scholarly journals Partial Automata and Finitely Generated Congruences: An Extension of Nerode's Theorem

1992 ◽  
Vol 21 (400) ◽  
Author(s):  
Dexter Kozen
Keyword(s):  
Polynomial Time ◽  
Term Rewriting ◽  
Equivalent System ◽  
Finitely Generated ◽  
Ground Term ◽  
One To One ◽  
Rewrite System

Let T_Sigma be the set of ground terms over a finite ranked alphabet Sigma. We define <em> partial autornata on</em> 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.

Linear pattern matching of compressed terms and polynomial rewriting

2018 ◽  
Vol 28 (8) ◽  
pp. 1415-1450 ◽  
Author(s):  
MANFRED SCHMIDT-SCHAUß
Keyword(s):  
Pattern Matching ◽  
Polynomial Time ◽  
Directed Acyclic Graph ◽  
Term Rewriting ◽  
Single Step ◽  
Worst Case ◽  
Linear Pattern ◽  
Input Size ◽  
Straight Line ◽  
Rewrite System

We consider term rewriting under sharing in the form of compression by singleton tree grammars (STG), which is more general than the term dags. Algorithms for the subtasks of rewriting are analysed: finding a redex for rewriting by locating a position for a match, performing a rewrite step by constructing the compressed result and executing a sequence of rewrite steps. The first main result is that locating a match of a linear termsin another termtcan be performed in polynomial time ifs,tare both STG-compressed. This generalizes results on matching of STG-compressed terms, matching of straight-line-program-compressed strings with character-variables, where every variable occurs at most once, and on fully compressed matching of strings. Also, for the case wheresis directed-acyclic-graph (DAG)-compressed, it is shown that submatching can be performed in polynomial time. The general case of compressed submatching can be computed in non-deterministic polynomial time, and an algorithm is described that may be exponential in the worst case, its complexity isnO(k), wherekis the number of variables with double occurrences insandnis the size of the input. The second main result is that in case there is an oracle for the redex position, a sequence ofmparallel or single-step rewriting steps under STG-compression can be performed in polynomial time. This generalizes results on DAG-compressed rewriting sequences. Combining these results implies that for an STG-compressed term rewrite system with left-linear rules,mparallel or single-step term rewrite steps can be performed in polynomial time in the input sizenandm.

CLOSED SUBGROUPS IN PRO-V TOPOLOGIES AND THE EXTENSION PROBLEM FOR INVERSE AUTOMATA

2001 ◽  
Vol 11 (04) ◽  
pp. 405-445 ◽  
Author(s):  
S. MARGOLIS ◽  
M. SAPIR ◽  
P. WEIL
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Finite Groups ◽  
Extension Problem ◽  
Membership Problem ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Finite Monoid ◽  
Finite Set ◽  
One To One

We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-V topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extension-closed. Turning to practical computations, we modify Ribes and Zalesskiĭ's algorithm to compute the pro-p closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups. Résumé: Nous établissons un lien entre le problème du calcul de l'adhéerence d'un sous-groupe finiment engendré du groupe libre dans la topologie pro-V, oú V est une pseudovariété de groupes finis, et un probléme d'extension pour les automates inversifs qui peut être énoncé de la faç con suivante: étant données des transformations partielles injectives d'un ensemble fini, peuvent-elles être étendues en des permutations qui engendrent un groupe dans V? Les deux problèmes sont équivalents si V est fermée par extensions. Nous intéressant ensuite aux calculs pratiques, nous modifions l'algorithme de Ribes et Zalesskiĭ pour calculer l'adhérence pro-p d'un sous-groupe finiment engendré du groupe libre en temps polynomial et pour calculer effectivement sa clôture pro-nilpotente. Enfin nous appliquons nos résultats à un problème de théorie des monoïdes finis, celui de de l'appartenance dans les pseudovariétés de monoïdes inversifs qui sont des produits de Mal'cev de demi-treillis et d'une pseudovariété de groupes.

scholarly journals The Church-Rosser property for ground term-rewriting systems is decidable

1987 ◽  
Vol 49 (1) ◽  
pp. 43-79 ◽  
Author(s):  
Michio Oyamaguchi
Keyword(s):  
Term Rewriting ◽  
Ground Term ◽  
Term Rewriting Systems ◽  
Rewriting Systems ◽  
The Church

scholarly journals ON THE COMPLEXITY OF THE WHITEHEAD MINIMIZATION PROBLEM

2007 ◽  
Vol 17 (08) ◽  
pp. 1611-1634 ◽  
Author(s):  
ABDÓ ROIG ◽  
ENRIC VENTURA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Minimization Problem ◽  
Finite Rank ◽  
Polynomial Algorithm ◽  
Minimum Size ◽  
Input Word ◽  
Finitely Generated ◽  
Factor Problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem — to decide whether a word is an element of some basis of the free group — and the free factor problem can also be solved in polynomial time.

scholarly journals COMPRESSED WORDS AND AUTOMORPHISMS IN FULLY RESIDUALLY FREE GROUPS

2010 ◽  
Vol 20 (03) ◽  
pp. 343-355 ◽  
Author(s):  
JEREMY MACDONALD
Keyword(s):  
Automorphism Group ◽  
Word Problem ◽  
Free Group ◽  
Polynomial Time ◽  
Free Groups ◽  
Finitely Generated

We show that the compressed word problem in a finitely generated fully residually free group ([Formula: see text]-group) is decidable in polynomial time, and use this result to show that the word problem in the automorphism group of an [Formula: see text]-group is decidable in polynomial time.

A new decidability technique for ground term rewriting systems with applications

2005 ◽  
Vol 6 (1) ◽  
pp. 102-123 ◽  
Author(s):  
Rakesh Verma ◽  
Ara Hayrapetyan
Keyword(s):  
Term Rewriting ◽  
Ground Term ◽  
Term Rewriting Systems ◽  
Rewriting Systems

scholarly journals Infinitely generated semigroups and polynomial complexity

2016 ◽  
Vol 26 (04) ◽  
pp. 727-750 ◽  
Author(s):  
J. C. Birget
Keyword(s):  
Polynomial Time ◽  
Time Complexity ◽  
Polynomial Complexity ◽  
Free Monoid ◽  
Functional Approach ◽  
Finitely Generated ◽  
Machine Model ◽  
Polynomial Time Complexity ◽  
Evaluation Functions ◽  
Separation Results

This paper continues the functional approach to the [Formula: see text]-vs.-[Formula: see text] problem, begun in [J. C. Birget, Semigroups and one-way functions, Internat. J. Algebra Comput. 25 (1–2) (2015) 3–36]. Here, we focus on the monoid [Formula: see text] of right-ideal morphisms of the free monoid, that have polynomial input balance and polynomial time-complexity. We construct a machine model for the functions in [Formula: see text], and evaluation functions. We prove that, [Formula: see text] is not finitely generated, and use this to show separation results for time-complexity.

scholarly journals The word problem for free adequate semigroups

2014 ◽  
Vol 24 (06) ◽  
pp. 893-907 ◽  
Author(s):  
Mark Kambites ◽  
Alexandr Kazda
Keyword(s):  
Word Problem ◽  
Polynomial Time ◽  
Normal Forms ◽  
Finitely Generated ◽  
Model Of Computation ◽  
Free Left ◽  
Computation Algorithms

We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and compute normal forms in each of these, and hence to test whether any given identity holds in the classes of left, right and/or two-sided adequate semigroups.

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