Set Constraints in Some Equational Theories

Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. Recently, several algorithms for solving general systems of positive set constraints have appeared. In this paper we consider systems of mixed positive and negative constraints, which are considerably more expressive than positive constraints alone. We show that it is decidable whether a given such system has a solution. The proof involves a reduction to a number-theoretic decision problem that may be of independent interest.

A positive set constraint is of the form exp1\(\subseteq\)exp2, a negative set constraint is of the form exp1\(\subseteq\)exp2 where exp1 and exp2 are set expressions constructed using set variables, function symbols, and the set union, intersection and complement symbols. Decision algorithms for satisfiability of systems of positive and negative set constraints were given by Gilleron et al. [GTT93b], Aiken et al. [AKW93], and Charatonik and Pacholski [CP94]. In this paper, we study properties of the set of solutions of such systems and properties of solutions of interest for applications. The main decidability results are: for positive and negative set constraints, equivalence of systems is decidable, it is decidable whether or not a system has a unique solution; for positive set constraints, it is decidable whether or not a system has a least solution, it is decidable whether or not a system has a finite solution (i.e. the interpretation maps each set variable on a finite set).

Introduction to set constraint-based program analysis

The present study pertains to the finding of the lower bound solution, formulating it as a non‐linear programming problem using the generalized method developed by Lysmer with certain variations to incorporate the non‐linear no‐yield condition constraints directly in the analysis. The method considers the family of plane stress fields having the property that all stresses vary linearly within each triangular element of some mesh which covers the soil mass under study. For this type of stress field it is possible to express all equilibrium conditions as a set of linear constraints and the no‐yield as a set of non‐linear constraints. The boundary condition constraints may be of linear equality or inequality type. By expressing some of the design variables in terms of the remaining variables the linear equality constraints are implicity satisfied. Such a technique minimizes the complexity of the problem by eliminating the equality constraints and reduces the dimensionality of the problem, saving much, computational effort. The optimal lower bound is isolated by formulating it as a non‐linear programming (NLP) problem subjected to both linear and non‐linear inequality constraints. The sequential unconstrained minimization technique using the extended penalty function method as suggested by Kavlie has been used to isolate the optimal lower bound. The method has successfully been applied to the passive earth pressure and bearing capacity problem.Numerical results are obtained and compared with Lysmer's solution to show the effectiveness of the present approach.

Resource-constrained scheduling problems are commonly found in various areas, such as project management, manufacturing, transportation, software engineering, computer networks, and supply chain management. Its problem models involve a large number of constraints and discrete decision variables, including binary and integer. In effect, the representation of resource allocation, for instance, is often expressed using binary or integer decision variables to form several constraints according to the respective scheduling problem. It significantly increases the number of decision variables and constraints as the problem scales; such kind of traditional approaches based on operations research is insufficient. Therefore, a hybrid approach to decision support for resource-constrained scheduling problems which combines operation research (OR) and constraint logic programming (CLP) is proposed. Unlike OR-based approaches, declarative CLP provides a natural representation of different types of constraints. This approach provides: (a) decision support through the answers to the general and specific questions, (b) specification of the problem based on a set of facts and constraints, (c) reduction to the combinatorial solution space. To evaluate efficiency and applicability of the proposed hybrid approach and implementation platform, implementation examples of job-shop scheduling problem are presented separately for the three environments, i.e., Mathematical Programming (MP), CLP, and hybrid implementation platform.

Evaluating position uncertainties is a fundamental problem of wireless sensor network localization. A constraint set, including both positive and negative constraints, is constructed to bound sensor position. By projecting the feasible region of this constraint set onto a 2D plane, the feasible scope of sensor position is computed to evaluate node position uncertainty. The projection result, called feasible geographic region (FGR), is approximated by its inner and outer polygon. The polygon approximation will converge to the actual FGR if we incrementally add more polygon vertices. A distributed algorithm is proposed to compute FGR. Finally, we study the impact of node position uncertainty upon a typical network application, target event detection. The feasible scope of target event position is computed even though the sensor position is not certain.

Over the years the constraint-based method has been successfully applied to a wide range of problems in program analysis, from invariant generation to termination and non-termination proving. Quite often the semantics of the program under study as well as the properties to be generated belong to difference logic, i.e., the fragment of linear arithmetic where atoms are inequalities of the form \(u - v \le k\). However, so far constraint-based techniques have not exploited this fact: in general, Farkas’ Lemma is used to produce the constraints over template unknowns, which leads to non-linear SMT problems. Based on classical results of graph theory, in this paper we propose new encodings for generating these constraints when program semantics and templates belong to difference logic. Thanks to this approach, instead of a heavyweight non-linear arithmetic solver, a much cheaper SMT solver for difference logic or linear integer arithmetic can be employed for solving the resulting constraints. We present encouraging experimental results that show the high impact of the proposed techniques on the performance of the VeryMax verification system.

The incorporation of global program analysis into recent compilers for Constraint Logic Programming (CLP) languages has greatly improved the efficiency of compiled programs. We present a global analyser based on abstract interpretation. Unlike traditional optimizers, whose designs tend to be ad hoc, the analyser has been designed with flexibility in mind. The analyser is incremental, allowing substantial program transformations by a compiler without requiring redundant re-computation of analysis data. The analyser is also generic in that it can perform a large number of different program analyses. Furthermore, the analyser has an object-oriented design, enabling it to be adapted to different applications easily and allowing it to be used with various CLP languages with simple modifications. As an example of this generality, we sketch the use of the analyser in two different applications involving two distinct CLP languages: an optimizing compiler for CLP(R) programs and an application for detecting occur-check problems in Prolog programs. © 1998 John Wiley & Sons Ltd.

