Set Unification

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature--and properly revisited and analyzed--and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI1 unification and a new simpler algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of Logic Programming (TPLP

Set-based Nondeterministic Declarative Programming in Singleton

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An Automatically Verified Prototype of the Tokeneer ID Station Specification

The Tokeneer project was an initiative set forth by the National Security Agency (NSA, USA) to be used as a demonstration that developing highly secure systems can be made by applying rigorous methods in a cost effective manner. Altran Praxis (UK) was selected by NSA to carry out the development of the Tokeneer ID Station. The company wrote a Z specification later implemented in the SPARK Ada programming language, which was verified using the SPARK Examiner toolset. In this paper, we show that the Z specification can be easily and naturally encoded in the {log} set constraint language, thus generating a functional prototype. Furthermore, we show that {log}'s automated proving capabilities can discharge all the proof obligations concerning state invariants as well as important security properties. As a consequence, the prototype can be regarded as correct with respect to the verified properties. This provides empirical evidence that Z users can use {log} to generate correct prototypes from their Z specifications. In turn, these prototypes enable or simplify some verificatio activities discussed in the paper

Combining Type Checking and Set Constraint Solving to Improve Automated Software Verification

In this paper we show how prescritive type checking and constraint solving can be combined to increase automation during software verification. We do so by defining a type system and implementing a typechecker for {log} (read `setlog'), a Constraint Logic Programming (CLP) language and satisfiability solver based on set theory. Hence, we proceed as follows: a) a type system for {log} is defined; b) the constraint solver is proved to be safe w.r.t. the type system; c) the implementation of a concrete typechecker is presented; d) the integration of type checking and set constraint solving to increase automation during software verification is discussed; and f) two industrial-strength case studies are presented where this combination is used with very good results

Declarative Programming with Intensional Sets in Java Using JSetL

Intensional sets are sets given by a property rather than by enumerating their elements. In previous work, we have proposed a decision procedure for a first-order logic language which provides Restricted Intensional Sets (RIS), i.e., a sub-class of intensional sets that are guaranteed to denote finite---though unbounded---sets. In this paper we show how RIS can be exploited as a convenient programming tool also in a conventional setting, namely, the imperative O-O language Java. We do this by considering a Java library, called JSetL, that integrates the notions of logical variable, (set) unification and constraints that are typical of constraint logic programming languages into the Java language. We show how JSetL is naturally extended to accommodate for RIS and RIS constraints, and how this extension can be exploited, on the one hand, to support a more declarative style of programming and, on the other hand, to effectively enhance the expressive power of the constraint language provided by the library

Proof Automation in the Theory of Finite Sets and Finite Set Relation Algebra

{log} ('setlog') is a satisfiability solver for formulas of the theory of finite sets and finite set relation algebra (FSTRA). As such, it can be used as an automated theorem prover (ATP) for this theory. {log} is able to automatically prove a number of FSTRA theorems, but not all of them. Nevertheless, we have observed that many theorems that {log} cannot automatically prove can be divided into a few subgoals automatically dischargeable by {log}. The purpose of this work is to present a prototype interactive theorem prover (ITP), called {log}-ITP, providing evidence that a proper integration of {log} into world-class ITP's can deliver a great deal of proof automation concerning FSTRA. An empirical evaluation based on 210 theorems from the TPTP and Coq's SSReflect libraries shows a noticeable reduction in the size and complexity of the proofs with respect to Coq

Automated Reasoning with Restricted Intensional Sets

Intensional sets, i.e., sets given by a property rather than by enumerating elements, are widely recognized as a key feature to describe complex problems (see, e.g., specification languages such as B and Z). Notwithstanding, very few tools exist supporting high-level automated reasoning on general formulas involving intensional sets. In this paper we present a decision procedure for a first-order logic language offering both extensional and (a restricted form of) intensional sets (RIS). RIS are introduced as first-class citizens of the language and set-theoretical operators on RIS are dealt with as constraints. Syntactic restrictions on RIS guarantee that the denoted sets are finite, though unbounded. The language of RIS, called L_RIS , is parametric with respect to any first-order theory X providing at least equality and a decision procedure for X-formulas. In particular, we consider the instance of L_RIS when X is the theory of hereditarily finite sets and binary relations. We also present a working implementation of this instance as part of the {log} tool and we show through a number of examples and two case studies that, although RIS are a subclass of general intensional sets, they are still sufficiently expressive as to encode and solve many interesting problems. Finally, an extensive empirical evaluation provides evidence that the tool can be used in practice

Integrating cardinality constraints into constraint logic programming with sets

Formal reasoning about finite sets and cardinality is important for many applications, including software verification, where very often one needs to reason about the size of a given data structure. The Constraint Logic Programming tool {log} (‘setlog’) provides a decision procedure for deciding the satisfiability of formulas involving very general forms of finite sets, although it does not provide cardinality constraints. In this paper we adapt and integrate a decision procedure for a theory of finite sets with cardinality into {log}. The proposed solver is proved to be a decision procedure for its formulas. Besides, the new CLP instance is implemented as part of the {log} tool. In turn, the implementation uses Howe and King’s Prolog SAT solver and Prolog’s CLP(Q) library, as an integer linear programming solver. The empirical evaluation of this implementation based on +250 real verification conditions shows that it can be useful in practice.Sociedad Argentina de Informática e Investigación Operativ

Pianeta Cuore 3.0 istruzioni per conoscerlo e mantenerlo sano

Il testo è caratterizzato dall’essenzialità, dalla schematicità e dalla chiarezza e nasce dalle domande più frequenti che i Pazienti e i familiari pongono sulle varie cardiopatie, prefazione all'edizione del 2010, e sugli aspetti diagnostico-terapeutici in Cardiologia. Indugia maggiormente sulla prevenzione cardiovascolare e le cardiopatie coronariche perché, attraverso la prevenzione è possibile ridurre i nuovi casi di malattia e le recidive di malattia, massimizzando i risultati delle procedure e tecnologie terapeutiche avanzate delle quali oggi disponiamo. Include un utilissimo glossario che illustra il significato delle parole più ricorrenti nel gergo cardiologico la cui comprensione non sempre è immediata