W. J. Quine, Concatenation as a basis for arithmetic, J. Symbolic Logic 11 (1946), 105--114.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... are all first order definable in , Thus, in case the first order theory of is decidable. Now suppose there are T . Then the structure can be interpreted by a first order formula in . Since the first order theory of is undecidable [56, 42, 21], we get Proposition 3.7. The first order theory of is decidable if and only if Id . Note that the undecidability arises from the monoid operation. Therefore, we will concentrate on reducts of where the signature is restricted to the other relations. For a scattered rewriting system we ....

W. V. O. Quine. Concatenation as a basis for arithmetic. Journal of Symbol Logic, 11(4):105--114, 1946. 42

....we can at best show undecidability of the 3 fragment of the theory. An improvement of the method using encodings of sequences instead of sets, and that can yield undecidability results for the 2 fragment of a theory, is given in [Tre92] 4. 2 Example: The Theory of Words Theorem 17 ( Qui46] The rst order theory of words is undecidable. We can show this classical result now easily by using Theorem 16. Proof: For simplicity we assume that the alphabet contains four letters a; b; c; d. The proof for a binary alphabet fa; bg is obtained by encoding letters over fa; b; c; dg as ....

W. V. Quine. Concatenation as a basis for arithmetic. Journal of Symbolic Logic, 11(4), 1946.

....space 2 2 cn for some c 0, and thus can be considered a useful alternative to the (unbounded) theory of finite trees. Although the analogy is not complete here, we would like to recall the similar situation with the full first order theory of binary concatenation 2 , which is undecidable [11, 13], and the theory of t bounded concatenation 3 [3, 2] which is decidable within elementary space and time if the function t is computable in elementary space. In Sections 2 and 3 we fix notation and give basic definitions. In Section 4 we introduce two definitionally equivalent versions of the ....

W. V. Quine. Concatenation as a basis for arithmetic. J. Symb. Logic, 11(4):105-- 114, 1946.

....theory fragment (Treinen 1996) used the proof; see Section 5 for details. On the other hand, Vorobyov 1995) presented a simple fixed rewrite rule system with undecidable theory of one step rewriting, by using a reduction from the undecidable theory of binary concatenation (free semigroup) (Quine 1946). We therefore distinguish between the weak undecidability, i.e. non existence of a general algorithm applicable to all systems uniformly, and strong undecidability, i.e. undecidability of the theories of fixed systems. It should be noted that both (Treinen 1996) and (Vorobyov 1995) con3 ....

....1997) where a fixed finite, simultaneously finitely terminating and linear system with undecidable theory of one step rewriting was constructed. The proof again was given by reduction from the theory of binary concatenation (finitely generated free semigroups) well known to be undecidable (Quine 1946). As a practical drawback compensating for the ease of reduction, the quantifier alternation of the sentences forming the undecidable class was quite high. Then (Marcinkowski 1997) showed that no algorithm is capable of deciding the 9 8 theory of one step rewriting of an arbitrary finite ....

Quine, W. V. (1946), `Concatenation as a basis for arithmetic', J. Symb. Logic 11(4), 105--114.

....of arbitrary length instead of just a single atomic value. This definition was essentially introduced in [14] an later used in e.g. 17] and [24] However, a brief excursion into the history books reveals for instance that Stockmeyer [34] was familiar with string relations. Earlier still Quine, [26] showed that first order logic over strings is undecidable. From the point of view of design, in addition to data extraction features, such as retrieve all palindromes, the string language needs also data restructuring constructs [14, 17, 24, 35] For example, given two unary relations, one ....

....userfriendly, and of low time complexity, without recursion and within logspace. The user friendly aspect was part of the motivation of the theory of range restriction, inspiring numerous works (see e.g. 1] Chap. 5.3 and 5. 4) In this paper we return to the spirit of the founding fathers ([34, 26]) In other words, we will use as string language relational calculus with an interpreted concatenation function. Our syntax will be called FO(ffl) where the symbol ffl stands for concatenation. We will define various semantics, one of them yielding our main language. Since according to a central ....

