Semi-decision Procedure Research Articles

Verifying parallel programs with dynamic communication structures

We address the verification problem of networks of communicating pushdown systems modeling communicating parallel programs with procedure calls. Processes in such networks can read the control state of the other processes according to a given communication structure (specifying the observability rights between processes). The reachability problem of such models is undecidable in general. First, we define a class of networks that effectively preserves recognizability (hence, its reachability problem is decidable). Then, we consider networks where the communication structure can change dynamically during the execution according to a phase graph. The reachability problem for these dynamic networks being undecidable in general, we define a subclass for which it becomes decidable. Then, we consider reachability when the switches in the communication structures are bounded. We show that this problem is undecidable even for one switch. We define a natural class of models for which this problem is decidable. This class can be used in the definition of an efficient semi-decision procedure for the analysis of the general model of dynamic networks. Our techniques allowed to find bugs in two versions of a Windows NT Bluetooth driver.

A decision procedure for detecting atomicity violations for communicating processes with locks

The problem of interest is to verify data consistency of a concurrent Java program. In particular, we present a new decision procedure for verifying that a class of data races caused by inconsistent accesses on multiple fields of an object cannot occur (so-called atomic-set serializability). Atomic-set serializability generalizes the ordinary notion of a data race (i.e., inconsistent coordination of accesses on a single memory location) to a broader class of races that involve accesses on multiple memory locations. Previous work by some of the authors presented a technique to abstract a concurrent Java program into an EML program, a modeling language based on pushdown systems and a finite set of reentrant locks. Our previous work used only a semi-decision procedure, and hence provides a definite answer only some of the time. In this paper, we rectify this shortcoming by developing a decision procedure for verifying data consistency, i.e., atomic-set serializability, of an EML program. When coupled with the previous work, it provides a decision procedure for verifying data consistency of a concurrent Java program. We implemented the decision procedure, and applied it to detect both single-location and multi-location data races in models of concurrent Java programs. Compared with the prior method based on a semi-decision procedure, not only was the decision procedure 34 times faster overall, but the semi-decision procedure timed out on about 50% of the queries, whereas the decision procedure timed out on none of the queries.

Verifying Multithreaded Recursive Programs with Integer Variables

We consider the verification problem of programs containing the following complex features: (1) dynamic creation of parallel threads, (2) synchronisation between parallel threads via global variables, (3) (possibly recursive) procedure calls, and (4) integer variables. The configurations of such programs are represented by terms, and their transitions by term rewriting systems. The novelty of our modeling w.r.t. other existing works consists in explicitely modeling integer variables in the terms. We propose a semi-decision procedure that, in case of termination, checks whether an infinite set of configurations, represented by a regular tree language, is reachable from an infinite set of initial configurations of the program (usually represented by a set of non ground terms). As fas as we know, this is the first time that reachability between non-ground terms and regular tree languages is considered. We implemented our techniques in a tool, and tested it successfully on several examples.

On Solvability of a Decentralized Supervisory Control Problem With Communication

We study a decentralized control problem of discrete event systems with communication. We first give a general automata-theoretic formalism that enables us to handle various types of communication, including delay. The decentralized control problem is shown to be undecidable under this formalism. Next we propose a semi-decision procedure for computing finite-state controllers to achieve a given specification in the sense of bisimilarity between the controlled system and a given specification. Using this procedure, we prove decidability of the problem for two special cases, one is the case in which the communication behavior is given as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -bounded-delay communication, and the other is the case in which any cycle in the state transition diagram of the system contains an event observable by all controllers. We also show a method for optimizing controllers based on a graph problem.

Higher-Order Syntax and Saturation Algorithms for Hybrid Logic

We present modal logic on the basis of the simply typed lambda calculus with a system of equational deduction. Combining first-order quantification and higher-order syntax, we can maintain modal reasoning in terms of classical logic by remarkably simple means. Such an approach has been broadly uninvestigated, even though it has notable advantages, especially in the case of Hybrid Logic.We develop a tableau-like semi-decision procedure and subsequently a decision procedure for an alternative characterization of HL(@), a well-studied fragment of Hybrid Logic.With regards to deduction, our calculus simplifies in particular the treatment of identities. Moreover, labeling and access information are both internal and explicit, while in contrast traditional modal tableau calculi either rely on external labeling mechanisms or have to maintain an implicit accessibility relation by equivalent formulas.With regards to computational complexity, our saturation algorithm is optimal. In particular, this proves the satisfiability problem for HL(@) to be in PSPACE, a result that was previously not achieved by the saturation approach.

