Two Recognizable String-Matching Problems Over Free Partially Commutative Monoids

Let C be a finite alphabet, and C* the free monoid generated by Z. Let x, y E C* and x = uyv for U, v E C*. Then y is a factor of x. Moreover if u = 1, then y is a prefix of x, and if u = 1, then y is a suffix of x. Here 2. is the null word. One of the typical string matching problems over C* is the following: given a text string XEX* and a pattern string VEC*, decide whether or not y is a factor of x. Many efficient algorithms for this string matching problem are known; see, e.g., Aho et al. (1974). Recently many contributions about free partially commutative monoids have also appeared; see, e.g., Perrin (1985). Let us first recall the definition of free partially commutative monoids briefly. Let 0 be an irreflexive symmetric binary relation over C. = B (or simply E, when 8 is understood) denotes the smallest equivalence relation over Z* such that for any x,yEC* , I = y if x = uabu and Y = ubav for some u, v E Z* and (a, b) E 0. Then = is a congruence relation. M(C, 0) denotes the quotient of Z* by the congruence =. M(C, 0) is the free partially commutative monoid generated by Z w.r.t. 8, and can be regarded as a model of concurrency control systems, or a model of any systems with finitely many partially commutative operations. For any x, y E Z*, if x = uyz) for some U, v E C*, then we call y a Q-factor of x: moreover if u = i, then y is a &prefix of x, and if u = ;I, then y is a B-suffix of x. This paper studies the following two string matching problems over