Combining Decision Procedures Research Articles

Combining decision procedures by (model-)equality propagation

Formal methods in software and hardware design often generate formulas that need to be validated, either interactively or automatically. Among the automatic tools, SMT (Satisfiability Modulo Theories) solvers are particularly suitable to discharge such proof obligations, as their input language is equational logic with symbols from various useful decidable fragments such as uninterpreted symbols, linear arithmetic, and usual data-structures like arrays or lists. In this paper, we present an approach to combine decision procedures and propositional solvers into an SMT-solver, based not only on the exchange of deducible equalities between decision procedures, but also on the generation of model equalities by decision procedures. This extends nicely the classical Nelson–Oppen combination procedure in a simple platform to smoothly combine convex and non-convex theories. We show the soundness and completeness of this approach using an original abstract framework to represent and reason about SMT-solvers. We then describe an algorithmic translation of this method, implemented in the kernel of the veriT solver (Bouton et al. (2009)) [9].

Symbolic protocol analysis in the union of disjoint intruder theories: Combining decision procedures

Most of the decision procedures for symbolic analysis of protocols are limited to a fixed set of algebraic operators associated with a fixed intruder theory. Examples of such sets of operators comprise XOR, multiplication, abstract encryption/decryption. In this report we give an algorithm for combining decision procedures for arbitrary intruder theories with disjoint sets of operators, provided that solvability of ordered intruder constraints, a slight generalization of intruder constraints, can be decided in each theory. This is the case for most of the intruder theories for which a decision procedure has been given. In particular our result allows us to decide trace-based security properties of protocols that employ any combination of the above mentioned operators with a bounded number of sessions.

Combining Decision Procedures by (Model-)Equality Propagation

SMT (Satisfiability Modulo Theories) solvers are automatic verification engines suitable to discharge important classes of proof obligations generated in applying formal construction of software and hardware designs. In this paper, we present a new approach to combine decision procedures and propositional solvers into an SMT-solver. This approach is based on the generation of model equalities by decision procedures. We show the soundness and completeness of the proposed approach using an original abstract framework to represent and reason about SMT-solvers. We then present an algorithmic version of the new SMT-solving approach and discuss practical aspects of our implementation.

CoLoSS: The Coalgebraic Logic Satisfiability Solver

CoLoSS, the Coalgebraic Logic Satisfiability Solver, decides satisfiability of modal formulas in a generic and compositional way. It implements a uniform polynomial space algorithm to decide satisfiability for modal logics that are amenable to coalgebraic semantics. This includes e.g. the logics K, KD, Pauly's coalition logic, graded modal logic, and probabilistic modal logic. Logics are easily integrated into CoLoSS by providing a complete axiomatisation of their coalgebraic semantics in a specific format. Moreover, CoLoSS is compositional: it synthesises decision procedures for modular combinations of logics that include the fusion of two modal logics as a special case. One thus automatically obtains reasoning support e.g. for logics interpreted over probabilistic automata that combine non-determinism and probabilities in different ways.

Combining decision procedures for the reals

We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which "local" decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let <em>Tadd[Q]</em> be the first-order theory of the real numbers in the language of ordered groups, with negation, a constant <em>1</em>, and function symbols for multiplication by rational constants. Let <em>Tmult[Q]</em> be the analogous theory for the multiplicative structure, and let <em>T[Q]</em> be the union of the two. We show that although <em>T[Q]</em> is undecidable, the universal fragment of <em>T[Q]</em> is decidable. We also show that terms of <em>T[Q]</em>can fruitfully be put in a normal form. We prove analogous results for theories in which <em>Q</em> is replaced, more generally, by suitable subfields <em>F</em> of the reals. Finally, we consider practical methods of establishing quantifier-free validities that approximate our (impractical) decidability results.

A new combination procedure for the word problem that generalizes fusion decidability results in modal logics

Previous results for combining decision procedures for the word problem in the non-disjoint case do not apply to equational theories induced by modal logics—which are not disjoint for sharing the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other types of equational theories. In this paper, we present a new approach for combining decision procedures for the word problem in the non-disjoint case that applies to equational theories induced by modal logics, but is not restricted to them. The known fusion decidability results for modal logics are instances of our approach. However, even for equational theories induced by modal logics our results are more general since they are not restricted to so-called normal modal logics.

An Efficient Nelson-Oppen Decision Procedure for Difference Constraints over Rationals

Nelson and Oppen provided a methodology for modularly combining decision procedures for individual theories to construct a decision procedure for a combination of theories. In addition to providing a check for satisfiability, the individual decision procedures need to provide additional functionalities, including equality generation.In this paper, we propose a decision procedure for a conjunction of difference constraints over rationals (where the atomic formulas are of the form x⩽y+c or x<y+c). The procedure extends any negative cycle detection algorithm (like the Bellman-Ford algorithm) to generate (1) equalities between all pair of variables, (2) produce proofs and (3) generates models that can be extended by other theories in a Nelson-Oppen framework. All the operations mentioned above can be performed with only a linear overhead to the cycle detection algorithm, in the average case.

Strategies for combining decision procedures

Implementing efficient algorithms for combining decision procedures has been a challenge and their correctness precarious. In this paper we describe an inference system that has the classical Nelson–Oppen procedure at its core and includes several optimizations: variable abstraction with sharing, canonization of terms at the theory level, and Shostak's streamlined generation of new equalities for theories with solvers. The transitions of our system are fine-grained enough to model most of the mechanisms currently used in designing combination procedures. In particular, with a simple language of regular expressions we are able to describe several combination algorithms as strategies for our inference system, from the basic Nelson–Oppen to the very highly optimized one recently given by Shankar and Rueß. Presenting the basic system at a high level of generality and non-determinism allows transparent correctness proofs that can be extended in a modular fashion when new features are introduced in the system.

