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Validity Checking for Combinations of Theories with Equality
, 1996
"... . An essential component in many verification methods is a fast decision procedure for validating logical expressions. This paper presents the algorithm used in the Stanford Validity Checker (SVC) which has been used to aid several realistic hardware verification efforts. The logic for this decision ..."
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Cited by 151 (23 self)
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. An essential component in many verification methods is a fast decision procedure for validating logical expressions. This paper presents the algorithm used in the Stanford Validity Checker (SVC) which has been used to aid several realistic hardware verification efforts. The logic for this decision procedure includes Boolean and uninterpreted functions and linear arithmetic. We have also successfully incorporated other interpreted functions, such as array operations and linear inequalities. The primary techniques which allow a complete and efficient implementation are expression sharing, heuristic rewriting, and congruence closure with interpreted functions. We discuss these techniques and present the results of initial experiments in which SVC is used as a decision procedure in PVS, resulting in dramatic speedups. 1 Introduction Decision procedures are emerging as a central component of formal verification systems. Such a procedure can be included as a component of a generalpurpos...
On the Complexity Analysis of Static Analyses
 Journal of the ACM
, 1999
"... . This paper argues that for many algorithms, and static analysis ..."
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Cited by 66 (3 self)
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. This paper argues that for many algorithms, and static analysis
On Shostak's Decision Procedure for Combinations of Theories
, 1996
"... Decision procedures are increasingly being employed for deciding or simplifying propositional combinations of ground equalities involving uninterpreted function symbols, linear arithmetic, arrays, and other theories. Two approaches for constructing decision procedures for combinations of ground theo ..."
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Cited by 53 (8 self)
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Decision procedures are increasingly being employed for deciding or simplifying propositional combinations of ground equalities involving uninterpreted function symbols, linear arithmetic, arrays, and other theories. Two approaches for constructing decision procedures for combinations of ground theories were pioneered in the late seventies. In the approach of Nelson and Oppen, decision procedures for two disjoint theories are combined by introducing variables to name subterms and iteratively propagating any deduced equalities between variables from one theory to another. Shostak employs a different approach that works far more efficiently in practice. He uses an optimized implementation of the congruence closure procedure for ground equality over uninterpreted function symbols to combine theories that are canonizable and algebraically solvable. Many useful theories have these properties. Shostak's algorithm is subtle and complex and his description of this procedure is lacking in ri...
Deconstructing Shostak
, 2002
"... Decision procedures for equality in a combination of theories are at the core of a number of verification systems. Shostak's decision procedure for equality in the combination of solvable and canonizable theories has been around for nearly two decades. Variations of this decision procedure have been ..."
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Cited by 36 (3 self)
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Decision procedures for equality in a combination of theories are at the core of a number of verification systems. Shostak's decision procedure for equality in the combination of solvable and canonizable theories has been around for nearly two decades. Variations of this decision procedure have been implemented in a number of systems including STP, Ehdm, PVS, STeP, and SVC. The algorithm is quite subtle and a correctness argument for it has remained elusive. Shostak's algorithm and all previously published variants of it yield incomplete decision procedures. We describe a variant of Shostak's algorithm along with proofs of termination, soundness, and completeness.
Shostak's Congruence Closure as Completion
 Proceedings of the 8th International Conference on Rewriting Techniques and Applications, volume 1232 of Lecture Notes in Computer Science
, 1997
"... . Shostak's congruence closure algorithm is demystified, using the framework of ground completion on (possibly nonterminating, nonreduced) rewrite rules. In particular, the canonical rewriting relation induced by the algorithm on ground terms by a given set of ground equations is precisely cons ..."
