The Representation of Many-One Degrees by the Word Problem for Thue Systems

Overbeek, Ross

doi:10.1112/plms/s3-26.1.184

Cited by 8 publications

(5 citation statements)

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“…M. ZIEGLER [11] then showed that the result could not be extended to btt-degrees, even for recursively presented groups. I n contrast with ZIEGLER'S result, OVERBEEK [7] (extending earlier work of BOONE [2] for Turing degrees) showed that every r e . m-degree does contain the word problem of a finitely presented semigroup.…”

Section: Boone's Proof Is In [3 41 and References Tocontrasting

confidence: 45%

Fine Degrees of Word Problems of Cancellation Semigroups

Jockusch¹

1980

Mathematical Logic Qtrly

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have shown independentIy that for each recursively enumerable (r.e.) Turing degree d there is a finitely presented group whose word problem has degree d. (BOONE'S proof is in [3, 41 and references to Dhr ot,her papers on the subject may be found in [ l , pp. 126-1281 or the bibliography of [ 5 ] . ) The object of the current paper is to refute some extensions of this result to finer reducibilities, such as many-one (m) reducibility and bounded truth-table (btt) reducibility, even when one considers cancellation semigroups instead of groups. For definitions and information on these reducihilities, see [9, Chapter 71 or [S]. We first sum up the known results in this area. I n [5], D. J. COLLINS extended the result for finitely presented groups mentioned above from Turing degrees to truthtable degrees. M. ZIEGLER [11] then showed that the result could not be extended to btt-degrees, even for recursively presented groups. I n contrast with ZIEGLER'S result, OVERBEEK [7] (extending earlier work of BOONE [2] for Turing degrees) showed that every r e . m-degree does contain the word problem of a finitely presented semigroup.In the present note we give an extremely simple proof that there is an r.c. m-degree nhich does not contain the word problem of any recursively presented cancellation semigroup. I n fact we show that any m-degree containing a simple set which is not hyprmimple has this property. We also sketch a priority argument for showing that there is an r.e. btt-degree which does not contain the word problem of any cancellation semigroup, thus extending ZIEGLER'S result from groups to cancellation semigroup. A kcy step common to both proofs is to specify a simple recursion-theoretic property of sets of natural numbers which holds of all sets encoding word problems of the sort we are considering. It is then a purely recursion-theoretic matter to construct m-degrees (btt-degrees) containing no sets with this property. We now define this property.recursive function f such that for all 5 E A and all natural numbers y, Definition. A set A of natural numbers is called ~~Z t~p l~c a~~v eif t,here is a binaryThis definition is applied to sets of words on a recursive alphahet by identifying words with their Godel numbers, and similarly for sets of pairs of words. I n our discussion we do not distinguish betweens words and t,heir Godel numbers. E x a m p l e s . The following sets are all multiplicative:(1) The set of words representing elements of H , where H is any fixed subgroup of a given recursively presented group G . l) This research was supported by National Science Foundation Grant MCS 76-07444. The aut,hor is indebted to W. W. BOOXE for helpful suggestions and information.

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Section: Boone's Proof Is In [3 41 and References Tocontrasting

confidence: 45%

Fine Degrees of Word Problems of Cancellation Semigroups

Jockusch¹

1980

Mathematical Logic Qtrly

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“…(I), (II), (III), (IV), and (V) were shown in [12], [7], [6], [9], and [8], respectively. (VI) and (VII) may be arrived at by combining the constructions of [3] with those of [7] and [9], respectively.…”

Section: Two-letter Alphabetsmentioning

confidence: 96%

“…More recent work has had as its objective the proof that these Turing degree results can be strengthened to many-one degrees. Examples of papers falling into this category are [6], [7], [8], [9], [10], and [12]. The constructions presented in each of the above lead to combinatorial systems in which the alphabets are finite but not uniformly bounded.…”

Section: Introductionmentioning

confidence: 97%

Combinatorial systems defined over one- and two-letter alphabets

Hughes

Singletary

1975

Arch math Logik

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In this paper we shall investigate some families of decision problems associated with a number of combinatorial systems whose alphabets are restricted to one and two letters. Our purpose is to try to gain a better understanding of the boundaries between classes of combinatorial systems whose decision problems are solvable and those whose decision problems are of any prescribed r.e. manyone degree of unsolvability. We show that, for one-letter alphabets, the decision problems considered here for semi-Thue systems, Thue systems, Post normal systems, Markov algorithms and Post correspondence classes are each solvable. In contrast, for two-letter alphabets, each such family of decision problems represents every r.e. many-one degree of unsolvability.

