Knuth-Bendix algorithm and the conjugacy problems in monoids

We present an algorithmic approach to the conjugacy problems in monoids, using rewriting systems. We extend the classical theory of rewriting developed by Knuth and Bendix to a rewriting that takes into account the cyclic conjugates.Comment: This is a new version of the paper 'The conjugacy problems in monoids and semigroups'. This version will appear in the journal 'Semigroup forum

Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids

There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by $\sim_p$, $\sim_o$, and $\sim_c$) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, $p$-conjugacy is "almost" transitive, $\sim_c$ is strictly included in $\sim_p$, and the $p$- and $c$-conjugacy problems are decidable with linear compexity. For other classes of monoids, the situation is more complicated. We show that there exists a monoid $M$ defined by a finite complete presentation such that the $c$-conjugacy problem for $M$ is undecidable, and that for finitely presented monoids, the $c$-conjugacy problem and the word problem are independent, as are the $c$-conjugacy and $p$-conjugacy problems.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1503.0091

On the equivalence problem for regular Thue systems

AbstractA decision procedure is presented for the equivalence problem for regular almost-confluent Thue systems. On the other hand, the equivalence problem for regular preperfect systems is shown to be undecidable

A complete transformation rule set and a minimal equation set for CNOT-based 3-qubit quantum circuits (Draft)

We introduce a complete transformation rule set and a minimal equation set for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum circuits that compute the same Boolean function are reduced to the same normal form. We can thus easily check the equivalence of circuits by comparing their normal forms. By applying the Knuth-Bendix completion algorithm to a set of modified 18 equations introduced by Iwama et al. 2002, we obtain a complete transformation rule set (i.e., a set of transformation rules with the properties of `termination' and `confluence'). Our transformation rule set consists of 114 rules. Moreover, we discovered a minimal combination of equations for the initial equation set

Pushouts in software architecture design

A classical approach to program derivation is to progressively extend a simple specification and then incrementally refine it to an implementation. We claim this approach is hard or impractical when reverse engineering legacy software architectures. We present a case study that shows optimizations and pushouts--in addition to refinements and extensions--are essential for practical stepwise development of complex software architectures.NSF CCF 0724979NSF CNS 0509338NSF CCF 0917167NSF DGE-1110007FCT SFRH/BD/47800/2008FCT UTAustin/CA/0056/200

Complexity Results for Confluence Problems

Abstract. We study the complexity of the confluence problem for re-stricted kinds of semi–Thue systems, vector replacement systems and general trace rewriting systems. We prove that confluence for length– reducing semi–Thue systems is P–complete and that this complexity reduces to NC2 in the monadic case. For length–reducing vector re-placement systems we prove that the confluence problem is PSPACE– complete and that the complexity reduces to NP and P for monadic sys-tems and special systems, respectively. Finally we prove that for special trace rewriting systems, confluence can be decided in polynomial time and that the extended word problem for special trace rewriting systems is undecidable.