Complete symbolic reachability analysis using back-and-forth narrowing

We propose a method called back-and-forth narrowing for solving reachability goals of the form ( ∃ x → ) . t 1 → * t 1 ′ ∧ ⋯ ∧ t n → * t n ′ in general term rewrite systems. The method is a complete semi-decision procedure in the sense that it is guaranteed to find a solution when one exists, but in general it may not terminate when there are no solutions. The completeness result is very general in that it makes no assumptions about the given term rewrite system. Specifically, the rewrite rules need not be linear, confluent, or terminating, and can even have extra-variables in the right-hand side. Such generality is often essential while modeling concurrent systems or axiomatizing inference systems as rewrite rules, and in such applications back-and-forth narrowing can be used as a sound and complete technique for symbolic reachability analysis or as a deductive procedure for proving existential formulae.

On the Recognition of Algorithm Templates

This paper deals with the problem of deciding whether a System of Affine Recurrent Equations (SARE) is an instantiation of a SARE template. A solution to this problem would be a step toward algorithm template recognition and open new perspectives in program analysis, optimization and parallelization. The problem is known to be undecidable and we show that there exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations. This is a non-effective process which has been extensively studied. We then describe the limitations of our algorithm and point to unsolved problems.

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.

Verifying time Petri nets by linear programming

We propose an approach for solving some verification problems of time Petri nets using linear programming. The approach is based on the observation that for loop-closed time Petri nets, it is only necessary to investigate a finite prefix of an untimed run of the underlying Petri net. Using the technique we give solutions to the following problems: reachability, bounded delay timing analysis, and satisfication of linear duration properties. For all three problems we give algorithms that are decision procedures for loop-closed time Petri nets, and semi-decision procedures for general time Petri nets.

Bisimulation Equivalence Is Decidable for All Context-Free Processes

A result originally due to Baeten, Bergstra, and Klop shows that strong bisimulation equivalence for normed BPA processes is decidable. On the other hand, Huynh and Tian, and Groote and Hüttel, have proved that all other standard equivalences are undecidable for normed BPA and thus for BPA in general, The open problem remaining has been whether bisimulation is decidable for the full BPA language. In this paper, we answer this question in the affirmative, using a proof technique based on the proof by Caucal of the decidability of language equivalence for simple algebraic grammars. The decision procedure relies on our main result, extending that of Caucal, that the maximal bisimulation of any BPA transition graph is finitely representable as a Thue congruence. The decision procedure consists of two semi-decision procedures, one testing for non-bisimilarity and one testing for bisimilarity.

Towards a foundation of completion procedures as semidecision procedures

Completion procedures, originated from the seminal work of Knuth and Bendix, are well-known as procedures for generating confluent rewrite systems, i.e. decision procedures for equational theories. In this paper we present a new abstract framework for the utilization of completion procedures as semidecision procedures for theorem proving. The key idea in our approach in that a semidecision process should be target-oriented, i.e. keep into account the target theorem to be proved. For the inference rules of a completion procedure, we present target-oriented schemes of contraction inference rules, i.e. inference rules that delete sentences which are redundant for proving the target. For the search plan, we give a target-oriented, definition of fairness, according to which not all critical pairs need to be considered. We prove that our notion of fairness, together with the refutational completeness of the inference rules, is sufficient for a completion procedure to be a semidecision procedure. By relaxing the requirement of considering all critical pairs, our target-oriented framework should be more suitable for designing efficient procedures for theorem proving. The generation of decision procedures is included as a special side-effect and all the results of the classical approach to completion are re-obtained in our framework. The application of completion to disprove inductive conjectures, i.e. the so-called inductionless induction method, is also covered as a semidecision process. Finally, we present according to our framework, some equational completion procedures based on unfailing Knuth-Bendix completion.

