M. O. Rabin, A simple method for undecidability proofs and some applications, Logic, Methodology and Philosophy of Science. I1, edited by Y. Bar-Hillel, North-Holland, Amsterdam, 1964, pp. 58-68.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....technique (which can be extended in this case to stationary logic) we get the interpretability of Th (K ) in Th. K) Now we prove the interpretability of the elementary theory of one binary irreflexive and symmetric relation in Th(K ) This is sufficient since this theory is undecidable (see [19]) At first we have to remember that there are many pairwise disjoint stationary subsets of (see [11] 29] Let A, v G ) be such sets. Now let (A, R) be an at most countable irreflexive and symmetric graph. Let B be the set of all elements of w 2 of the form ( w) n (for n G w) and ....

M. O. Rabin, A simple method for undecidability proofs and some applications, Logic, Methodology and Philosophy of Science. I1, edited by Y. Bar-Hillel, North-Holland, Amsterdam, 1964, pp. 58-68.

....do not have an induced subgraph which is planar and of type Q. Before this conjecture can be proved the following result is useful. Fact 2. For each natural number m let Wm be the class of all graphs whose tree width is m. The (weak) monadic theory of Wm is interpretable in the sense of [36] in. the (weak) monadic theory of the class of all trees. Hence both theories are decidable. To find the wished interpretation one simply has to code the treedecomposition of the graphs into trees using standard and folklore ideas of interpretability. The interpretation can in fact be done ....

M.O. Rabin, A simple method for undecidability proofs and some applications, in: Y. Bar-Hillel, ed., Logic, Methodology and Philosophy of Science (North-Holland, Amsterdam, 1965) 58-68.

....in a programming language. For program correctness, we would use Hoare logic to show that the store transformations leave the graph specifications satisfied. In this paper we consider restricted graph transformations, called transductions, which are based on the method of semantic interpretation [7] and studied in [3] Given logical graph specifications A and B and a transduction, we address the problem of verifying what we call transductional correctness: for any graph satisfying A, any graph resulting from the transduction satisfies B. This informal definition omits the difficulty of ....

M. Rabin. A simple method for undecidability proofs and some applications. In Logic, Methodology and Philosophy of Science II, pages 58--68. North-Holland, 1965. This article was processed using the L a T E X macro package with LLNCS style

....in a programming language. For program correctness, we would use Hoare logic to show that the store transformations leave the graph specifications satisfied. In this paper we consider restricted graph transformations, called transductions, which are based on the method of semantic interpretation [7] and studied in [3] Given logical graph specifications A and B and a transduction, we address the problem of verifying what we call transductional correctness: for any graph satisfying A, any graph resulting from the transduction satisfies B. This informal definition omits the difficulty of ....

M. Rabin. A simple method for undecidability proofs and some applications. In Logic, Methodology and Philosophy of Science II, pages 58--68. North-Holland,

....2 A is undecidable as well. Proof. To prove the former claim, it suffices to observe that FM[N] is powerful enough to interpret the theory of graphs (i.e. the theory of structures hW; Ri, where R is a symmetric and reflexive binary relation on W ) which is known to be hereditarily undecidable [23]. Indeed, let (x; y) be the formula ffi(x; y) 1 ffi(x; y) 0: Given a graph hW; Ri, we can define a metric space hW; di by taking, for all a; b 2 W , d(a; b) 8 : 0; if a = b, 1; if a 6= b and aRb, 2; if not aRb. We then clearly have hW; di ffl [a; b] iff aRb. ii) follows from ....

M. O. Rabin. A simple method for undecidability proofs and some applications. In Y. Bar-Hillel, editor, Logic and Methodology of Sciences, pages 58--68. North-Holland, 1965.

....the last assertion, the (2,1) de nability is an easy consequence of the de nition of a line graph. See [23] for the negative part of this assertion. 2 4. 2 The fundamental property of de nable transductions The following proposition is the basic fact behind the notion of semantic interpretation ([53]) It says that if T = def (S; i.e. if T is de ned in (S; by , then the monadic second order properties of T can be expressed as monadic second order properties of (S; The usefulness of de nable transductions is based on this proposition. Let = 1 ; k ; w ) w2Q k ....

: RABIN M., A simple method for undecidability proofs and some applications, in \Logic, Methodology and Philosophy of Science II", Y. Bar-Hillel ed., North-Holland, Amsterdam, 1965, pp.58-68.

