scholarly journals Reasoning with Triggers

10.29007/3c1n ◽  
2018 ◽  
Author(s):  
Claire Dross ◽  
Sylvain Conchon ◽  
Johannes Kanig ◽  
Andrei Paskevich
Keyword(s):  
Decision Procedure ◽  
Decision Procedures ◽  
Description Language ◽  
Smt Solver ◽  
First Order ◽  
Smt Solvers ◽  
Precise Sense ◽  
Universal Quantifiers ◽  
Internal Knowledge

SMT solvers can decide the satisfiability of ground formulas modulo a combination ofbuilt-in theories. Adding a built-in theory to a given SMT solver is a complex and time consuming task that requires internal knowledge of the solver. However, many theories can be easily expressed using first-order formulas. Unfortunately, since universal quantifiers are not handled in a complete way by SMT solvers, these axiomatics cannot be used as decision procedures.In this paper, we show how to extend a generic SMT solver to accept a custom theory description and behave as a decision procedure for that theory, provided that the described theory is complete and terminating in a precise sense. The description language consists of first-order axioms with triggers, an instantiation mechanism that is found in many SMT solvers. This mechanism, which usually lacks a clear semantics in existing languages and tools, is rigorously defined here; this definition can be used to prove completeness and termination of the theory. We demonstrate using the theory of arrays, how such proofs can be achieved in our formalism.

scholarly journals Complete decision procedure for the theory of bounded pointer arithmetic based on quantifier instantiation and SMT

2021 ◽  
Vol 33 (4) ◽  
pp. 177-194
Author(s):  
Rafael Faritovich Sadykov ◽  
Mikhail Usamovich Mandrykin
Keyword(s):  
Program Verification ◽  
Source Code ◽  
Decision Procedures ◽  
C Programs ◽  
Smt Solver ◽  
Smt Solvers ◽  
Relevant Logics ◽  
Soundness And Completeness ◽  
C Program ◽  
Quantifier Instantiation

The process of developing C programs is quite often prone to errors related to the uses of pointer arithmetic and operations on memory addresses. This promotes a need in developing various tools for automated program verification. One of the techniques frequently employed by those tools is invocation of appropriate decision procedures implemented within existing SMT-solvers. But at the same time both the SMT standard and most existing SMT-solvers lack the relevant logics (combinations of logical theories) for directly and precisely modelling the semantics of pointer operations in C. One of the possible ways to support these logics is to implement them in an SMT solver, but this approach can be time-consuming (as requires modifying the solver’s source code), inflexible (introducing any changes to the theory’s signature or semantics can be unreasonably hard) and limited (every solver has to be supported separately). Another way is to design and implement custom quantifier instantiation strategies. These strategies can be then used to translate formulas in the desired theory combinations to formulas in well-supported decidable logics such as QF_UFLIA. In this paper, we present an instantiation procedure for translating formulas in the theory of bounded pointer arithmetic into the QF_UFLIA logic. We formally proved soundness and completeness of our instantiation procedure in Isabelle/HOL. The paper presents an informal description of this proof of the proposed procedure. The theory of bounded pointer arithmetic itself was formulated based on known errors regarding the correct use of pointer arithmetic operations in industrial code as well as the semantics of these operations specified in the C standard. Similar procedure can also be defined for a practically relevant fragment of the theory of bit vectors (monotone propositional combinations of equalities between bitwise expressions). Our approach is sufficient to obtain efficient decision procedures implemented as Isabelle/HOL proof methods for several decidable logical theories used in C program verification by relying on the existing capabilities of well-known SMT solvers, such as Z3 and proof reconstruction capabilities of the Isabelle/HOL proof assistant.

scholarly journals Experiments on the feasibility of using a floating-point simplex in an SMT solver

10.29007/j7x4 ◽  
2018 ◽  
Author(s):  
Diego Caminha Barbosa de Oliveira ◽  
David Monniaux
Keyword(s):  
Simplex Algorithm ◽  
Decision Procedures ◽  
Linear Constraints ◽  
Decision Problems ◽  
Floating Point ◽  
Protocol Verification ◽  
Smt Solver ◽  
Smt Solvers ◽  
State Of Art ◽  
Floating Point Implementation

SMT solvers use simplex-based decision procedures to solve decision problems whose formulas are quantifier-free and atoms are linear constraints over the rationals. State-of-art SMT solvers use rational (exact) simplex implementations, which have shown good performance for typical software, hardware or protocol verification problems over the years.Yet, most other scientific and technical fields use (inexact) floating-point computations, which are deemed far more efficient than exact ones.It is therefore tempting to use a floating-point simplex implementation inside an SMT solver, though special precautions must be taken to avoid unsoundness.In this work, we describe experimental results, over common benchmarks (SMT-LIB) of the integration of a mature floating-point implementation of the simplex algorithm (GLPK) into an existing SMT solver (OpenSMT).We investigate whether commonly cited reasons for and against the use of floating-point truly apply to real cases from verification problems.

