E. L. Post. Recursive unsolvability of a problem of thue. Journal of Symbolic Logic, 12:1--11, 1947.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[4] also called string rewriting systems) received a lot of attention in mathematics and theoretical computer science and are still an active field of research. Historically, rewriting systems were introduced to solve word problems in certain structures [63] By the work of Markov [43] and Post [54], this hope vanished as they presented semi Thue systems with an undecidable word problem. Despite this result, there are plenty of rewriting systems with a decidable word problem, the most famous class being that of confluent and terminating systems [32] In general, also confluence and ....

.... is defined as usual. Theorem 3.12. There exists a fixed permutation rewriting system finite alphabet such that the first order theory of the model is undecidable. Furthermore we may assume that for every variable that occurs in some , Proof. By [43, 54] there exists a fixed semi Thue system over some with an undecidable word problem, i.e. it is undecidable for given words whether or not , where denotes the transitive and symmetric closure of . 1 where . Define semi Thue rules (i.e, elements of a semi Thue ....

E. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12(1):1--11, 1947.

....position i, executes one step (in forward or backward direction) Also, the polynomial Gamma and the polynomials g j are homogeneous, the g j of degree say 3 and Gamma of degree roughly n. Now, M accepts m iff Gamma is in the ideal generated by the g j : As already noted in [48], the fact that we have replaced the semi Thue system underlying M by a Thue system does not hurt us since M was assumed to be deterministic. We also remark that the exponential space lower bound for PIMP also holds if we replace Q by an infinite field of finite characteristic, say 2. The reason ....

E. Post. Recursive unsolvability of a problem of Thue. J. Symbolic Logic, 12:1--11, 1947.

....that since every countable semigroup embeds into End(R) by Theorem 3, every universal sentence true in End(R) will be true in all countable semigroups and, by the Lowenheim Skolem Theorem (see [8] in all semigroups. It is well known that the universal theory of semigroups is undecidable. See [12, 13, 14]. This is implied by the existence of a semigroup with undecidable word problem. Hence, the universal theory of End(R) is undecidable. 3. The full transformation monoid If X is a nonempty set, then a self mapping of X is a mapping from X to X. The full transformation monoid, written T (X) is ....

E.L. Post, Recursive unsolvability of a problem of Thue, J. Symbolic Logic 12 (1947), 1--11.

....of the PCP (noted VPCP) is to find a sequence i 1 ; i n 2 f1; kg with n 1 such that: 81 j n ju i 1 : u i j j jv i 1 : v i j j This condition imposes the u i always being ahead of the v i . The PCP and the VPCP are well known to be undecidable problems [13, 11]. A picture word is a word over the alphabet Pi = fu; r; d; lg where each letter codes a unit move up for u, to the right, down and to the left for r, d and l respectively. For each unit move we plot the covered segment [12] For instance, the word = u rdl describes the picture: The ....

E. L. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12:1--11, 1947.

....1 Introduction Word problems for nite semi Thue systems, or more precisely word problems for monoids presented by nite semi Thue systems, received a lot of attention in mathematics and theoretical computer science and are still an active eld of research. Since the work of Markov [18] and Post [22] it is known that there exists a xed semi Thue system with an undecidable word problem. This has motivated the search for classes of semi Thue systems with decidable word problems and the investigation of the computational complexity of these word problems, see e.g. 9, 7, 16] In [1] Adjan has ....

E. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12(1):1-11, 1947.

....the same element as 3 As usual, we write B # for the Kleene closure of some set B. 8 aaaaab is an instance of the word problem. In this particular example the answer is yes given that, aaaaab = aaa)aab = aaab = a(aab) abb The word problem for semigroups is undecidable in the general case [Pos47, Gur66] However, if the set of constraints E is empty (k = 0) then the problem reduces to pattern matching and becomes decidable. It is also decidable if the set E is just a singleton. Clearly, the word problem is also a decision problem when the underlying mathematical structure is a monoid ....