....the following fact. 39 Theorem 8.6 The SUCCESS problem for nonrecursive logic programs over S 0 is undecidable. 2 We leave a detailed proof to the reader. The idea is that it is possible to interpret the theory of a finite binary relation (Kalmar Traktenbrot VaughtRabin) cf. Vaught 1960, Rabin 1964). Indeed, using the infinite atomic domain D we can express a finite set of any size (as any finite set S of elements of D) and interpret an arbitrary binary relation as a set of pairs fa; bg with elements in D. Analogously, if the atomic domain is finite, but variables and the membership ....

....universe Similarly, we obtain Theorem 8.7 The SUCCESS problems for nonrecursive logic programs over the untyped universe and the colored universe are undecidable. 2 This follows from the undecidability of the theory of a binary relation (Kalmar, Traktenbrot, Vaught, Rabin) cf. Vaught 1960, Rabin 1964), or from Gandy s theorem (Barwise 1975) 40 9 Summary of results The complexity of the success problem for nonrecursive logic programs over trees are summarized in the following table. In all cases we have completeness in the corresponding complexity class, except for NONELEMENTARY (in this ....

Rabin, M. O. (1964), A simple method for undecidability proofs and some applications, in Y. Bar-Hillel, ed., `Proc. Intern. Congress on Logic, Methodology and Philosophy of Science', Studies in Logic and the Foundations of Mathematics, pp. 58--68.

....this case we have the following fact. Theorem 8.6 The SUCCESS problem for nonrecursive logic programs over S 0 is undecidable. 2 We leave a detailed proof to the reader. The idea is that it is possible to interpret the theory of a finite binary relation (Kalmar Traktenbrot Vaught Rabin) cf. [57, 48]. Indeed, using the infinite atomic domain D we can express a finite set of any size (as any finite set S of elements of D) and interpret an arbitrary binary relation as a set of pairs fa; bg with elements in D. Analogously, if the atomic domain is finite, but variables and the membership ....

....8.3 Untyped universe Similarly, we obtain Theorem 8.7 The SUCCESS problems for nonrecursive logic programs over the untyped universe and the colored universe are undecidable. 2 This follows from the undecidability of the theory of a binary relation (Kalmar, Trakhtenbrot, Vaught, Rabin) cf. [57, 48], or from Gandy s theorem [8] Our Theorems 8.6 and 8.7 on undecidability of the SUCCESS problem for nonrecursive logic programs over untyped sets should be contrasted to the result of [28] who show that their set based query language over untyped sets without while is expressively equivalent ....

M. O. Rabin. A simple method for undecidability proofs and some applications. In Y. Bar-Hillel, editor, Proc. Intern. Congress on Logic, Methodology and Philosophy of Science, Studies in Logic and the Foundations of Mathematics, pages 58--68, 1964.

....lower bounds techniques we apply to prove our main Theorem 7. In 1936 L. Kalmar proved that the first order theory of a binary relation is undecidable, which greatly simplified undecidability proofs, as compared to those based on straightforward encodings of Turing machines, see, e.g. M. Rabin [13]. B. Trakhtenbrot [19] and later R. Vaught [20] proved even stronger Theorem 10 . Let L be the first order language with the unique binary relation symbol. The set of valid sentences of L and the set of sentences of L refutable by some finite model are recursively inseparable. 2 Two sets are ....

....no recursive sets containing one and disjoint with the other. Notice that recursive inseparability is stronger than simple undecidability: both of recursively inseparable sets are undecidable. Usually Theorem 10 is applied in conjunction with the method of interpretations, extensively discussed in [13, 4]. Recently, K. Compton and C. Henson [2] refined the above inseparability idea for proving hereditary lower complexity bounds for logical theories. ComptonHenson s method is based on interpretations, in a given theory, of all finite binary relations up to a certain size, by means of short ....

M. O. Rabin. A simple method for undecidability proofs and some applications. In Y. Bar-Hillel, editor, Proc. Intern Congress on Logic, Methodology and Philosophy of Science, Studies in Logic and the Foundations of Mathematics, pages 58--68. North-Holland, 1964.

....new notion of dependency preserving refinements of database schemes which is meaningful for arbitrary first order dependencies. The basic underlying notion is a first order translation scheme. It is based on the classical syntactic notion of interpretability from logic made explicit by M. Rabin in [Rab65]. More recently, translation schemes have been used in descriptive complexity theory under the name of first order reductions [Dah83] and [Imm87] and in descriptive graph theory [Cou94] A systematic survey of their use to analyze complexities of logical theories, cf. CH90] In section 2 we ....

M.A. Rabin. A simple method for undecidability proofs and some applications. In Y. Bar Hillel, editor, Logic, Methodology and Philosophy of Science II, Studies in Logic, pages 58--68. North Holland, 1965.