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W. V. Quine. Concatenation as a basis for arithmetic. Journal of Symbolic Logic, 11(4):105-114, 1946

....one outermost universal quantifier to a context equation (1) leads to the Pi 0 1 hard class of formulas, where Pi 0 1 is the class of all co recursively enumerable sets. For comparison, the following undecidability results are known about quantified fragments of context unification. Quine [10] showed that the full firstorder theory of free semigroups is undecidable (this corresponds to context unification in unary signatures) Durnev [3] improved it to the undecidability of 989 3 positive (without negation, but with and ) theory of free semigroups. Marchenkov [8] improved it to the ....

W. V. Quine. Concatenation as a basis for arithmetic. J. Symb. Logic, 11(4):105-- 114, 1946.

....basis of our decision and complexity analysis method. We carry over this machinery to the case of infinite signatures. Although the analogy is not complete here, we would like to recall a rather similar situation with the full first order theory of binary concatenation 1 , which is undecidable [Qui46, Smu61], and the theory of t bounded concatenation 2 [BM80, Ber80] which is decidable within elementary space and time if the function t is computable in elementary space. Venkataraman in [Ven87] showed that the first order theory of finite trees with the subtree predicate (s t meaning that s is a ....

W. V. Quine. Concatenation as a basis for arithmetic. J. Symb. Logic, 11(4):105--114, 1946.

....unified decision method for many problems in rewriting. The decidability of the problem was widely believed and conjectured. We present a finite term rewriting system R such that the dyadic word concatenation 1 is expressible via the one step rewriting relation in R. Consequently (Quine [5], Smullyan [6] the elementary theory of onestep rewriting is undecidable, in general. 1 Dyadic Concatenation Let f0; 1g be the set of finite words over a two letter alphabet f0; 1g. Consider the first order model hf0; 1g ; Conci of signature fConcg with f0; 1g as a carrier, and the ....

....Conc interpreted as follows: Conc(x; y; z) is true for x; y; z 2 f0; 1g if and only if x Delta y = z, i.e. the concatenation of x and y equals z. Recall that the elementary theory of a model M of signature Sigma is the set of all first order sentences of signature Sigma true in M . Quine [5] and Smullyan [6] showed that the recursion theory can be developed independently of arithmetic, in terms of dyadic concatenation. As a consequence, Theorem1 ( 5, 6] The elementary theory of the model hf0; 1g ; Conci is undecidable. ut We will present a finite term rewriting system R such ....

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W. V. Quine. Concatenation as a basis for arithmetic. J. Symb. Logic, 11(4):105-- 114, 1946.

....and the dyadic word concatenation (i.e. concatenation of words in a two letter alphabet) can be represented by means of formulas of L for an appropriate system R with desirable properties. The undecidability of the theory of one step rewriting in R then follows by the well known Theorem 1 (Quine, [1]) The first order theory of dyadic concatenation is undecidable. ut We employ the same reduction as in [3] However, to detect properties of terms by means of a one step reducibility relation in a linear finitely terminating system we use a different approach based on rewriting diagrams. Very ....

W. V. Quine. Concatenation as a basis for arithmetic. J. Symb. Logic, 11(4):105-- 114, 1946.

....[2, 3] In a recent paper [4] we demonstrated that the full first order theory of one step rewriting in one particular finite linear finitely terminating systems is undecidable. We used the reduction from the first order theory of binary concatenation (free semigroups) known to be undecidable [1]. In the current note, by applying the reduction from the Post correspondence problem we show that there are finite linear finitely terminating systems with undecidable 989 theories of one step rewriting. Using PCPs is quite usual in undecidability proofs, cf. e.g. 2] but our construction of ....

W. V. Quine. Concatenation as a basis for arithmetic. J. Symb. Logic, 11(4):105-- 114, 1946.

....one outermost universal quantifier to a context equation (1) leads to the Pi 0 1 hard class of formulas, where Pi 0 1 is the class of all co recursively enumerable sets. For comparison, the following undecidability results are known about quantified fragments of context unification: 1. (Quine 1946) showed that the full first order theory of free semigroups 1 is undecidable. 2. Durnev 1973) improved it to undecidability of 989 3 positive (without negation, but with and ) theory of free semigroups. 3. Marchenkov 1982) improved it to undecidability of 89 4 positive theory of free ....

Quine, W. V. (1946), `Concatenation as a basis for arithmetic', J. Symb. Logic 11(4), 105--114.

