phenomena from a graphic point of view.
A Comprehensive Framework for Combined Decision Procedures – p. 32/4
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r r r r r r r r
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X ∧ Y ¬X ∧ Y ¬X ∧ ¬Y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
r r r r r r r r r r r r
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X ∧ Y ¬X ∧ Y ¬X ∧ ¬Y
{w | ∃v(R1(w, v) ∧ Y (v)} = ∅ {w | Y (w)} = {w | ∀z(R2(w, z) → X(z)))}
A Comprehensive Framework for Combined Decision Procedures – p. 33/4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... X ∧ Y ¬X ∧ Y ¬X ∧ ¬Y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... X ∧ Y ¬X ∧ Y ¬X ∧ ¬Y
{w | ∃v(R1(w, v) ∧ Y (v)} = ∅ {w | Y (w)} = {w | ∀z(R2(w, z) → X(z)))}
After taking a large disjoint copy...
A Comprehensive Framework for Combined Decision Procedures – p. 33/4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... ... ... X ∧ Y ¬X ∧ Y ¬X ∧ ¬Y
{w | ∃v(R1(w, v) ∧ Y (v)} = ∅ {w | Y (w)} = {w | ∀z(R2(w, z) → X(z)))}
After taking a large disjoint copy...
A Comprehensive Framework for Combined Decision Procedures – p. 33/4
The moral of all this:
A Comprehensive Framework for Combined Decision Procedures – p. 34/4
The moral of all this: arguments through disjoint unions (like the above one) are common folklore in semantically oriented literature: we just showed that they fit the requirements for Nelson-Oppen to apply;
A Comprehensive Framework for Combined Decision Procedures – p. 34/4
The moral of all this: arguments through disjoint unions (like the above one) are common folklore in semantically oriented literature: we just showed that they fit the requirements for Nelson-Oppen to apply; semantic and algebraic approaches to combined modal constraints are much more close than suspected: they both use the Nelson-Oppen schema, they only use different isomorphism theorems to prove completeness;
A Comprehensive Framework for Combined Decision Procedures – p. 34/4
The moral of all this: arguments through disjoint unions (like the above one) are common folklore in semantically oriented literature: we just showed that they fit the requirements for Nelson-Oppen to apply; semantic and algebraic approaches to combined modal constraints are much more close than suspected: they both use the Nelson-Oppen schema, they only use different isomorphism theorems to prove completeness; it can be shown that also results on decidability of monodic modal/temporal fragments can be rationally reconstructed by using Nelson-Oppen schema plus (another) isomorphism theorem;
A Comprehensive Framework for Combined Decision Procedures – p. 34/4
using just the two isomorphisms theorems explained in these slides,
combinations of modal/guarded/two-variable/A-Box/T-Box fragments
equational theories, etc. (see the FroCoS ’05 Proceedings or our extended Technical Report for details);
A Comprehensive Framework for Combined Decision Procedures – p. 35/4
using just the two isomorphisms theorems explained in these slides,
combinations of modal/guarded/two-variable/A-Box/T-Box fragments
equational theories, etc. (see the FroCoS ’05 Proceedings or our extended Technical Report for details); however one has to be very careful with types match, with the precise formulation of the fragments to be combined and with the choice of the appropriate isomorphism theorem ...
A Comprehensive Framework for Combined Decision Procedures – p. 35/4
using just the two isomorphisms theorems explained in these slides,
combinations of modal/guarded/two-variable/A-Box/T-Box fragments
equational theories, etc. (see the FroCoS ’05 Proceedings or our extended Technical Report for details); however one has to be very careful with types match, with the precise formulation of the fragments to be combined and with the choice of the appropriate isomorphism theorem ... ...this is why a rigorous comprehensive approach is needed!
