Thue Systems Research Articles

Combinatorial systems I. Cylindrical problems

The reduction of one combinatorial decision problem PI to another P2 can be achieved by constructing an effective map r : Pa --~ P2 such that each problem p in /'1 is reducible to the corresponding problem r in P2 (el. [4]). Judicious choice of ~b can sometimes ensure that p ~,~ r If, in addition, all the problems in P2, are cylinders [2] then p 41 r so that the reduction of P1 to P2 is best possible. This is because any cylinder has maximum one-one degree within its many-one degree--two cylinders which are many-one equivalent are actually one-one equivalent. The primary aim of this paper is, therefore, to identify some combinatorial problems which are always cylinders. Our investigation is effected by considering those functions on the natural numbers whose values are finite sets of natural numbers--system functions--which possess certain formal properties in common with those arising naturally from G6del numbering of many kinds of combinatorial system. This approach also enables us to construct system functions with noncylindrical problems--a matter reserved for a later paper. Many aspects of combinatorial systems (partial propositional calculi, Markov algorithms semi-Thue systems, etc.) are comprehended in Tarski's definition of deductive systems [5]. Accordingly we first expound a general method of recognizing cylindrical combinatorial problems of deductive systems and then apply it to system functions. For an explanation of the concepts of recursive-function theory as used in this paper the reader is referred to [2]; the notion of a cylinder, which is central to our discussion, and its elementary properties are presented on p. 89. The idea of using system functions in the theory of combinatorial systems is due to W. E. Singletary. The points in the ensuing development at which I am indebted to him are too numerous to mention.

Word Problems and Recursively Enumerable Degrees of Unsolvability

In [2] it was shown that for every r.e. (i.e., recursively enumerable) Turing degree of unsolvability D there exists a Thue system with word problem of degree D. In [3] the analogous result was shown for finite presentations of groups. Except for minor slips set aright at the end, this corrective note is concerned only with footnote 8, p. 523 of [2] and footnote 4, p. 50 of [3]. In these footnotes it is claimed that, in effect, somewhat sharper results had been proved, viz., the corresponding results for tt (i.e., unbounded truth-table) degrees instead of Turing degrees. Two points should be made: these footnotes do not bear upon the validity of the body of [2] and [3] as they were not used, indeed not even referred to, elsewhere in these papers; moreover, these stronger results about tt degrees are in fact valid, and for presentations having the particular form discussed. What is seriously amiss is the implied claim that both these tt results could be verified by the interested reader once he had understood [2] and [3]. For the weaker result about Thue systems and tt degrees this still seems possible' to the author; but certainly not for the stronger result about finite presentations of groups and tt degrees no matter how dedicated the reader! (In a letter to the present author, A. A. Fridman has explained that his methods of [6] and [7] do not settle the question about finitely presented groups and truth-tables either.) Fortunately all the required arguments have now been supplied by Donald J. Collins in DJC.2' Remark (1) of footnote 8 of [2] claims that Thue system Zs i.e., Z, of [2], is tt-reducible to the r.e. set S of natural numbers. From this follows the existence of a Thue system with word problem of arbitrarily preassigned r.e. tt degree. Remark (1) is correct and can be verified by a straightforward systematic reworking of [2], much like writing a computer program; this

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented

Some remarks on derivations in general rewriting systems

We study a class of general rewriting system derivations called canonical derivations. For context-free rewriting systems, these derivations are in one-to-one correspondence with the usual structural descriptions. Two derivations are similar if one can be obtained from the other by trivial rule rearrangement. We show that every similarity class of derivations includes at most one canonical derivation and that only in general rewriting systems allowing nontrivial derivations from A to A, does there fail to exist a canonical derivation for each similarity class. Whether nontrivial A to A derivations are possible in a general rewriting system is, in general, undecidable, but they are never possible in semi-Thue systems. Transformations preserving canonical derivations are considered, one having the interesting property that, in a certain sense, it modifies a general rewriting system so that only its canonical derivations remain. We also obtain a representation of each recursively enumerable set as the homomorphic image of the intersection of a context-free language and the right-cancelling language. A few remarks about parsing are included.

A Remark on Post Normal Systems

Let N ( T ) be the normal system of Post which corresponds to the Thue system, T , as in Martin Davis, Computability and Unsolvability (McGraw-Hill, New York, 1958), pp. 98-100. It is proved that for any recursively enumerable degree of unsolvability, D , there exists a normal system, N T ( D ) , such that the decision problem for N T ( D ) is of degree D . Define a generalized normal system as a normal system without initial assertion. For such a GN the decision problem is to determine for any enunciations A and B whether or not A and B are equivalent in GN . Thus the generalized system corresponds more naturally to algebraic problems. It is proved that for any recursively enumerable degree of unsolvability, D , there exists a generalized normal system, GN T ( D ) , such that the decision problem for GN T ( D ) , such that the decision problem for GN T ( D ) is of degree D .

Word Problems and Recursively Enumerable Degrees of Unsolvability. A Sequel on Finitely Presented Groups

The present paper is a sequel to [B4.2 We specify an explicit grouptheoretic construction based on techniques of Britton [D] which, if added to the Thue system constructions of [B], results in the exhibition of a finitely presented group with word problem of arbitrary recursively enumerable degree of unsolvability. One may be given the arbitrary degree as the word problem of a given Thue system. Only the results of [B] are needed, not the details, except for two definitions of notation.3 The exhibition of a finitely generated, recursively related group of arbitrary recursively enumerable degree of unsolvability (as described in [B3])2 is briefly noted. Related independent lines of development are the announcements by Cejtin [B9] (for Thue systems), and by Fridman [B13] (for finitely presented groups), as well as the detailed arguments by Clapham [Bl0] (for finitely presented groups), and by Shepherdson [B251 (for Thue systems). (See footnote 6.) Like [BI, this work is dedicated to the memory of Thoralf Skolem.

Word Problems and Recursively Enumerable Degrees of Unsolvability. A First Paper on Thue Systems

Word Problems and Recursively Enumerable Degrees of Unsolvability. A First Paper on Thue Systems

Note on the boolean properties of context free languages

A language is considered to be any set of sentences (strings) made up from a finite vocabulary (alphabet). A grammar is a device which enumerates a language, e.g., a Turing machine with any set of input signals, a finite automaton, or a Post system. One often considers the family of languages which have grammars meeting certain specifications or restrictions, e.g., the finite state languages (the languages generatable by finite automata) . A major question of interest is whether such a family is a closed system with respect to the Boolean operations: set-theoretic union, intersection, and difference. The family of recursively enumerable languages is well known to be closed under union and intersection, but not under difference. I t is also well known that the family of finite state languages is closed under all three operations. In this note we answer this question for another family of languages, called type 2 or by Chomsky (1959). For the sake of having a convenient reference we shall present in modified form Chomsky's definitions. I t will be seen that a context free grammar is essentially a special case of a semi-Thue system, cf. Davis (1958). We consider a context free (CF) grammar to be a finite set G of rewriting a --~ ~, where a is a single symbol and ~ is a finite string of symbols from a finite alphabet (vocabulary) V. V contains precisely the symbols appearing in these rules plus the boundary symbol ~, which does not appear in these rules. Rules of the form a --~ a (which

The word problem for semigroups with two generators

Emil Post has shown that the word problem for Thue systems is unsolvable. A Thue system is a system of strings of letters x1, x2, …, xr, where a string or word is either void or a succession of letters a1a2 … at with each ai some xi. We are given a finite number of pairs of strings Ai, Bii = 1, …, m. Operations in the Thue system consist of the replacements or “productions.”