Divergence of F⩽ type checking

System F ⩽ is an extension of second-order typed lambda calculus, where a subtype hierarchy among types is defined, and bounded second-order lambda abstraction is allowed. This language is a basis for much of the current research on integration of typed functional languages with subtypes and inheritance. An algorithm to perform type checking for F ⩽ expressions has been known since the language Fun was defined. The algorithm has been proved complete, by the author and Curien, which means that it is a semi-decision procedure for the type-checking problem. In this paper we show that this algorithm is not a decision procedure, by exhibiting a term which makes it diverge. This result was the basis of Pierce's proof of undecidability of typing for F ⩽. We study the behavior of the algorithm to show that our diverging judgement is in some sense contained in any judgement which makes the algorithm diverge. On the basis of this result, and of other results in the paper, we claim that the chances that the algorithm will loop while type-checking a “real program” are negligible. Hence, the undecidability of F ⩽ type-checking should not be considered as a reason to prevent the adoption of F ⩽ as a basis for defining programming languages of practical interest. Finally, we show the undecidability of an important subsystem of F ⩽.

Type inference with partial types

As a partial solution to the problem of combining static and dynamic typing in a language with parametric polymorphism, this paper introduces a new form of type expressions which represent partial type information. These expressions are meant to capture the type information statically derivable from heterogeneous objects. The new ground types form a semilattice of subtypes and require type inference based on inclusion constraints. We discuss the existence and form of principal types under this extension and present a semi-decision procedure for the complete type inference problem.

A rewrite-based type discipline for a subset of computer algebra

This paper is concerned with the type structure of a system including polymorphism, type properties and subtypes. This type system originates from computer algebra but it is not intended to be the solution of all type problems in this area. Types (or sets of types) are denoted by terms in some order-sorted algebra. ‡ We consider a rewrite relation in this algebra, which is intended to express subtyping. The relations between the semantics and the axiomatization are investigated. It is shown that the problem of type inference is undecidable but a narrowing strategy for semi-decision procedures is described and studied.

Inference rules and proof procedures for inequations

The negation of equality is an important relation that arises naturally in the study of equational programming languages and logic programming with equality. Proving and solving equations and inequations may also constitute subtasks in constraint logic programming. In this paper, we give forward (i.e., nonrefutational) techniques for proving the negation of equality in a theory. We develop a complete inference system to check whether an inequation is a logical consequence of a given system of equations and inequations. The inference system is used to develop a goal-directed semidecision procedure which uses a narrowing technique for proving inequations. A decision procedure is obtained when certain additional conditions are satisfied. The semidecision procedure for proving inequations is also modified to obtain a semidecision procedure for solving inequations in a theory, i.e., finding a substitution such that the corresponding instance of the given inequation is a logical consequence of the given system.

Decision procedures for elementary sublanguages of set theory VIII. A semidecision procedure for finite satisfiability of unqualified set‐theoretic formulae

Decision procedures for elementary sublanguages of set theory VIII. A semidecision procedure for finite satisfiability of unqualified set‐theoretic formulae

Resolution Strategies as Decision Procedures

The resolution principle, an automatic inference technique, is studied as a possible decision procedure for certain classes of first-order formulas. It is shown that most previous resolution strategies do not decide satisfiability even for “simple” solvable classes. Two new resolution procedures are described and are shown to be complete (i.e. semidecision procedures) in the general case and, in addition, to be decision procedures for successively wider classes of first-order formulas. These include many previously studied solvable classes. The proofs that a complete resolution procedure will always halt (without producing the empty clause) when applied to satisfiable formulas in certain classes provide new, and in some cases more enlightening, demonstrations of the solvability of these classes. A technique for constructing a model for a formula shown satisfiable in this way is also described.

On simplifying the matrix of a wff

In [3], [4], and [5] Joyce Friedman formulated and investigated certain rules which constitute a semi-decision procedure for wffs of first order predicate calculus in closed prenex normal form with prefixes of the form ∀x1 … ∀xκ∃y1 … ∃ym∀z1 … ∀zn. Given such a wff QM, where Q is the prefix and M is the matrix in conjunctive normal form, Friedman's rules can be used, in effect, to construct a matrix M* which is obtained from M by deleting certain conjuncts of M.

Joyce Friedman. A semi-decision procedure for the functional calculus. Journal of the Association for Computing Machinery, vol. 10 (1963), pp. 1–24. - Joyce Friedman. A computer program for a solvable case of the decision problem. Journal of the Association for Computing Machinery, vol. 10 (1963), pp. 348–356.

Joyce Friedman. A semi-decision procedure for the functional calculus. Journal of the Association for Computing Machinery, vol. 10 (1963), pp. 1–24. - Joyce Friedman. A computer program for a solvable case of the decision problem. Journal of the Association for Computing Machinery, vol. 10 (1963), pp. 348–356. - Volume 29 Issue 2