A Comprehensive Framework for Combined Decision Procedures – p. 35/4
§ 8. Algebraic Fragments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 37 § 9. Comprehensive Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 41 § 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 45
A Comprehensive Framework for Combined Decision Procedures – p. 36/4
We need a language L for higher-order (Church’s style) type theory. This language should:
A Comprehensive Framework for Combined Decision Procedures – p. 37/4
We need a language L for higher-order (Church’s style) type theory. This language should: contain a stock of atomic types (including the truth-value type Ω);
A Comprehensive Framework for Combined Decision Procedures – p. 37/4
We need a language L for higher-order (Church’s style) type theory. This language should: contain a stock of atomic types (including the truth-value type Ω); be able to build types from atomic types by exponentiation;
A Comprehensive Framework for Combined Decision Procedures – p. 37/4
We need a language L for higher-order (Church’s style) type theory. This language should: contain a stock of atomic types (including the truth-value type Ω); be able to build types from atomic types by exponentiation; contain a stock of typed constants (including Boolean connectives like ⊤ : Ω, ¬ : Ω → Ω, ∧ : Ω → (Ω → Ω) and typed equalities =τ τ → (τ → Ω));
A Comprehensive Framework for Combined Decision Procedures – p. 37/4
We need a language L for higher-order (Church’s style) type theory. This language should: contain a stock of atomic types (including the truth-value type Ω); be able to build types from atomic types by exponentiation; contain a stock of typed constants (including Boolean connectives like ⊤ : Ω, ¬ : Ω → Ω, ∧ : Ω → (Ω → Ω) and typed equalities =τ τ → (τ → Ω)); contain an infinite supply of variables for each type;
A Comprehensive Framework for Combined Decision Procedures – p. 37/4
We need a language L for higher-order (Church’s style) type theory. This language should: contain a stock of atomic types (including the truth-value type Ω); be able to build types from atomic types by exponentiation; contain a stock of typed constants (including Boolean connectives like ⊤ : Ω, ¬ : Ω → Ω, ∧ : Ω → (Ω → Ω) and typed equalities =τ τ → (τ → Ω)); contain an infinite supply of variables for each type; be able to build typed terms from typed variables and constants by applying functional evaluation and λ-abstraction.
A Comprehensive Framework for Combined Decision Procedures – p. 37/4
Terms φ : Ω of type Ω are called formulae (quantifiers can be explicitly defined); if φ : Ω, we abbreviate λxτφ : τ → Ω as {xτ | φ} : τ → Ω. Types are often omitted for notational simplicity, if confusion does not arise.
A Comprehensive Framework for Combined Decision Procedures – p. 38/4
Terms φ : Ω of type Ω are called formulae (quantifiers can be explicitly defined); if φ : Ω, we abbreviate λxτφ : τ → Ω as {xτ | φ} : τ → Ω. Types are often omitted for notational simplicity, if confusion does not arise. Set-theoretic interpretations (called L-structures) for a typed language L can be defined in the obvious way. Of course, valid formulae are not axiomatizable, however this does not concern us because the idea is to restrict quite soon to more tractable fragments.
A Comprehensive Framework for Combined Decision Procedures – p. 38/4
Terms φ : Ω of type Ω are called formulae (quantifiers can be explicitly defined); if φ : Ω, we abbreviate λxτφ : τ → Ω as {xτ | φ} : τ → Ω. Types are often omitted for notational simplicity, if confusion does not arise. Set-theoretic interpretations (called L-structures) for a typed language L can be defined in the obvious way. Of course, valid formulae are not axiomatizable, however this does not concern us because the idea is to restrict quite soon to more tractable fragments. We won’t even need any deductive formalism (notice however that for instance βη-conversion is often required to apply out tool to concrete examples).
A Comprehensive Framework for Combined Decision Procedures – p. 38/4
Terms φ : Ω of type Ω are called formulae (quantifiers can be explicitly defined); if φ : Ω, we abbreviate λxτφ : τ → Ω as {xτ | φ} : τ → Ω. Types are often omitted for notational simplicity, if confusion does not arise. Set-theoretic interpretations (called L-structures) for a typed language L can be defined in the obvious way. Of course, valid formulae are not axiomatizable, however this does not concern us because the idea is to restrict quite soon to more tractable fragments. We won’t even need any deductive formalism (notice however that for instance βη-conversion is often required to apply out tool to concrete examples). Our philosophy is that of using higher-order formalism as a specification/interface language and to delegate deductive problems to specific modules.