In this paper we present Cardinal, a general finite sets constraint solver just made publicly available in ECLiPSe Constraint System, suitable for combinatorial problem solving by exploiting inferences over sets cardinality. In fact, to deal with set variables and set constraints, existing set constraint solvers are not adequate to handle a number of problems, as they do not actively use important information about the cardinality of the sets, a key feature in such problems. Cardinal is formally presented as a set of rewriting rules on a constraint store and we illustrate its efficiency with experimental results. We show the importance of propagating constraints on sets cardinality, by comparing Cardinal with other solvers. Another contribution of this paper is on modelling: we focus essentially on digital circuits problems, for which we present new modelling approaches and prove that sets alone can be used to model these problems in a clean manner and solve them efficiently using Cardinal. Results on a set of diagnostic problems show that Cardinal obtains a speed up of about two orders of magnitude over Conjunto, a previous available set constraint solver, which uses a more limited amount of constraint propagation on cardinalities. Additionally, to further extend modelling capabilities and efficiency, we generalized Cardinal to actively consider constraints over set functions other than cardinality. The Cardinal version just released allows declaring union, minimum and maximum functions on set variables, and easily constraining those functions, letting Cardinal especial inferences efficiently take care of different problems. We describe such extensions and discuss its potentialities, which promise interesting research directions.

Constraint Logic Programming (CLP) and Hereditary Harrop formulas (HH) are two well known ways to enhance the expressivity of Horn clauses. In this paper, we present a novel combination of these two approaches. We show how to enrich the syntax and proof theory of HH with the help of a given constraint system, in such a way that the key property of HH as a logic programming language (namely, the existence of uniform proofs) is preserved. We also present a procedure for goal solving, showing its soundness and completeness for computing answer constraints. As a consequence of this result, we obtain a new strong completeness theorem for CLP that avoids the need to build disjunctions of computed answers, as well as a more abstract formulation of a known completeness theorem for HH.

A method of automatically constructing a system model or set of constraints from domain principles for solving the dynamic behavior of a mechanical system is presented. The following issues are emphasized: (1) the extraction of a necessary and sufficient set of system constraints from the fundamental principles and laws of physics based solely on the physical description of a given mechanical system, and (2) the modification of system constraints through time by detecting and identifying 'system discontinuities' due to collisions, separations. and other critical states associated with each system object. The first is accomplished by describing a mechanical system by a collection of object and interconnection primitives. which allows direct invocations and instantiations of all the relevant physics laws from the knowledge base. In consequence, the final constraints are selected on the basis of the A* algorithm. with heuristics providing the problem-solving expertise. The second is accomplished by monitoring whether the states of any individual subsystems evolve to system discontinuities represented by intra/inter subsystem critical states. >

Automated program repair relying on static analysis complements test-driven repair, since it does not require failing tests to repair a bug, and it avoids test-overfitting by considering program properties. Due to the rich variety and complexity of program analyses, existing static program repair techniques are tied to specific analysers, and thus repair only narrow classes of defects. To develop a general-purpose static program repair framework that targets a wide range of properties and programming languages, we propose to integrate program repair with Datalog-based analysis. Datalog solvers are programmable fixed point engines which can be used to encode many program analysis problems in a modular fashion. The program under analysis is encoded as Datalog facts, while the fixed point equations of the program analysis are expressed as recursive Datalog rules. In this context, we view repairing the program as modifying the corresponding Datalog facts. This is accomplished by a novel technique, symbolic execution of Datalog, that evaluates Datalog queries over a symbolic database of facts, instead of a concrete set of facts. The result of symbolic query evaluation allows us to infer what changes to a given set of Datalog facts repair the program so that it meets the desired analysis goals. We developed a symbolic executor for Datalog called Symlog, on top of which we built a repair tool SymlogRepair. We show the versatility of our approach on several analysis problems --- repairing null pointer exceptions in Java programs, repairing data leaks in Python notebooks, and repairing four types of security vulnerabilities in Solidity smart contracts.

Decision procedures underlie many program analysis problems. Traditional program analysis algorithms attempt to prove some property about a single, statically-defined program by generating a single constraint. Accordingly, traditional decision procedures take single constraints as input. Extending these traditional program analysis algorithms to reason about potentially infinite languages of programs (as generated by a given metaprogram) requires a new class of decision procedures that reason about languages of constraints. This paper introduces the parameterized class of validity checking problems that take as input a language generator $\mathcal{A}$. The parameters are: (1) the language formalism for $\mathcal{A}$, (2) the theory under which each string in the language of $\mathcal{A}$ is interpretted, and (3) the quantification (existential/universal) of the constraints in the language to which the validity property applies. We introduce such decision problems by presenting an algorithm that decides whether a given finite state automaton $\mathcal{A}$ generates any valid linear arithmetic constraints.

Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.KeywordsWord ProblemInverse SemigroupCayley GraphDependence GraphDependence RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Type Inference for Recursively Constrained Types and its Application to OOP