Some Non‐Recursive Classes of Thue Systems With Solvable Word Problem

Abstract

We take as our general setting decision problems regarding certain combinatorial properties of THUE systems and algebraic properties of the semi-groups they present. Let the class of all THUE systems be divided up as follows:THUE systems with unsolvable word problem}, Wl = {all THUE systems for which the number of equivalence classes of words is finite}, V2 = {all THUE systems for which the cardinality of each of its equivalence classes of g3 = {all THUE systems not in %, , w Vl w V 2 } .Section 1 of this paper is mainly an expository discussion of the setting with reference to the four classes. Ideas implicite in [17], which will be discussed in section 2 , are used in section 3 to prove that there is no algorithm to decide of an arbitrary THUE system whether or not it is in g2. I n section 4 we show that g i w g j , 0 5 i 5 3 , is also not recursive. I n section 5 there are a few comments about the kinds of THUE systems in V2 and V3. words is finite}, Section 1Familiarity with the standard terminology of TURING machines, partial recursive functions, THUE systems, finite presentations of semi-groups, solvable and unsolvable word problems is assumed. See for example [ 5 ] , [8] or [ZO]. Given a finite set of symbols 2, let 2* denote the set of all words on 2. (Words will always have finite length here.) We assume available for the entire paper a fixed inifinite set of symbols and that by a finitme set of symbols we mean a subset of the fixed set. Let A , 3, U , V , W (perhaps with subscripts) denote words. The defining relations (rules, productions) of a THUE system T with alphabet . Z partition Z* into equivalence classes; if W is a word in Z*, let [WIT denote the equivalence class of W with respect to T. A THUE system is then the finite presentation of a semi-group 9 ' T whose elements are the equivalence classes of Z* determined by T and whose operation is defined byThe class of all THUE systems can be partitioned into equivalence classes by: T r~ T' if and only if Y T g Y T , , (r for isomorphism); let the elements be denoted by [TI and the corresponding abstract semi-groups by 9 [ T ] . The term combinatorial property will be used loosely to refer to properties of THUE systems (e.g. number of generators, number of defining relations, number of equivalence classes, cardinality of equivalence class, solvable word problem, etc.), and the term algebraic property (of semi-groups) will be used, as usual, to refer to any property of semi-groups which is preserved under isomorphism. If 9 ' is a combinatorial property define 4 to be {T I there exists a T' €9 such that Y T z Y T , } .Obviously B is an algebraic property. Notice that