Introduction to algorithms
A Computing Procedure for Quantification Theory
Journal of the ACM (JACM)
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
Fast Decision Procedures Based on Congruence Closure
Journal of the ACM (JACM)
Variations on the Common Subexpression Problem
Journal of the ACM (JACM)
An algorithm for reasoning about equality
Communications of the ACM
A machine program for theorem-proving
Communications of the ACM
Chaff: engineering an efficient SAT solver
Proceedings of the 38th annual Design Automation Conference
Shostak's Congruence Closure as Completion
RTA '97 Proceedings of the 8th International Conference on Rewriting Techniques and Applications
Checking Satisfiability of First-Order Formulas by Incremental Translation to SAT
CAV '02 Proceedings of the 14th International Conference on Computer Aided Verification
A SAT Based Approach for Solving Formulas over Boolean and Linear Mathematical Propositions
CADE-18 Proceedings of the 18th International Conference on Automated Deduction
Abstract Congruence Closure and Specializations
CADE-17 Proceedings of the 17th International Conference on Automated Deduction
Knuth-Bendix completion of theories of commuting group endomorphisms
Information Processing Letters
SMT(CLU): a step toward scalability in system verification
Proceedings of the 2006 IEEE/ACM international conference on Computer-aided design
Electronic Notes in Theoretical Computer Science (ENTCS)
Type checking with open type functions
Proceedings of the 13th ACM SIGPLAN international conference on Functional programming
From KSAT to Delayed Theory Combination: Exploiting DPLL Outside the SAT Domain
FroCoS '07 Proceedings of the 6th international symposium on Frontiers of Combining Systems
Combining Proof-Producing Decision Procedures
FroCoS '07 Proceedings of the 6th international symposium on Frontiers of Combining Systems
CAV '08 Proceedings of the 20th international conference on Computer Aided Verification
Ground Interpolation for the Theory of Equality
TACAS '09 Proceedings of the 15th International Conference on Tools and Algorithms for the Construction and Analysis of Systems: Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2009,
Automated deduction for verification
ACM Computing Surveys (CSUR)
Combination of convex theories: Modularity, deduction completeness, and explanation
Journal of Symbolic Computation
A scalable decision procedure for fixed-width bit-vectors
Proceedings of the 2009 International Conference on Computer-Aided Design
Mining Propositional Simplification Proofs for Small Validating Clauses
Electronic Notes in Theoretical Computer Science (ENTCS)
Knuth--Bendix completion of theories of commuting group endomorphisms
Information Processing Letters
A tutorial on satisfiability modulo theories
CAV'07 Proceedings of the 19th international conference on Computer aided verification
Challenges in satisfiability modulo theories
RTA'07 Proceedings of the 18th international conference on Term rewriting and applications
Deciding equality in the constructor theory
TYPES'06 Proceedings of the 2006 international conference on Types for proofs and programs
QuickSpec: guessing formal specifications using testing
TAP'10 Proceedings of the 4th international conference on Tests and proofs
An interpolating decision procedure for transitive relations with uninterpreted functions
HVC'09 Proceedings of the 5th international Haifa verification conference on Hardware and software: verification and testing
Decision procedures for SAT, SAT modulo theories and beyond. the barcelogictools
LPAR'05 Proceedings of the 12th international conference on Logic for Programming, Artificial Intelligence, and Reasoning
Inference systems for logical algorithms
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Outsidein(x) modular type inference with local assumptions
Journal of Functional Programming - Dedicated to ICFP 2009
The algebra of equality proofs
RTA'05 Proceedings of the 16th international conference on Term Rewriting and Applications
From propositional satisfiability to satisfiability modulo theories
SAT'06 Proceedings of the 9th international conference on Theory and Applications of Satisfiability Testing
A progressive simplifier for satisfiability modulo theories
SAT'06 Proceedings of the 9th international conference on Theory and Applications of Satisfiability Testing
Modular SMT proofs for fast reflexive checking inside coq
CPP'11 Proceedings of the First international conference on Certified Programs and Proofs
A pearl on SAT and SMT solving in Prolog
Theoretical Computer Science
Journal of Automated Reasoning
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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 $\mathit{Explain}$ operation. First, two variations of union-find data structures with $\mathit{Explain}$ are introduced. Then, these are applied inside a congruence closure algorithm with $\mathit{Explain}$, where a k-step proof can be recovered in almost optimal time (quasi-linear in k), without increasing the overall O(n log n) runtime of the fastest known congruence closure algorithms. This non-trivial (ground) equational reasoning result has been quite intensively sought after (see, e.g., [SD99,dMRS04,KS04]), and moreover has important applications to verification.