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We give our crucial definitions:
A Comprehensive Framework for Combined Decision Procedures – p. 39/4
We give our crucial definitions: Definition 2. An algebraic fragment is a pair L, T, where L is a language and T is a recursive set of L-terms satisfying the following conditions:
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We give our crucial definitions: Definition 2. An algebraic fragment is a pair L, T, where L is a language and T is a recursive set of L-terms satisfying the following conditions: T is closed under composition, i.e. if u ∈ T, then u σ ∈ T, where σ : {x1 → t1, . . . , xn → tn} is a (type conformal) substitution such that ti ∈ T for all i = 1, . . . , n;
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We give our crucial definitions: Definition 2. An algebraic fragment is a pair L, T, where L is a language and T is a recursive set of L-terms satisfying the following conditions: T is closed under composition, i.e. if u ∈ T, then u σ ∈ T, where σ : {x1 → t1, . . . , xn → tn} is a (type conformal) substitution such that ti ∈ T for all i = 1, . . . , n; T contains domain variables, i.e. if τ is a type such that some variable of type τ occurs free in a term t ∈ T, then every variable of type τ belongs to T;
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We give our crucial definitions: Definition 2. An algebraic fragment is a pair L, T, where L is a language and T is a recursive set of L-terms satisfying the following conditions: T is closed under composition, i.e. if u ∈ T, then u σ ∈ T, where σ : {x1 → t1, . . . , xn → tn} is a (type conformal) substitution such that ti ∈ T for all i = 1, . . . , n; T contains domain variables, i.e. if τ is a type such that some variable of type τ occurs free in a term t ∈ T, then every variable of type τ belongs to T; T contains codomain variables, i.e. if t : τ belongs to T, then every variable of type τ belongs to T.
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Definition 3. An interpreted algebraic fragment is a triple Φ=L, T, S, where L, T is an algebraic fragment and S is a class of L-structures closed under isomorphisms.
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Definition 3. An interpreted algebraic fragment is a triple Φ=L, T, S, where L, T is an algebraic fragment and S is a class of L-structures closed under isomorphisms. Definition 4. Given an algebraic fragment Φ, a Φ-atom is an equation t1 = t2 between Φ-terms t1, t2 of the same type; a Φ-literal is a Φ-atom or a negation of a Φ-atom, a Φ-constraint is a finite conjunction of Φ-literals.
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Definition 3. An interpreted algebraic fragment is a triple Φ=L, T, S, where L, T is an algebraic fragment and S is a class of L-structures closed under isomorphisms. Definition 4. Given an algebraic fragment Φ, a Φ-atom is an equation t1 = t2 between Φ-terms t1, t2 of the same type; a Φ-literal is a Φ-atom or a negation of a Φ-atom, a Φ-constraint is a finite conjunction of Φ-literals. Definition 5. The constraint satisfiability problem for an interpreted algebraic fragment Φ=L, T, S is the problem of deciding whether a Φ-constraint is satisfiable in some structure A ∈ S.
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The above definitions allow the development of a Nelson-Oppen combination methodology.