Combining Non-stably Infinite, Non-first Order Theories

A crucial step in the assertional verification of concurrent programs is deciding whether some sets of literals are satisfiable or not. In this context, the Nelson-Oppen combination scheme is often used. This scheme combines decision procedures for two disjoint theories into a decision procedure for the union of these theories. However, the standard version of the Nelson-Oppen technique tackles only one-sorted, stably infinite first-order theories. The scheme has previously been adapted to a many-sorted framework [C. Tinelli and C. G. Zarba. Combining decision procedures for theories in sorted logics. Technical Report 04-01, Department of Computer Science, The University of Iowa, Feb. 2004], and to handle non-stably infinite theories [C. Tinelli and C. G. Zarba. Combining non-stably infinite theories, in: I. Dahn and L. Vigneron, editors, Proceedings of the 4th International Workshop on First Order Theorem Proving, FTP'03 (Valencia, Spain), Electronic Notes in Theoretical Computer Science 86.1 (2003), Elsevier Science Publishers]. Those two enhancements were presented independently. We propose a unifying version in the continuity of both previous ones, which further relaxes the stably infinite requirement. Notably, some non-stably infinite theories can now be combined with the theory of arrays. Also, the combination scheme is presented here using a semantic notion of theory, allowing to handle non-first order theories.

Simplify: a theorem prover for program checking

This article provides a detailed description of the automatic theorem prover Simplify, which is the proof engine of the Extended Static Checkers ESC/Java and ESC/Modula-3. Simplify uses the Nelson--Oppen method to combine decision procedures for several important theories, and also employs a matcher to reason about quantifiers. Instead of conventional matching in a term DAG, Simplify matches up to equivalence in an E-graph, which detects many relevant pattern instances that would be missed by the conventional approach. The article describes two techniques, error context reporting and error localization, for helping the user to determine the reason that a false conjecture is false. The article includes detailed performance figures on conjectures derived from realistic program-checking problems.

Canonization for disjoint unions of theories

If there exist efficient procedures (canonizers) for reducing terms of two first-order theories to canonical form, can one use them to construct such a procedure for terms of the disjoint union of the two theories? We prove this is possible whenever the original theories are convex. As an application, we prove that algorithms for solving equations in the two theories (solvers) can not be combined in a similar fashion. These results are relevant to the widely used Shostak’s method for combining decision procedures for theories. They provide the first rigorous answers to the questions about the possibility of directly combining canonizers and solvers.

Accurate Theorem Proving for Program Verification

Symbolic software verification engines such as Slam and ESC/Java often use automatic theorem provers to implement forms of symbolic simulation. The theorem provers that are used, such as Simplify, usually combine decision procedures for the theories of uninterpreted functions, linear arithmetic, and sometimes bit vectors using techniques proposed by Nelson-Oppen or Shostak. Programming language constructs such as pointers, structures and unions are not directly supported by the provers, and are often encoded imprecisely using axioms and uninterpreted functions. In this paper we describe a more direct and accurate approach towards providing symbolic infrastructure for program verification engines. We propose the use of a theorem prover called Cogent, which provides better accuracy for ANSI-C expressions with the possibility of nested logic quantifiers. The prover’s implementation is based on a machinelevel interpretation of expressions into propositional logic. Cogent’s translation allows the program verification tools to better reason about finite machine-level variables, bit operations, structures, unions, references, pointers and pointer arithmetic. This paper also provides experimental evidence that the proposed approach is practical when applied to industrial program verification.

Quantifier Elimination and Provers Integration

We exploit quantifier elimination in the global design of combined decision and semi-decision procedures for theories over non-disjoint signatures, thus providing in particular extensions of Nelson-Oppen results.

Deciding the Word Problem in the Union of Equational Theories

The main contribution of this paper is a new method for combining decision procedures for the word problem in equational theories. In contrast to previous methods, this method is based on transformation rules. Furthermore, it is not limited to theories with disjoint signatures but it also applies to theories sharing constructors .

Tractable and intractable instances of combination problems for unification and disunification

The algorithms for combining decision procedures for (dis)unification in the union of arbitrary disjoint equational theories known from the literature generate an exponential number of 'pure' output problems over the component theories. Despite several optimization techniques, no general results are known that describe in which cases a polynomial-time combination procedure is (im)possible. In the first part of this paper we characterize a large class of equational theories E where E-unification with free function symbols is NP-hard and where a polynomial-time optimization of the combination algorithm is impossible (unless P = NP). This supports the conjecture that for the theories in this class there is no polynomial-time algorithm for combining E-unification with constants with free unification. In the second part of the paper we introduce another class of equational theories where (dis)unification algorithms can be combined deterministically and in polynomial time. Under reasonable assumptions on the form of the given unification procedure, each unitary, regular and collapse-free equational theory with a polynomial-time unification algorithm belongs to this class.

Simplification by Cooperating Decision Procedures

A method for combining decision procedures for several theories into a single decision procedure for their combination is described, and a simplifier based on this method is discussed. The simplifier finds a normal form for any expression formed from individual variables, the usual Boolean connectives, the equality predicate =, the conditional function if-then-else, the integers, the arithmetic functions and predicates +, -, and ≤, the Lisp functions and predicates car, cdr, cons, and atom, the functions store and select for storing into and selecting from arrays, and uninterpreted function symbols. If the expression is a theorem it is simplified to the constant true, so the simplifier can be used as a decision procedure for the quantifier-free theory containing these functions and predicates. The simplifier is currently used in the Stanford Pascal Verifier.