A BDD-representation for the logic of equality and uninterpreted functions (a full version with proofs).

The logic of equality and uninterpreted functions (EUF) has been proposed for processor verification. This paper presents a new data structure called Binary Decision Diagrams for representing EUF formulas (EUF-BDDs). We define EUF-BDDs similar to BDDs, but we allow equalities between terms as labels instead of Boolean variables. We provide an approach to build a reduced ordered EUF-BDD (EUF-ROBDD) and prove that every path to a leaf is satisfiable by construction. Moreover, EUF-ROBDDs are logically equivalent representations of EUF-formulae, so they can also be used to represent state spaces in symbolic model checking with dat

New results on rewrite-based satisfiability procedures

Program analysis and verification require decision procedures to reason on theories of data structures. Many problems can be reduced to the satisfiability of sets of ground literals in theory T. If a sound and complete inference system for first-order logic is guaranteed to terminate on T-satisfiability problems, any theorem-proving strategy with that system and a fair search plan is a T-satisfiability procedure. We prove termination of a rewrite-based first-order engine on the theories of records, integer offsets, integer offsets modulo and lists. We give a modularity theorem stating sufficient conditions for termination on a combinations of theories, given termination on each. The above theories, as well as others, satisfy these conditions. We introduce several sets of benchmarks on these theories and their combinations, including both parametric synthetic benchmarks to test scalability, and real-world problems to test performances on huge sets of literals. We compare the rewrite-based theorem prover E with the validity checkers CVC and CVC Lite. Contrary to the folklore that a general-purpose prover cannot compete with reasoners with built-in theories, the experiments are overall favorable to the theorem prover, showing that not only the rewriting approach is elegant and conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page

Two solutions to incorporate zero, successor and equality in binary decision diagrams

In this article we extend BDDs (binary decision diagrams) for plain propositional logic to the fragment of first order logic, consisting of quantifier free logic with equality, zero and successor. We insert equations with zero and successor in BDDs, and call these objects (0,S,=)-BDDs. We extend the notion of {em Ordered} BDDs in the presence of equality, zero and successor. (0,S,=)-BDDs can be transformed to equivalent Ordered (0,S,=)-BDD s by applying a number of rewrite rules. All paths in these extended OBDDs are satisfiable. The major advantage of transforming a formula to an equivalent Ordered (0,S,=)-BDD is that on the latter it can be observed in constant time whether the formula is a tautology, a contradiction, or just satisfiable

Path constraints in semistructured data

International audienceWe consider semistructured data as multirooted edge-labelled directed graphs, and path inclusion constraints on these graphs. A path inclusion constraint pnot precedes, equalsq is satisfied by a semistructured data if any node reached by the regular query p is also reached by the regular query q. In this paper, two problems are mainly studied: the implication problem and the problem of the existence of a finite exact model. - We give a new decision algorithm for the implication problem of a constraint pnot precedes, equalsq by a set of bounded path constraints pinot precedes, equalsui where p, q, and the pi's are regular path expressions and the ui's are words, improving in this particular case, the more general algorithms of S. Abiteboul and V. Vianu, and N. Alechina et al. In the case of a set of word equalities ui≡vi, we provide a more efficient decision algorithm for the implication of a word equality u≡v, improving the more general algorithm of P. Buneman et al. We prove that, in this case, implication for nondeterministic models is equivalent to implication for (complete) deterministic ones. - We introduce the notion of exact model: an exact model of a set of path constraints Click to view the MathML source satisfies the constraint pnot precedes, equalsq if and only if this constraint is implied by Click to view the MathML source. We prove that any set of constraints has an exact model and we give a decidable characterization of data which are exact models of bounded path inclusion constraints sets

Deciding the Word Problem for Ground Identities with Commutative and Extensional Symbols

The word problem for a finite set of ground identities is known to be decidable in polynomial time using congruence closure, and this is also the case if some of the function symbols are assumed to be commutative. We show that decidability in P is preserved if we add the assumption that certain function symbols f are extensional in the sense that f(s1,…,sn) ≈ f(t1,…,tn) implies s1 ≈ t1,…,sn ≈ tn. In addition, we investigate a variant of extensionality that is more appropriate for commutative function symbols, but which raises the complexity of the word problem to coNP

Проблема проверки выполнимости формул разрешимых теорий (обзор)

Данная работа посвящена анализу современного состояния исследований проблемы проверки выполнимости формул разрешимых теорий 1-го порядка на основе ѕленивого подходаї, т.е. на интеграции SAT-решателей с T -решателями. Охарактеризована структура SAT-решателя, построенного на основе управляющей конфликтами DPLL-процедуре. Рассмотрены основные понятия и принципы, используемые в процессе построения современных T -решателей. Изложение иллюстрируется на примере решателя, предназначенного для анализа выполнимости формул линейной целочисленной арифметики. Охарактеризованы методы организации взаимодействия SAT-решателей и T -решателей.Дану статтю присв’ячено аналiзу сучасного стану дослiджень проблеми перевiрки здiйсненостi формул теорiй 1-го порядку на основi ѕледащого пiдходуї, тобто на iнтеграцiї SAT-вирiшувачiв з T -вирiшувачами. Охарактеризовано структуру SAT-вирiшувача, який побудовано на основi керуючою конфлiктами DPLL-процедури. Розглянуто основнi поняття та принципи, якi використуються при побудовi сучасних T -вирiшувачiв. Викладення iлюструється на прикладi вирiшувача, який призначено для перевiрки здiйсненостi формул лiнiйної арифметики цiлих чисел. Охарактеризовано методи iнтеграцiї SAT-вирiшувачiв з T -вирiшувачами.Given paper is devoted to analysis of the state of the art for investigations of the problem of checking for satisfiability of formulae in decidable first-order theories on the base of the lazy approach, i.e. on integration of SAT-solvers with T -solvers. The structure of SAT-solver designed on the base of conflict driven DPLL procedure is characterized. Basic notions and principles applied in the process of elaboration of modern T -solvers are considered. They are presented in detail for example of a solver intended for checking of satisfiability for formulae of linear integer arithmetic. Methods of integration of SAT-solvers with T -solvers are characterized