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The above definitions allow the development of a Nelson-Oppen combination methodology. The combined fragment of the interpreted algebraic fragments Φ1 = L1, T1, S1 and Φ2 = L2, T2, S2 is the interpreted algebraic fragment Φ1 ⊕ Φ2 = L1 ⊕ L2, T1 ⊕ T2, S1 ⊕ S2 where
A Comprehensive Framework for Combined Decision Procedures – p. 41/4
The above definitions allow the development of a Nelson-Oppen combination methodology. The combined fragment of the interpreted algebraic fragments Φ1 = L1, T1, S1 and Φ2 = L2, T2, S2 is the interpreted algebraic fragment Φ1 ⊕ Φ2 = L1 ⊕ L2, T1 ⊕ T2, S1 ⊕ S2 where – L1 ⊕ L2 is L1 ∪ L2;
A Comprehensive Framework for Combined Decision Procedures – p. 41/4
The above definitions allow the development of a Nelson-Oppen combination methodology. The combined fragment of the interpreted algebraic fragments Φ1 = L1, T1, S1 and Φ2 = L2, T2, S2 is the interpreted algebraic fragment Φ1 ⊕ Φ2 = L1 ⊕ L2, T1 ⊕ T2, S1 ⊕ S2 where – L1 ⊕ L2 is L1 ∪ L2; – T1 ⊕ T2 is the smallest set of L1 ∪ L2-terms containing T1 ∪ T2 and making L1 ∪ L2, T1 ⊕ T2 an algebraic fragment;
A Comprehensive Framework for Combined Decision Procedures – p. 41/4
The above definitions allow the development of a Nelson-Oppen combination methodology. The combined fragment of the interpreted algebraic fragments Φ1 = L1, T1, S1 and Φ2 = L2, T2, S2 is the interpreted algebraic fragment Φ1 ⊕ Φ2 = L1 ⊕ L2, T1 ⊕ T2, S1 ⊕ S2 where – L1 ⊕ L2 is L1 ∪ L2; – T1 ⊕ T2 is the smallest set of L1 ∪ L2-terms containing T1 ∪ T2 and making L1 ∪ L2, T1 ⊕ T2 an algebraic fragment; – S1 ⊕ S2 contains those L1 ∪ L2-structures whose L1- and L2-reducts are in S1 and in S2, respectively.
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The remaining parts of our program are developed according to the expected lines. Given a constraint in a combined fragment, we run Nelson-Oppen on it.
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The remaining parts of our program are developed according to the expected lines. Given a constraint in a combined fragment, we run Nelson-Oppen on it. Purification : the definition of an algebraic fragment is designed precisely for purification to goes through (but little care is needed, see the extended Technical Report for details);
A Comprehensive Framework for Combined Decision Procedures – p. 42/4
The remaining parts of our program are developed according to the expected lines. Given a constraint in a combined fragment, we run Nelson-Oppen on it. Purification : the definition of an algebraic fragment is designed precisely for purification to goes through (but little care is needed, see the extended Technical Report for details); Propagation : this can be implemented in the usual ways provided the shared fragment is locally finite (the shared fragment contains the common terms in the common signature and is interpreted in the structures which are either a reduct of a structure from S1 or a reduct
A Comprehensive Framework for Combined Decision Procedures – p. 42/4
The remaining parts of our program are developed according to the expected lines. Given a constraint in a combined fragment, we run Nelson-Oppen on it. Purification : the definition of an algebraic fragment is designed precisely for purification to goes through (but little care is needed, see the extended Technical Report for details); Propagation : this can be implemented in the usual ways provided the shared fragment is locally finite (the shared fragment contains the common terms in the common signature and is interpreted in the structures which are either a reduct of a structure from S1 or a reduct
Completeness : here we need an isomorphism theorem induced by an extensible family of semantic operations (see Conference Paper/TR again for precise definitions of these intuitive notions).
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We can finally state a comprehensive combination result (we formulate it here in a weakened version for simplicity):
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We can finally state a comprehensive combination result (we formulate it here in a weakened version for simplicity): Theorem 6. Suppose that:
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We can finally state a comprehensive combination result (we formulate it here in a weakened version for simplicity): Theorem 6. Suppose that: the interpreted algebraic fragments Φ1, Φ2 have decidable constraint satisfiability problems;
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We can finally state a comprehensive combination result (we formulate it here in a weakened version for simplicity): Theorem 6. Suppose that: the interpreted algebraic fragments Φ1, Φ2 have decidable constraint satisfiability problems; the shared fragment Φ0 is effectively locally finite;
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We can finally state a comprehensive combination result (we formulate it here in a weakened version for simplicity): Theorem 6. Suppose that: the interpreted algebraic fragments Φ1, Φ2 have decidable constraint satisfiability problems; the shared fragment Φ0 is effectively locally finite; there is a family of semantic operations on Φ0 which are all Φ1- and Φ2-extensible and induce a Φ0-isomorphism theorem.