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Cited by 33 (3 self)
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. Shostak's congruence closure algorithm is demystified, using the framework of ground completion on (possibly nonterminating, nonreduced) rewrite rules. In particular, the canonical rewriting relation induced by the algorithm on ground terms by a given set of ground equations is precisely constructed. The main idea is to extend the signature of the original input to include new constant symbols for nonconstant subterms appearing in the input. A byproduct of this approach is (i) an algorithm for associating a confluent rewriting system with possibly nonterminating ground rewrite rules, and (ii) a new quadratic algorithm for computing a canonical rewriting system from ground equations. 1 Introduction Equality reasoning has been found critical in many applications including compiler optimization, functional languages, and reasoning about data bases, most importantly, reasoning about different aspects of software and hardware  circuits, programs and specifications. Signific...
Proofproducing Congruence Closure
 16th International Conference on Rewriting Techniques and Applications
, 2005
"... www.lsi.upc.es/{~roberto,~oliveras} Abstract. Many applications of congruence closure nowadays require the ability of recovering, among the thousands of input equations, the small subset that caused the equivalence of a given pair of terms. For this purpose, here we introduce an incremental congruen ..."
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Cited by 32 (2 self)
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www.lsi.upc.es/{~roberto,~oliveras} Abstract. Many applications of congruence closure nowadays require the ability of recovering, among the thousands of input equations, the small subset that caused the equivalence of a given pair of terms. For this purpose, here we introduce an incremental congruence closure algorithm that has an additional Explain operation. First, two variations of unionfind data structures with Explain are introduced. Then, these are applied inside a congruence closure algorithm with Explain, whereakstep proof can be recovered in almost optimal time (quasilinear in k), without increasing the overall O(n log n)runtime of the fastest known congruence closure algorithms. This nontrivial (ground) equational reasoning result has been quite intensively sought after (see, e.g., [SD99, dMRS04, KS04]), and moreover has important applications to verification.
New Results on Local Inference Relations
 In Principles of Knolwedge Representation and Reasoning: Proceedings of the Third International Conference
, 1992
"... We consider the concept of a local set of inference rules. A local rule set can be automatically transformed into a rule set for which bottom up evaluation terminates in polynomial time. The local rule set transformation gives polynomial time evaluation strategies for a large variety of rule sets th ..."
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Cited by 28 (9 self)
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We consider the concept of a local set of inference rules. A local rule set can be automatically transformed into a rule set for which bottom up evaluation terminates in polynomial time. The local rule set transformation gives polynomial time evaluation strategies for a large variety of rule sets that can not be given terminating evaluation strategies by any other known automatic technique. This paper discusses three new results. First, it is shown that every polynomial time predicate can be defined by an (unstratified) local rule set. Second, a new machine recognizable subclass of the local rule sets is identified. Finally we show that locality, as a property of rule sets, is undecidable in general. This paper appeared in KR92. A postscript electronic source for this paper can be found in ftp.ai.mit.edu:/pub/dam/kr92.ps. A bibtex reference can be found in internet file ftp.ai.mit.edu:/pub/dam/dam.bib. 1 INTRODUCTION Under what conditions does a given set of inference rules define ...
Combining Shostak Theories
, 2002
"... Ground decision procedures for combinations of theories are used in many systems for automated deduction. There are two basic paradigms for combining decision procedures. The NelsonOppen method combines decision procedures for disjoint theories by exchanging equality information on the shared vari ..."
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Cited by 25 (2 self)
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Ground decision procedures for combinations of theories are used in many systems for automated deduction. There are two basic paradigms for combining decision procedures. The NelsonOppen method combines decision procedures for disjoint theories by exchanging equality information on the shared variables. In Shostak's method, the combination of the theory of pure equality with canonizable and solvable theories is decided through an extension of congruence closure that yields a canonizer for the combined theory. Shostak's original presentation, and others that followed it, contained serious errors which were corrected for the basic procedure by the present authors. Shostak also claimed that it was possible to combine canonizers and solvers for disjoint theories. This claim is easily verifiable for canonizers, but is unsubstantiated in the case of solvers. We show how our earlier procedure can be extended to combine multiple disjoint canonizable, solvable theories within the Shostak framework.