Logical aspects in the study of tree languages
Proc. of the conference on Ninth colloquium on trees in algebra and programming
Automatic proofs by induction in theories without constructors
Information and Computation
Automating inductionless induction using test sets
Journal of Symbolic Computation
Handbook of theoretical computer science (vol. B)
Testing for the ground (co-)reducibility property in term-rewriting systems
CAAP '90 Selected papers of the conference on Fifteenth colloquium on trees in algebra and programming
Deduction and declarative programming
Deduction and declarative programming
Linearizing term rewriting systems using test sets
Journal of Symbolic Computation - Special issue on conditional term rewriting systems
Decidability of regularity and related properties of ground normal form languages
Information and Computation
Automata for reduction properties solving
Journal of Symbolic Computation
Automated theorem proving by test set induction
Journal of Symbolic Computation
Semigroups and Combinatorial Applications
Semigroups and Combinatorial Applications
Automata, Languages, and Machines
Automata, Languages, and Machines
On proving inductive properties of abstract data types
POPL '80 Proceedings of the 7th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Equality and Disequality Constraints on Direct Subterms in Tree Automata
STACS '92 Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science
Mechanized Inductive Proof of Properties of a Simple Code Optimizer
TAPSOFT '95 Proceedings of the 6th International Joint Conference CAAP/FASE on Theory and Practice of Software Development
Computing Ground Reducability and Inductively Complete Positions
RTA '89 Proceedings of the 3rd International Conference on Rewriting Techniques and Applications
Towards an Efficient Construction of Test Sets for Deciding Ground Reducability
RTA '95 Proceedings of the 6th International Conference on Rewriting Techniques and Applications
Proving Inductive Theorems Based on Term Rewriting Systems
Proceedings of the International Workshop on Algebraic and Logic Programming
Proof by Induction Using Test Sets
Proceedings of the 8th International Conference on Automated Deduction
Proceedings of the 10th International Conference on Automated Deduction
Induction using Term Orderings
CADE-12 Proceedings of the 12th International Conference on Automated Deduction
Ground Reducibility is EXPTIME-complete
LICS '97 Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science
Regular Tree Languages And Rewrite Systems
Fundamenta Informaticae
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Finite test sets are a useful tool for deciding the membership problem for the universal closure of a given tree language, that is, for deciding whether a term has all its ground instances in the given language. A uniform test set for the universal closure must serve the following purpose: In order to decide membership of a term, it is sufficient to check whether all its test set instances belong to the underlying language. A possible application, and our main motivation, is ground reducibility, an essential concept for many approaches to inductive reasoning. Ground reducibility modulo some rewrite system is membership in the universal closure of the set of reducible ground terms. Here, test sets always exist, and several algorithmic approaches are known. The resulting sets, however, are often unnecessarily large. In this paper we consider regular languages and linear closure operators. We prove that universal as well as existential closure, defined analogously, preserve regularity. By relating test sets to tree automata and to appropriate congruence relations, we show how to characterize, how to compute, and how to minimize ground and non-ground test sets. In particular, optimal solutions now replace previous ad hoc approximations for the ground reducibility problem.