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We can finally state a comprehensive combination result (we formulate it here in a weakened version for simplicity): Theorem 6. Suppose that: the interpreted algebraic fragments Φ1, Φ2 have decidable constraint satisfiability problems; the shared fragment Φ0 is effectively locally finite; there is a family of semantic operations on Φ0 which are all Φ1- and Φ2-extensible and induce a Φ0-isomorphism theorem. Then the Nelson-Oppen procedure decides constraint satisfiability in the combined fragment Φ1 ⊕ Φ2.
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In the above formulation, Theorem 6 is more a schema for a combination result than a combination result itself. However many concrete combination problems falls within Theorem 6 (for instance, practically all the problems investigated or just mentioned in these slides).
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In the above formulation, Theorem 6 is more a schema for a combination result than a combination result itself. However many concrete combination problems falls within Theorem 6 (for instance, practically all the problems investigated or just mentioned in these slides). The amount of work needed to check Theorem 6 hypotheses to a concrete combination problem is variable:
A Comprehensive Framework for Combined Decision Procedures – p. 44/4
In the above formulation, Theorem 6 is more a schema for a combination result than a combination result itself. However many concrete combination problems falls within Theorem 6 (for instance, practically all the problems investigated or just mentioned in these slides). The amount of work needed to check Theorem 6 hypotheses to a concrete combination problem is variable: we have closely investigated two rather easy (but significant) cases in Parts I and II;
A Comprehensive Framework for Combined Decision Procedures – p. 44/4
In the above formulation, Theorem 6 is more a schema for a combination result than a combination result itself. However many concrete combination problems falls within Theorem 6 (for instance, practically all the problems investigated or just mentioned in these slides). The amount of work needed to check Theorem 6 hypotheses to a concrete combination problem is variable: we have closely investigated two rather easy (but significant) cases in Parts I and II; slightly more work is needed to extend the modally inspired notion of a ‘fusion’ to families of guarded/two-variable fragments;
A Comprehensive Framework for Combined Decision Procedures – p. 44/4
In the above formulation, Theorem 6 is more a schema for a combination result than a combination result itself. However many concrete combination problems falls within Theorem 6 (for instance, practically all the problems investigated or just mentioned in these slides). The amount of work needed to check Theorem 6 hypotheses to a concrete combination problem is variable: we have closely investigated two rather easy (but significant) cases in Parts I and II; slightly more work is needed to extend the modally inspired notion of a ‘fusion’ to families of guarded/two-variable fragments; substantial work is needed to recover decidability results for monodic modal/temporal fragments as transfer decidability results to combined interpreted algebraic fragments in our sense.
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We succeeded in giving a quite general formulation of Nelson-Oppen combination schema:
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We succeeded in giving a quite general formulation of Nelson-Oppen combination schema: a unique relatively simple Nelson-Oppen interface can accept as black boxes large kinds of specific modules;
A Comprehensive Framework for Combined Decision Procedures – p. 45/4
We succeeded in giving a quite general formulation of Nelson-Oppen combination schema: a unique relatively simple Nelson-Oppen interface can accept as black boxes large kinds of specific modules; in the extended Conference/TR version we provide also a modified Nelson-Oppen schema that works fairly even in case local finiteness
inputs ‘optimized’ residue enumerators and may not be terminating);
A Comprehensive Framework for Combined Decision Procedures – p. 45/4
We succeeded in giving a quite general formulation of Nelson-Oppen combination schema: a unique relatively simple Nelson-Oppen interface can accept as black boxes large kinds of specific modules; in the extended Conference/TR version we provide also a modified Nelson-Oppen schema that works fairly even in case local finiteness
inputs ‘optimized’ residue enumerators and may not be terminating); whereas soundness of the modified procedure is always guaranteed, completeness relies again on isomorphisms theorems.
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Thanks For Your Attention
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