Partial Automata and Finitely Generated Congruences: An Extension of Nerode’s Theorem

Let T_Sigma be the set of ground terms over a finite ranked alphabet Sigma. We define partial autornata on T_Sigma and prove that the finitely generated congruences on T_Sigma are in one-to one correspondence (up to isomorphism) with the finite partial automata on Sigma with no inaccessible and no inessential states. We give an application in term rewriting: every ground term rewrite system has a canonical equivalent system that can be constructed in polynomial time.

[1]  Eugene M. Luks.  Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time , 1980, FOCS.

[2]  A. Nerode,et al.  Linear automaton transformations , 1958 .

[3]  Walter Scott Brainerd,et al.  TREE GENERATING SYSTEMS AND TREE AUTOMATA , 1967 .

[4]  Walter S. Brainerd,et al.  The Minimalization of Tree Automata , 1968, Inf. Control..

[5]  Michael A. Arbib,et al.  Algebra Automata I: Parallel Programming as a Prolegomena to the Categorical Approach , 1968, Inf. Control..

[6]  Sándor Vágvölgyi,et al.  Congruential tree languages are the same as recognizable tree languages - a proof for a theorem of D. Kozen , 1989, Bull. EATCS.

[7]  George Gratzer,et al.  Universal Algebra , 1979 .

[8]  Jesse B. Wright,et al.  Automata in General Algebras , 1967, Inf. Control..

[9]  Dexter Kozen.  On the Myhill-Nerode Theorem for Trees , 1992 .

[10]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[11]  Sándor Vágvölgyi,et al.  Ground Term Rewriting Rules for the Word Problem of Ground Term Equations , 1991, Bull. EATCS.

[12]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[13]  A. Nerode,et al.  Concurrent Programs as Strategies in Games , 1992 .

[14]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[15]  Wayne Snyder,et al.  Efficient Ground Completion: An O(n log n) Algorithm for Generating Reduced Sets of Ground Rewrite Rules Equivalent to a Set of Ground Equations E , 1989, RTA.

[16]  Dexter Kozen,et al.  Complexity of finitely presented algebras , 1977, STOC '77.