A New Decision Procedure for Finite Sets and Cardinality Constraints in SMT

2016 ◽  
pp. 82-98 ◽  
Author(s):  
Kshitij Bansal ◽  
Andrew Reynolds ◽  
Clark Barrett ◽  
Cesare Tinelli
Keyword(s):  
Decision Procedure ◽  
Finite Sets ◽  
Cardinality Constraints

scholarly journals Integrating Cardinality Constraints into Constraint Logic Programming with Sets

2021 ◽  
pp. 1-33
Author(s):  
MAXIMILIANO CRISTIÁ ◽  
GIANFRANCO ROSSI
Keyword(s):  
Logic Programming ◽  
Decision Procedure ◽  
Software Verification ◽  
Empirical Evaluation ◽  
Constraint Logic Programming ◽  
Theory And Practice ◽  
Finite Sets ◽  
Cardinality Constraints ◽  
Programming Tool ◽  
Verification Conditions

Abstract 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 $$\{ log\} $$ 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. Under consideration in Theory and Practice of Logic Programming (TPLP)

On completely recursively enumerable classes and their key arrays

1956 ◽  
Vol 21 (3) ◽  
pp. 304-308 ◽  
Author(s):  
H. G. Rice
Keyword(s):  
Decision Procedure ◽  
Cardinal Number ◽  
Maximum Amount ◽  
Infinite Class ◽  
Finite Sets ◽  
Amount Of Information ◽  
Recursively Enumerable ◽  
Finite Set ◽  
Partial Recursive Functions ◽  
Set Up

The two results of this paper (a theorem and an example) are applications of a device described in section 1. Our notation is that of [4], with which we assume familiarity. It may be worth while to mention in particular the function Φ(n, x) which recursively enumerates the partial recursive functions of one variable, the Cantor enumerating functions J(x, y), K(x), L(x), and the classes F and Q of r.e. (recursively enumerable) and finite sets respectively.It is possible to “give” a finite set in a way which conveys the maximum amount of information; this may be called “giving explicitly”, and it requires that in addition to an effective enumeration or decision procedure for the set we give its cardinal number. It is sometimes desired to enumerate effectively an infinite class of finite sets, each given explicitly (e.g., [4] p. 360, or Dekker [1] p. 497), and we suggest here a device for doing this.We set up an effective one-to-one correspondence between the finite sets of non-negative integers and these integers themselves: the integer , corresponds to the set αi, = {a1, a2, …, an} and inversely. α0 is the empty set. Clearly i can be effectively computed from the elements of αi and its cardinal number.

A Decision Procedure for Bit-Vector Arithmetic

1998 ◽  
Author(s):  
Clark W. Barrett ◽  
David L. Dill ◽  
Jeremy R. Levitt
Keyword(s):  
Decision Procedure ◽  
Bit Vector

scholarly journals Algebraic singularities of scattering amplitudes from tropical geometry

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios
Keyword(s):  
Scattering Amplitudes ◽  
Natural Language ◽  
Tropical Geometry ◽  
Cluster Algebras ◽  
Finite Alphabet ◽  
Square Root ◽  
Finite Sets ◽  
Yang Mills ◽  
Square Roots ◽  
Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

An Automata-Theoretic Decision Procedure for Propositional Temporal Logic with Since and Until1

1992 ◽  
Vol 17 (3) ◽  
pp. 271-282
Author(s):  
Y.S. Ramakrishna ◽  
L.E. Moser ◽  
L.K. Dillon ◽  
P.M. Melliar-Smith ◽  
G. Kutty
Keyword(s):  
Temporal Logic ◽  
Decision Procedure ◽  
Linear Time ◽  
Computer Systems ◽  
Hierarchical Abstraction ◽  
Soundness And Completeness

We present an automata-theoretic decision procedure for Since/Until Temporal Logic (SUTL), a linear-time propositional temporal logic with strong non-strict since and until operators. The logic, which is intended for specifying and reasoning about computer systems, employs neither next nor previous operators. Such operators obstruct the use of hierarchical abstraction and refinement and make reasoning about concurrency difficult. A proof of the soundness and completeness of the decision procedure is given, and its complexity is analyzed.

Representation and manipulation of finite sets

1980 ◽  
Vol 10 (4) ◽  
pp. 8-12 ◽  
Author(s):  
B. L. McAllister
Keyword(s):  
Finite Sets

Deciding Clause Classes by Semantic Clash Resolution

1993 ◽  
Vol 18 (2-4) ◽  
pp. 163-182
Author(s):  
Alexander Leitsch
Keyword(s):  
Decision Procedure ◽  
Decision Procedures ◽  
General Concept ◽  
Complexity Measure ◽  
Syntactical Structure

It is investigated, how semantic clash resolution can be used to decide some classes of clause sets. Because semantic clash resolution is complete, the termination of the resolution procedure on a class Γ gives a decision procedure for Γ. Besides generalizing earlier results we investigate the relation between termination and clause complexity. For this purpose we define the general concept of atom complexity measure and show some general results about termination in terms of such measures. Moreover, rather than using fixed resolution refinements we define an algorithmic generator for decision procedures, which constructs appropriate semantic refinements out of the syntactical structure of the clause sets. This method is applied to the Bernays – Schönfinkel class, where it gives an efficient (resolution) decision procedure.

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