Machine Configuration and Word Problems of Given Degree of Unsolvability

Shepherdson, J. C.

doi:10.1002/malq.19650110210

Cited by 42 publications

(38 citation statements)

References 9 publications

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“…This reduction is performed in two steps. A semi-Thue system is constructed which, in the manner of Shepherdson [14], uses left, right, beginning and end symbols to simulate the restricted Markov algorithm in such a way as to preserve unbounded truth table degrees for the word and confluence problems. Productions are then added which allow the leftmost end symbol to arbitrarily destroy or create any symbols on its right.…”

Section: Many-one Equivalence Of Some Decision Problems 471mentioning

confidence: 99%

The many-one equivalence of some general combinatorial decision problems

Hughes¹,

Overbeek²,

Singletary³

1971

Bull. Amer. Math. Soc.

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Introduction.A decision problem for a combinatorial system shall denote a pair ($, S) where is a specified kind of decision problem (e.g. derivability problem, halting problem, etc.) and 5 is a combinatorial system. Two decision problems (<£i, Si), ($ 2 , £2) are sa^d t0 be of the same many-one degree (of unsolvability) if there exist effective many-one mappings ƒ and g such that each instance of ($1, Si) is reducible to an instance of ($2, S2) via ƒ and each instance of (02, S 2 ) is reducible to an instance of (i, Si) via g.A general combinatorial decision problem, i.e., a decision problem for a class of combinatorial systems, shall denote a pair (0, C) where 0 is a specified kind of decision problem and C is a class of combinatorial systems (e.g. Turing machines, semi-Thue systems, etc.). A general combinatorial decision problem (0i, G) is many-one reducible to another general combinatorial problem (0 2 , C 2 ) if there exists an effective one-one mapping \f/ of the problems p associated with (0i, Ci) into the problems associated with (0 2 , C2) such that p is of the same many-one degree as \p(p). (0i, G) and (0 2 , C 2 ) are said to be many-one equivalent if each is many-one reducible to the other.The reduction of one general combinatorial decision problem to another has been investigated by numerous authors. In particular, W. E. Singletary [15] has combined results of his own and those of others in such a way as to provide an effective proof of the (r.e.) equivalence of a number of general combinatorial decision problems. This former work has lead W. W. Boone to suggest that a stronger form of equivalence might exist between at least some subset of the problems considered. Our aim is to show that a number of these general problems are many-one equivalent. In addition, we indicate that these are, in a sense, best possible results.

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Section: Many-one Equivalence Of Some Decision Problems 471mentioning

confidence: 99%

The many-one equivalence of some general combinatorial decision problems

Hughes¹,

Overbeek²,

Singletary³

1971

Bull. Amer. Math. Soc.

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“…(See [7] and [S].) I n fact, the proofs of all the following theorems are based on the same idea and, indeed, they are all proved either explicitely or implicitely in [17]. Theorem I(d) below is also proved in [ l ] .…”

Section: Sectionmentioning

confidence: 97%

“…A proof, fairly easy because of the LRM feature mentioned, is implicite in [17]. A similar theorem holds for SRM.…”

mentioning

confidence: 89%

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

Yasuhara

1974

Mathematical Logic Qtrly

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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

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“…This reduction has been carried out by Shepherdson [12]. The idea is to construct a "large scale" machine with derivability problem equivalent to the given definition problem and then perform successive reductions to limited register, single register and finally to Turing machines maintaining the equivalence of the derivability problems at each stage.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

The equivalence of some general combinatorial decision problems

Singletary¹

1967

Bull. Amer. Math. Soc.

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Introduction.A decision problem for a combinatorial system shall denote a pair (<ï>, S) where $ is a specified kind of decision problem (e.g. word problem, halting problem, etc.) and S is a combina* torial system. Likewise, a general combinatorial decision problem, i.e. a decision problem for a class of combinatorial systems, shall denote a pair (<ï >> C), where * is a specified kind of decision problem and C is a general class of combinatorial systems (e.g. Turing machines, semi-Thue systems, etc.). Clearly, each general combinatorial decision problem P has a class of decision problems for combinatorial systems associated with it. We shall refer to these problems simply as the problems associated with P.There are many papers in the literature which deal with the reduction of one general combinatorial decision problem to another. These papers fall into two general groups. The first group consists of unsolvability proofs such as . The general format of these proofs is the following: Two general combinatorial decision problems Pi and P 2 are considered, where Pi is known to have an associated problem of each r.e. degree of unsolvability. Then an effective one-one mapping \j/ of the problems p associated with Pi into the problems associated with P 2 and uniformly effective reductions of p to \p(p) and of \p(p) to p are given.Our aim here is to link several of these reductions together in such a way as to provide an effective proof of the equivalence of a number of general combinatorial decision problems. Furthermore, all of our reductions will conform to the second format given above and hence for each pair P»-, Pj of general combinatorial decision problems con-

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Machine Configuration and Word Problems of Given Degree of Unsolvability

Cited by 42 publications

References 9 publications

The many-one equivalence of some general combinatorial decision problems

The many-one equivalence of some general combinatorial decision problems

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

The equivalence of some general combinatorial decision problems

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