E. L. Post. Recursive unsolvability of a problem of thue. Journal of Symbolic Logic, 12:1--11,

....nach gegebenen Regeln (1914) which is, in fact, a continuation of the paper to be considered here, Thue introduces word rewriting systems and poses the word problem for them. This problem was shown to be undecidable by Emil Post in his paper Recursive unsolvability of a problem of Thue [Post47], where the terms semi Thue system and Thue system were also introduced. While these three papers of Thue were taken up after some time by others and led to the foundation of new subjects in discrete mathematics, the fourth paper, well hidden behind its empty title Die L osung eines ....

E.L. Post, Recursive unsolvability of a problem of Thue, J. Symb. Logic 12 (1), 1-11, reprinted in: M. Davis (Ed.), The Collected Works of Emil L. Post, Birkhauser, Boston 1994, pp. 503-513.

...., 6 the problem that decides whether Kx9x represents the same element as K x is an instance of the word problem. In this particular example the answer is yes given that, K x2 : 5 : xS2 KxS20 V Kx =2 xBx The word problem for semigroups is undecidable in the general case [Pos47, Gur66] However, if the set of constraints is empty ( 2 ) then the problem reduces to pattern matching and becomes decidable. It is also decidable if the set is just a singleton. Clearly, the word problem is also a decision problem when the underlying mathematical structure is a ....

E. L. Post. Recursive unsolvability of a problem of thue. Journal of Symbolic Logic, 12:1--11, 1947.

....a consequence of the termination of E normalized completion, so the result is much stronger: every E ground theory has an E normalized rewrite system. The undecidability of the word problem for ground theories modulo associativity was proved independently by Post and Markov in 1947 (Markov, 1947; Post, 1947), for group theory it is a result of Novikov in 1955 (Novikov, 1955; Stillwell, 1982) and for ground theories modulo ACD it is a recent result (March e, 1992) As a conclusion, we have obtained theoretical results: the unification and the generalization of decidability results, and a new ....

Post, E. L. (1947). Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 13:1--11.

....a, aab = bb , 6 the problem that decides whether abb represents the same element as aaaaab is an instance of the word problem. In this particular example the answer is yes given that, aaaaab = aaa)aab = aaab = a(aab) abb The word problem for semigroups is undecidable in the general case [Pos47, Gur66] However, if the set of constraints E is empty (k = 0) then the problem reduces to pattern matching and becomes decidable. It is also decidable if the set E is just a singleton. Clearly, the word problem is also a decision problem when the underlying mathematical structure is a monoid ....

E. L. Post. Recursive unsolvability of a problem of thue. Journal of Symbolic Logic, 12:1--11, 1947.

....is to solve word problems for nitely presented monoids. But since there exists a xed semi Thue system R such that the word problem for the monoid presented by the set of equations that corresponds to R (in the following we will brie y speak of the word problem for R) is undecidable [Mar47,Pos47] also semi Thue systems cannot help for the e ective solution of arbitrary word problems. This motivates the investigation of restricted classes of semi Thue systems which give rise to decidable word problems. One of the most prominent class of semi Thue systems with a decidable word problem is ....

E. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12(1):1-11, 1947.

....elements are the equivalence classes of R. If Sigma is finite, a semigroup isomorphic to Sigma = R is said to have a finite presentation h Sigma j Ri. The word problem for semigroups is the problem to decide u Rv for given words u; v 2 Sigma . The problem was shown undecidable in [44]. Gurevich has later strengthened the result, extending it to finite semigroups [33] In the sequel we assume that a word w = w 1 w 2 : wn in Sigma is written as a dot separated sequence of symbols w 1 .w 2 . wn , to avoid clumsy notation for the path associated to w. We prove the ....

Emil L. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12:1--11, 1947.

....[44, 46] deal with word problems for finitely presented semigroups (these papers contain the definition of what is now called a Thue system ) He was able to solve the word problem in special cases. It was only in 1947 that the general case was shown to be unsolvable independently by E. L. Post [28] and A. A. Markov [24] The other two papers [43, 45] deal with repetitions in finite and infinite words. Perhaps because these papers were published in a journal with restricted availability (this is guessed by G. A. Hedlund [20] this work of Thue was widely ignored during a long time, and ....

