Introduction to algorithms
Fast Decision Procedures Based on Congruence Closure
Journal of the ACM (JACM)
Deciding Combinations of Theories
Journal of the ACM (JACM)
CHARME '99 Proceedings of the 10th IFIP WG 10.5 Advanced Research Working Conference on Correct Hardware Design and Verification Methods
Deciding Equality Formulas by Small Domains Instantiations
CAV '99 Proceedings of the 11th International Conference on Computer Aided Verification
ICS: Integrated Canonizer and Solver
CAV '01 Proceedings of the 13th International Conference on Computer Aided Verification
CVC: A Cooperating Validity Checker
CAV '02 Proceedings of the 14th International Conference on Computer Aided Verification
Checking Satisfiability of First-Order Formulas by Incremental Translation to SAT
CAV '02 Proceedings of the 14th International Conference on Computer Aided Verification
Automatic verification of Pipelined Microprocessor Control
CAV '94 Proceedings of the 6th International Conference on Computer Aided Verification
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
Checking validities and proofs with cvc and flea
Checking validities and proofs with cvc and flea
Proof-producing congruence closure
RTA'05 Proceedings of the 16th international conference on Term Rewriting and Applications
Knuth-Bendix completion of theories of commuting group endomorphisms
Information Processing Letters
Fast congruence closure and extensions
Information and Computation
Combining Proof-Producing Decision Procedures
FroCoS '07 Proceedings of the 6th international symposium on Frontiers of Combining Systems
Combination of convex theories: Modularity, deduction completeness, and explanation
Journal of Symbolic Computation
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
Challenges in satisfiability modulo theories
RTA'07 Proceedings of the 18th international conference on Term rewriting and applications
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
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Proofs of equalities may be built from assumptions using proof rules for reflexivity, symmetry, and transitivity. Reflexivity is an axiom proving x=x for any x; symmetry is a 1-premise rule taking a proof of x=y and returning a proof of y=x; and transitivity is a 2-premise rule taking proofs of x=y and y=z, and returning a proof of x=z. Define an equivalence relation to hold between proofs iff they prove a theorem in common. The main theoretical result of the paper is that if all assumptions are independent, this equivalence relation is axiomatized by the standard axioms of group theory: reflexivity is the unit of the group, symmetry is the inverse, and transitivity is the multiplication. Using a standard completion of the group axioms, we obtain a rewrite system which puts equality proofs into canonical form. Proofs in this canonical form use the fewest possible assumptions, and a proof can be canonized in linear time using a simple strategy. This result is applied to obtain a simple extension of the union-find algorithm for ground equational reasoning which produces minimal proofs. The time complexity of the original union-find operations is preserved, and minimal proofs are produced in worst-case time $O(n^{\textit{log}_2 3})$, where n is the number of expressions being equated. As a second application, the approach is used to achieve significant performance improvements for the CVC cooperating decision procedure.