scholarly journals AVATAR Modulo Theories

10.29007/k6tp ◽  
2018 ◽  
Author(s):  
Giles Reger ◽  
Nikolaj Bjorner ◽  
Martin Suda ◽  
Andrei Voronkov
Keyword(s):  
Theorem Proving ◽  
Satisfiability Modulo Theories ◽  
Theorem Prover ◽  
Ground Theory ◽  
Smt Solver ◽  
First Order ◽  
Proof Search ◽  
Smt Solvers ◽  
Sat Solver ◽  
A New Technique

This paper introduces a new technique for reasoning with quantifiers and theories. Traditionally, first-order theorem provers (ATPs) are well suited to reasoning with first-order problems containing many quantifiers and satisfiability modulo theories (SMT) solvers are well suited to reasoning with first-order problems in ground theories such as arithmetic. A recent development in first-order theorem proving has been the AVATAR architecture which uses a SAT solver to guide proof search based on a propositional abstraction of the first-order clause space. The approach turns a single proof search into a sequence of proof searches on (much) smaller sub-problems. This work extends the AVATAR approach to use a SMT solver in place of the SAT solver, with the effect that the first-order solver only needs to consider ground-theory-consistent sub-problems. The new architecture has been implemented using the Vampire theorem prover and Z3 SMT solver. Our experimental results, and the results of recent competitions, show that such a combination can be highly effective.

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.

scholarly journals Making Automatic Theorem Provers more Versatile

10.29007/n6j7 ◽  
2018 ◽  
Author(s):  
Simon Cruanes
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Research Directions ◽  
First Order ◽  
Theorem Provers ◽  
Smt Solvers ◽  
Automatic Theorem Provers ◽  
Automatic Theorem

We argue that automatic theorem provers should become more versatile and should be able to tackle problems expressed in richer input formats. Salient research directions include (i) developing tight combinations of SMT solvers and first-order provers; (ii) adding better handling of theories in first-order provers; (iii) adding support for inductive proving; (iv) adding support for user-defined theories and functions; and (v) bringing to the provers some basic abilities to deal with logics beyond first-order, such as higher-order logic.

Trial and error predicates and the solution to a problem of Mostowski

1965 ◽  
Vol 30 (1) ◽  
pp. 49-57 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Finite Number ◽  
Correct Answer ◽  
Decision Procedure ◽  
Finite Sequence ◽  
Decision Procedures ◽  
Turing Machines ◽  
Trial And Error ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Empirical Means

The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the following question: we know what sets are “decidable” — namely, the recursive sets (according to Church's Thesis). But what happens if we modify the notion of a decision procedure by (1) allowing the procedure to “change its mind” any finite number of times (in terms of Turing Machines: we visualize the machine as being given an integer (or an n-tuple of integers) as input. The machine then “prints out” a finite sequence of “yesses” and “nos”. The last “yes” or “no” is always to be the correct answer.); and (2) we give up the requirement that it be possible to tell (effectively) if the computation has terminated? I.e., if the machine has most recently printed “yes”, then we know that the integer put in as input must be in the set unless the machine is going to change its mind; but we have no procedure for telling whether the machine will change its mind or not.The sets for which there exist decision procedures in this widened sense are decidable by “empirical” means — for, if we always “posit” that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)

A NOTE ON THE ASYMPTOTIC PROBABILITIES OF EXISTENTIAL SECOND-ORDER MINIMAL GÖDEL SENTENCES WITH EQUALITY

1995 ◽  
Vol 06 (04) ◽  
pp. 339-351
Author(s):  
WIESŁAW SZWAST
Keyword(s):  
Dense Subset ◽  
Unit Interval ◽  
Second Order ◽  
Existential Quantifier ◽  
First Order ◽  
Order Part ◽  
Universal Quantifiers

The minimal Gödel class is the class of first-order prenex sentences whose quantifier prefix consists of two universal quantifiers followed by just one existential quantifier. We prove that asymptotic probabilities of existential second-order sentences, whose first-order part is in the minimal Gödel class, form a dense subset of the unit interval.

scholarly journals A Decision Procedure for (Co)datatypes in SMT Solvers

2016 ◽  
Vol 58 (3) ◽  
pp. 341-362 ◽  
Author(s):  
Andrew Reynolds ◽  
Jasmin Christian Blanchette
Keyword(s):  
Decision Procedure ◽  
Smt Solvers
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