....is decidable. For the algebra of rational trees with lists, the positive existential fragment of the theory is decidable. Problems in the existential fragment may be traced back to a difficult question about solvability of word equations with length constraints for variables. 1 Introduction Quine [9] has shown that the theory of concatenation is undecidable. The existential fragment of the theory was shown to be decidable by Buchi and Senger [3] building up on Makanin s decidability result for solvability of word equations [7] Concatenation, in the sense of Quine, is an operation acting on ....

W.V. Quine, "Concatenation as a Basis for Arithmetic," J. Symbolic Logic 11 (1946), pp.105-114.

....a finite set of equational axioms E, generating a congruence =E on the term algebra. For example, some symbols can be assumed commutative, or associative and commutative. The first order theory of a quotient algebra T (F ) E quickly becomes undecidable: a single associative symbol suffices [41], or an associativecommutative symbol [44] Decidability results include the case where E is a set of flat permutative axioms [35] ground axioms [12] and E is a set of shallow equations, a class which encompasses the two previous ones [17] 3.3 Feature constraints Feature trees have been ....

W. V. Quine. Concatenation as a basis for arithmetic. Journal of Symbolic Logic, 11(4), 1946.

....prefix a string with a fixed symbol. With the representation of strings as terms, this kind of left concatenation corresponds to the application of a unary function symbol. Sketch of the proof: We encode the concatenation of words, whose first order theory is known to be undecidable (see e.g. [23]) We use an additional symbol # and successively express the following properties: x# z, where x contains no #: OE 1 (x; z) def = x z 8y(#y z y = z = x#y (and x; y are # free) OE 2 (x; y; z) def = # 6 x # 6 y ## 6 z 8u[## 6 u (z u (OE 1 (x; u) #y u) This reads: z is ....

W. V. Quine. Concatenation as a basis for arithmetic. Journal of Symbolic Logic, 11(4), 1946.

....prefix a string with a fixed symbol. With the representation of strings as terms, this kind of left concatenation corresponds to the application of a unary function symbol. Sketch of the proof: We encode the concatenation of words, whose first order theory is known to be undecidable (see e.g. [23]) We use an additional symbol # and successively express the following properties: x# z, where x contains no #: OE 1 (x; z) def = x z 8y(#y z y = z = x#y (and x; y are # free) OE 2 (x; y; z) def = # 6 x # 6 y ## 6 z 8u[## 6 u (z u (OE 1 (x; u) #y u) This reads: z is ....

W. V. Quine. Concatenation as a basis for arithmetic. Journal of Symbolic Logic, 11(4), 1946.

....free function A New Method for Undecidability Proofs of First Order Theories 3 symbols is decidable. The extension by the axiom of idempotency to ACI in Example (B) is straightforward. A related result is the undecidability of the theory of ground terms modulo associativity alone (Example (F) Quine (1946) showed already the undecidability of the theory of concatenation. He gives a translation of number theory to the theory of concatenation that yields a Sigma 6 sentence for an instance of Hilbert s Tenth Problem, using the undecidability of Hilbert s Tenth Problem (Matijacevic (1970) this ....

....codings of the configurations. Another popular candidate for reduction is complete number theory. One may use the result of Matijacevic (1970) on the unsolvability of Hilberts Tenth Problem and reduce the Sigma 1 fragment hoping to obtain a formula solvable P in a pretty small fragment. In fact Quine (1946) gives a reduction of complete number theory to the theory of concatenation of strings over the alphabet fa; bg. The number n is coded by the string consisting of n a s, 18 Ralf Treinen such that addition of numbers corresponds to the concatenation of strings. Multiplication is expressed with ....

Quine, W. V. (1946). Concatenation as a basis for arithmetic. Journal of Symbol Logic, 11(4):105--114.

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W. J. Quine, Concatenation as a basis for arithmetic, J. Symbolic Logic 11 (1946), 105--114.

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Quine, W. V. (1946), Concatenation as a basis for arithmetic, J. Symbolic Logic 11(4), 105--114.

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W. V. Quine. Concatenation as a basis for arithmetic. J. Symb. Logic, 11(4):105-114, 1946.

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W.V. Quine, 1946, Concatenation as a basis for arithmetic, J. Symbolic Logic 11, 105--114.

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