Collections, Cardinalities, and Relations (original) (raw)
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On Decision Procedures for Collections, Cardinalities, and Relations
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Logics that involve collections (sets, multisets), and cardinality constraints are useful for reasoning about unbounded data structures and concurrent processes. To make such logics more useful in verification this paper extends them with the ability to compute direct and inverse relation and function images. We establish decidability and complexity bounds for the extended logics.
Set Theory for Verification: I. From Foundations to Functions
A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations, and functions and discusses interactive proofs of Cantor''s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey''s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics.
Simulation and Verification of Synchronous Set Relations in Rewriting Logic
Formal Methods, Foundations and Applications, 2011
This paper presents a mathematical foundation and a rewriting logic infrastructure for the execution and property verification of synchronous set relations. The mathematical foundation is given in the language of abstract set relations. The infrastructure consists of an ordersorted rewrite theory in Maude, a rewriting logic system, that enables the synchronous execution of a set relation provided by the user. By using the infrastructure, existing algorithm verification techniques already available in Maude for traditional asynchronous rewriting, such as reachability analysis and model checking, are automatically available to synchronous set rewriting. The use of the infrastructure is illustrated with an executable operational semantics of a simple synchronous language and the verification of temporal properties of a synchronous system. the first set can be transformed into the second set by parallel atomic transformations. The selection of the redexes in the source set is done by the strategy. Strategies can be defined using priorities, which solve conflicts arising from the overlapping of atomic transitions. Section 2 presents, in an abstract setting, definitions of synchronous set relations, strategies, and priorities.
Automatic Decidability: A Schematic Calculus for Theories with Counting Operators
Many verification problems can be reduced to a satisfiability problem modulo theories. For building satisfiability procedures the rewriting-based approach uses a general calculus for equational reasoning named paramodulation. Schematic paramodulation, in turn, provides means to reason on the derivations computed by paramodulation. Until now, schematic paramodulation was only studied for standard paramodulation. We present a schematic paramodulation calculus modulo a fragment of arithmetics, namely the theory of Integer Offsets. This new schematic calculus is used to prove the decidability of the satisfiability problem for some theories equipped with counting operators. We illustrate our theoretical contribution on theories representing extensions of classical data structures, e.g., lists and records. An implementation within the rewriting-based Maude system constitutes a practical contribution. It enables automatic decidability proofs for theories of practical use. © Elena Tushkanov...
ArXiv, 2021
As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms x = y \ z, x 6= y \ z, and z = {x }, where x, y, z stand for variables ranging over the von Neumann universe of sets, into unquantified Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness and inclusion. Moreover, the result of each translation is a conjunction of literals of the forms x = y \ z, x 6= y \ z and of implications whose antecedents are isolated literals and whose consequents are either inclusions (strict or non-strict) between variables, or equalities between variables. Besides reflecting a simple and natural semantics, which ensures satisfiability-preservation, the proposed translation has quadratic algorithmic time-complexity, and bridges two languages both of w...
Counter Machines: Decidable Properties and Applications to Verification Problems
Lecture Notes in Computer Science, 2000
We study various generalizations of reversal-bounded multicounter machines and show that they have decidable emptiness, infiniteness, disjointness, containment, and equivalence problems. The extensions include allowing the machines to perform linear-relation tests among the counters and parameterized constants (e.g., "Is ¿Ü Ý ¾ ½• ¾ ½¾?", where Ü Ý are counters, and ½ ¾ are parameterized constants). We believe that these machines are the most powerful machines known to date for which these decision problems are decidable. Decidability results for such machines are useful in the analysis of reachability problems and the verification/debugging of safety properties in infinite-state transition systems. For example, we show that (binary, forward, and backward) reachability, safety, and invariance are solvable for these machines.
On the Decidability of Query Containment Under Constraints
Proceedings of the …, 1998
Query containment under constraints is the problem of checking whether for every database satisfying a given set of constraints, the result of one query is a subset of the result of another query. Recent research points out that this is a central problem in several database applications, and we address it within a setting where constraints are speci ed in the form of special inclusion dependencies over complex expressions, built by using intersection and di erence of relations, special forms of quanti cation, regular expressions over binary relations, and cardinality constraints. These types of constraints capture a great variety of data models, including the relational, the entity-relational, and the object-oriented model.
Decidability and proof systems for language-based noninterference relations
ACM SIGPLAN Notices, 2006
Noninterference is the basic semantical condition used to account for confidentiality and integrity-related properties in programming languages. There appears to be an at least implicit belief in the programming languages community that partial approaches based on type systems or other static analysis techniques are necessary for noninterference analyses to be tractable. In this paper we show that this belief is not necessarily true. We focus on the notion of strong low bisimulation proposed by Sabelfeld and Sands. We show that, relative to a decidable expression theory, strong low bisimulation is decidable for a simple parallel while-language, and we give a sound and relatively complete proof system for deriving noninterference assertions. The completeness proof provides an effective proof search strategy. Moreover, we show that common alternative noninterference relations based on traces or input-output relations are undecidable. The first part of the paper is cast in terms of multi-level security. In the second part of the paper we generalize the setting to accommodate a form of intransitive interference. We discuss the model and show how the decidability and proof system results generalize to this richer setting.
Verification of Data-Aware Processes via Array-Based Systems (Extended Version)
arXiv (Cornell University), 2018
We study verification over a general model of artifact-centric systems, to assess (parameterized) safety properties irrespectively of the initial database instance. We view such artifact systems as array-based systems, which allows us to check safety by adapting backward reachability, establishing for the first time a correspondence with model checking based on Satisfiability-Modulo-Theories (SMT). To do so, we make use of the modeltheoretic machinery of model completion, which surprisingly turns out to be an effective tool for verification of relational systems, and represents the main original contribution of this paper. In this way, we pursue a twofold purpose. On the one hand, we reconstruct (restricted to safety) the essence of some important decidability results obtained in the literature for artifact-centric systems, and we devise a genuinely novel class of decidable cases. On the other, we are able to exploit SMT technology in implementations, building on the well-known MCMT model checker for array-based systems, and extending it to make all our foundational results fully operational.