678 research outputs found
Deciding KAT and Hoare Logic with Derivatives
Kleene algebra with tests (KAT) is an equational system for program
verification, which is the combination of Boolean algebra (BA) and Kleene
algebra (KA), the algebra of regular expressions. In particular, KAT subsumes
the propositional fragment of Hoare logic (PHL) which is a formal system for
the specification and verification of programs, and that is currently the base
of most tools for checking program correctness. Both the equational theory of
KAT and the encoding of PHL in KAT are known to be decidable. In this paper we
present a new decision procedure for the equivalence of two KAT expressions
based on the notion of partial derivatives. We also introduce the notion of
derivative modulo particular sets of equations. With this we extend the
previous procedure for deciding PHL. Some experimental results are also
presented.Comment: In Proceedings GandALF 2012, arXiv:1210.202
Order-Sorted Equality Enrichments Modulo Axioms
Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationally-defined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationally-defined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to effectively define a theory transformation epsilon bar right arrow epsilon(similar to) that extends an algebraic specification epsilon to a specification epsilon(similar to) having an equationally-defined equality predicate? This paper answers this question in the affirmative for a broad class of order-sorted conditional specifications epsilon that are sort-decreasing, ground confluent, and operationally terminating modulo axioms B and have a subsignature of constructors. The axioms B can consist of associativity, or commutativity, or associativity-commutativity axioms, so that the constructors are free modulo B. We prove that the transformation epsilon bar right arrow epsilon(similar to) preserves all the just-mentioned properties of epsilon. The transformation has been automated in Maude using reflection and is used as a component in many Maude formal tools. (C) 2014 Elsevier B.V. All rights reserved.This work has been supported in part by NSF Grants CCF 09-05584 and CNS 13-19109, the EU (FEDER) and the Spanish MINECO under Grants TIN 2010-21062-C02 and TIN 2013-45732-C4-1-P, and by the Generalitat Valenciana, ref. PROMETEO/2011/052. Raul Gutierrez is also partially supported by a Juan de la Cierva Fellowship from the Spanish MINECO, ref. JCI-2012-13528.Gutiérrez Gil, R.; Meseguer, J.; Rocha, C. (2015). Order-Sorted Equality Enrichments Modulo Axioms. Science of Computer Programming. 99:235-261. https://doi.org/10.1016/j.scico.2014.07.003S2352619
Rewriting Modulo SMT and Open System Analysis
This paper proposes rewriting modulo SMT, a new technique that
combines the power of SMT solving, rewriting modulo theories, and model checking.
Rewriting modulo SMT is ideally suited to model and analyze reachability
properties of infinite-state open systems, i.e., systems that interact with a nondeterministic
environment. Such systems exhibit both internal nondeterminism,
which is proper to the system, and external nondeterminism, which is due to the
environment. In a reflective formalism, such as rewriting logic, rewriting modulo
SMT can be reduced to standard rewriting. Hence, rewriting modulo SMT naturally
extends rewriting-based reachability analysis techniques, which are available
for closed systems, to open systems. The proposed technique is illustrated
with the formal analysis of: (i) a real-time system that is beyond the scope of
timed-automata methods and (ii) automatic detection of reachability violations in
a synchronous language developed to support autonomous spacecraft operations.NSF Grant CNS 13-19109 and NASA Research Cooperative Agreement No. NNL09AA00AOpe
Tactics for Reasoning modulo AC in Coq
We present a set of tools for rewriting modulo associativity and
commutativity (AC) in Coq, solving a long-standing practical problem. We use
two building blocks: first, an extensible reflexive decision procedure for
equality modulo AC; second, an OCaml plug-in for pattern matching modulo AC. We
handle associative only operations, neutral elements, uninterpreted function
symbols, and user-defined equivalence relations. By relying on type-classes for
the reification phase, we can infer these properties automatically, so that
end-users do not need to specify which operation is A or AC, or which constant
is a neutral element.Comment: 16
Constrained completion: Theory, implementation, and results
The Knuth-Bendix completion procedure produces complete sets of reductions but can not handle certain rewrite rules such as commutativity. In order to handle such theories, completion procedure were created to find complete sets of reductions modulo an equational theory. The major problem with this method is that it requires a specialized unification algorithm for the equational theory. Although this method works well when such an algorithm exists, these algorithms are not always available and thus alternative methods are needed to attack problems. A way of doing this is to use a completion procedure which finds complete sets of constrained reductions. This type of completion procedure neither requires specialized unification algorithms nor will it fail due to unorientable identities.
We present a look at complete sets of reductions with constraints, developed by Gerald Peterson, and the implementation of such a completion procedure for use with HIPER - a fast completion system. The completion procedure code is given and shown correct along with the various support procedures which are needed by the constrained system. These support procedures include a procedure to find constraints using the lexicographic path ordering and a normal form procedure for constraints.
The procedure has been implemented for use under the fast HIPER system, developed by Jim Christian, and thus is quick. We apply this new system, HIPER- extension, to attack a variety of word problems. Implementation alternatives are discussed, developed, and compared with each other as well as with the HIPER system.
Finally, we look at the problem of finding a complete set of reductions for a ternary boolean algebra. Given are alternatives to attacking this problem and the already known solution along with its run in the HIPER-extension system --Abstract, page iii
Formalization of Universal Algebra in Agda
In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin
Variant-Based Satisfiability
Although different satisfiability decision procedures
can be combined by algorithms such as those of Nelson-Oppen or
Shostak, current tools typically can only support a finite number of
theories to use in such combinations. To make SMT solving more
widely applicable, generic satisfiability algorithms that can
allow a potentially infinite number of decidable theories to be
user-definable, instead of needing to be built in by the
implementers, are highly desirable. This work studies how
folding variant narrowing, a generic
unification algorithm that offers
good extensibility in unification theory, can be extended to
a generic variant-based satisfiability algorithm for the initial
algebras of its user-specified input theories when such theories
satisfy Comon-Delaune's finite variant property (FVP) and some
extra conditions. Several, increasingly larger infinite classes of
theories whose initial algebras enjoy decidable variant-based satisfiability
are identified, and a method based on descent maps to bring other theories
into these classes and to improve the generic
algorithm's efficiency is proposed and illustrated with examples.Partially supported by NSF Grant CNS 13-19109.Ope
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