E. L. Post, Recursive unsolvability of a problem of Thue, J. Symbolic Logic 11 (1947), 1--11.

....(section 3, compressed rules) has this property. The constructions in [7] and also here can be viewed as attempts to convert what is essentially a reversible nondeterministic computing device (as represented by a Thue system) into a deterministic one such constructions go back to Post [9]. This paper is organized as follows: Section 2 establishes some terminology. In section 3, we describe a commutative Thue system S 0 that counts up to d 2 n , and section 4 proves 2 See Notes Added in Proof. that S 0 has certain uniqueness properties. Sections 5 and 6 give successive ....

....possibly w = w 0 here. In this paper, for each system S, we are interested in derivations from a distinguished word w 0 explicitly associated with S. We call w 0 the initial assertion of S and words derivable from the initial assertion are called assertions of S. This terminology is from Post[9]. 3 Compressed Rules for a Counter We describe a simple construction of a quadratic counter that will be improved upon in later sections. Constants. Throughout this construction, we fix the integers n 1 and d 2. We also define e d (k) e(k) d 2 k for all k 0. Observe that e(k 1) ....

Emil L. Post, "Recursive unsolvability of a problem of Thue", J. of Symbolic Logic 12(1947)1--11.

....decidable, this is a consequence of the termination of E normalised completion, so the result is much stronger: every E ground theory has a E normalised rewrite system. The undecidability of word problem for ground theories modulo associativity was proved independently by Post and Markov in 1947 [28, 33], for groups theory it is a result of Novikov in 1955 [31, 34] and for ground theories modulo ACD it is a recent result [26] As a conclusion, we have obtained theoretical results: the unification and the generalisation of decidability results, and a new completion algorithm, which generalises the ....

E. L. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 13:1--11, 1947.

....v 1 = 1 and u 2 = 1 or v 2 = 1, since Delta is cancellative. 3 The word problem for semi Thue systems The word problem for semigroups was the first undecidability result in algebra. That there are (presentations of) semigroups with an undecidable word problem was proved independently by Post [13] and Markov [8] The first concrete example of a finitely presented semigroup with an undecidable word problem was given by Markov in 1947. This example has 13 generators and 33 relators. This was later improved by Tzeitin [15] who proved that the semigroup S 0 = ha; b; a; b; e j R 0 i with five ....

E. Post, Recursive unsolvability of a problem of Thue, J. Symb. Logic 12 (1947), 1 -- 11.

....the history of combinatory mathematics second only to that of the formulation of natural number. Emil Post (1944) Abstract The first undecidability results in rewriting were proved by reduction to undecidable problems of Turing machines [7] More recently Post s Correspondence Problem [19] has been used (e.g. 14] The most natural candidate for proofs regarding term rewriting, however, is Recursion Theory, a direction we promote in this paper. We present some undecidability results for primitive term rewriting systems, which encode primitive recursive definitions. We also ....

Emil L. Post. Recursive unsolvability of a problem of Thue. J. of Symbolic Logic, 13:1--11, 1947.

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E. L. Post. Recursive unsolvability of a problem of thue. Journal of Symbolic Logic, 12:1--11, 1947.

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E.L. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12:1--11, 1947.

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E.L. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12:1--11, 1947.

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E.L. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12:1--11, 1947.

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E. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12(1):1-- 11, 1947.

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E. Post. Recursive unsolvability of a problem of Thue. J. Symbolic Logic, 12:1--11, 1947.

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E. Post. Recursive unsolvability of a problem of thue. In M. Davis, editor, The Undecidable, pages 292-303. Raven Press, Hewlett, New York, 1965.

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E. Post. Recursive unsolvability of a problem of thue. The Journal of Symbolic Logic, 12, 1947. Reprinted with discussion in [18].

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