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“…many-one degrees : the representation theorem for the decision problem of partial implicational propositional calculi in two variables (see HUGHES [36]), for the halting problem of Tag systems (AANDERAA, BELSNES [l]) with many-one equivalent word and halting probleni (HUGHES [32]), for the Post correspondence problem of correspondence classes resp. with arioni (CUDIA, SINGLETARY [16], HUGHES, SINGLETARY [39, chapter XI), for the word probleiri of Thue systems (OVERBEEK [53]), for the decision problem of recursive classes of first-order logical formulae with equality as only predicate symbol; the triple repraenfation theorem for halting, word and confluence problem of Turing machines (OVER-BEEK i52]), iMarkov algorithms without concluding rules (HUGHES [30]), Xarkov algorithms in Swanson…”

Section: %(Ptm)mentioning

confidence: 99%

“…Following BOONE'S suggestion prompted by this former work1) that such equivalence theorems might hold for stronger notions of degree reducibility, HUGHES, OVERBEEK, SINGLETARY [37] generalized tthe results of SINGLETARY I661 and others obtained in the meanwhile2) by showing that equivalence holds indeed with respect to many-one degrees (but in most cases not m y more for oneone degrees) for all the general decision problems mentioned above and many others as for example decision problems for Tag systems and Markov algorithms. Following that paper [37] up to now there has been a prolific production of proofs of new such equiva.lences including AANDERAA, BELSNES [ [53], SINGLETARY [67]. We want to present here a new approach to such equivalence proofs which permits to unify the myriad individual methods in the literature cited by bringing them back to some few and simple fundamental ideas explicitely formulated in five lemmata.…”

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confidence: 99%

A New General Approach to the Theory of the Many‐One Equivalence of Decision Problems for Algorithmic Systems

Börger¹

1979

Mathematical Logic Qtrly

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Starting from POST'S [55] different notions of effective reducibility and from the fact proved by FRIEDBERG [23] and MUONIK [49] that there are recursively enumerable (r.e.) sets A , of natural numbers neither of whose decision problems 1.x . x E A , is (Turing-)reducible to the other giving incomparable r.e. degrees of unsolvability, many authors investigated the natural question as to what extent known unsolvability results could be generalized to degree analogues of them. Attention was focused on proving or disproving theorems of the following type: let A be a set of objects (for example natural numbers, machines, algorithms, logical deductive systems) and D,, . . . , D,, functions containing for every element a E A so called decision problems, i.e. pairs (a, D,(a)) with sets D,(a) (as for example halting problems, correctness problems, deducibility problems) ; one can give an effective procedure which associates to every n-tuple of degrees d,, . . . , d,, an object a E A such that D,(a) has degree d , (1 5 i n).One expresses such a fact often also by saying that every n-tuple of degrees can be effectively represented by the decision problems D,, . . ., D,& of elements of A . So we will refer to such a theorem briefly as n-tuple representation theorem for the degrees of the decision problems D, , . . . , D, of elements of A , where the particular notion of degree used has to be specified. I n the case n = 1 we omit "1-tuple". Call a pair ( A , D ) as specified above a general decision problem, i.e. a class of particular decision problems (a, D(a)). As in the case of unsolvability proofs (see for example BOONE [5], MARKOV [43], MATIJASEVIE [46], NOVIKOV [51], POST [54, [56], 1571, TURING [69 I), the main method for proving a representation theorem for the degrees of a general decision problem ( A , D ) is the effective degree preserving reduction t o ( A , D) of another general decision problem (A', D') for which the representation theorem is already known. This means that one gives an effective (normally one-one) mapping f : A' -+ A such that the corresponding decision problems D'(a') and D(f(a')) have the same degree. One then says that (A', D') is degree reducible to ( A , D ) (for the particular notion of degree considered) ; if each of them is degree reducible t o the other one they are said to be degree equivalent. SLUGLETARY [66] linked several effective Turing degree preserving reductions published in the literature for certain specific general decision problems3) together and combined them with own such reductions providing a proof of the Turing equivalence of a great number of general decision problems for algorithmic systems such as partial recursive This work is partly based on and extends the author's Habilitationsschrift [8] presented to the 2, The reader who is not acquainted with notions used in this introduction may look at $ 2 where 3, For exact references see the bibliography op. cit. Fachbereich Mathematik of the University of Miinster i. W. definitions or references to the literature are gi...

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The Representation of Many-One Degrees by the Word Problem for Thue Systems

Cited by 8 publications

References 2 publications

Fine Degrees of Word Problems of Cancellation Semigroups

Fine Degrees of Word Problems of Cancellation Semigroups

Combinatorial systems defined over one- and two-letter alphabets

A New General Approach to the Theory of the Many‐One Equivalence of Decision Problems for Algorithmic Systems

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