....and dependency preserving refinements of database schemes which is meaningful for arbitrary first order dependencies. The basic underlying notion is a first order translation scheme. It is based on the classical syntactic notion of interpretability from logic made explicit by M. Rabin in [Rab65]. More recently, translation schemes have been used in descriptive complexity theory under the name of first order reductions [Dah83] and [Imm87] and in descriptive graph theory [Cou94] A systematic survey of their use to analyze complexities of logical theories, cf. CH90] The paper is ....

M.A. Rabin. A simple method for undecidability proofs and some applications. In Y. Bar Hillel, editor, Logic, Methodology and Philosophy of Science II, Studies in Logic, pages 58--68. North Holland, 1965.

....in a programming language. For program correctness, we would use Hoare logic to show that the store transformations leave the graph specifications satisfied. In this paper we consider restricted graph transformations, called transductions, which are based on the method of semantic interpretation [7] and studied in [3] Given logical graph specifications A and B and a transduction, we address the problem of verifying what we call transductional correctness: for any graph satisfying A, any graph resulting from the transduction satisfies B. This informal definition omits the difficulty of ....

M. Rabin. A simple method for undecidability proofs and some applications. In Logic, Methodology and Philosophy of Science II, pages 58--68. North-Holland, 1965. This article was processed using the L a T E X macro package with LLNCS style

....K determines its semantics. We first generalize the notion of first order reductions as introduced by Immerman and Dahlhaus [Dah82, Dah83, Imm87] to any regular logic L. We first presented this concept in [MP93] Our definition is very close to Rabin s notion of interpretability as described in [Rab65]. A similar notion is also used in [Cou92] and, for general logics, in [Daw94] Definition18 (L reducibility) Let K 1 ; K 2 be classes of 1 ; 2 structures closed under isomorphisms and L be a regular logic. i) Let 2 = fR 1 ; Rm g and let ae(R i ) be the arity of R i . Let Phi ....

M.A. Rabin. A simple method for undecidability proofs and some applications. In Y. Bar Hillel, editor, Logic, Methodology and Philosophy of Science II, Studies in Logic, pages 58--68. North Holland, 1965.

....a simple instance of such a translation scheme. More complicated examples are given in section 6. 4 Translation Schemes In this section we introduce the general framework for syntactically defined translation schemes. We introduce also the notion of abstract translation schemes according to Rabin [Rab65]. N. Immerman and E. Dahlhaus have used translation schemes in analyzing the expressive power of various logics over finite structures, Imm87, Dah82] B. Courcelle has used such translation schemes extensively in his studies of Monadic Second Order Properties of graphs under the name of ....

M.A. Rabin. A simple method for undecidability proofs and some applications. In Y. Bar Hillel, editor, Logic, Methodology and Philosophy of Science II, Studies in Logic, pages 58--68. North Holland, 1965.

....and dependency preserving refinements of database schemes which is meaningful for arbitrary first order dependencies. The basic underlying notion is a first order translation scheme. It is based on the classical syntactic notion of interpretability from logic made explicit by M. Rabin in [Rab65]. More recently, translation schemes have been used in descriptive complexity theory under the name of first order reductions [Dah83] and [Imm87] and in descriptive graph theory [Cou94] A systematic survey of their use to analyze complexities of logical theories, cf. CH90] The paper is ....

M.A. Rabin. A simple method for undecidability proofs and some applications. In Y. Bar Hillel, editor, Logic, Methodology and Philosophy of Science II, Studies in Logic, pages 58--68. North Holland, 1965.

....in T . In order to show relative weak interpretability of T 0 in T one has to find first order formulas defining the universe and operations of T 0 in some consistent extension of T . Hence the correspondence between the theories is expressed completely within the logic. The method of Rabin (1965) does not require a finite axiomatization of the underlying undecidable theory. Rabin (1965) summarizes his proof principle as follows: If T 0 is an undecidable theory and T is a theory such that by using appropriate formulas of T to represent the universe of T 0 and the non logical constants ....

....first order formulas defining the universe and operations of T 0 in some consistent extension of T . Hence the correspondence between the theories is expressed completely within the logic. The method of Rabin (1965) does not require a finite axiomatization of the underlying undecidable theory. Rabin (1965) summarizes his proof principle as follows: If T 0 is an undecidable theory and T is a theory such that by using appropriate formulas of T to represent the universe of T 0 and the non logical constants of T 0 , every model of T 0 is obtained from some model of T , then T is also ....

Rabin, M. O. (1965). A simple method for undecidability proofs and some applications. In Bar-Hillel, Y., editor, Logic, Methodology and Philosophy of Science, pages 58--68. North-Holland.

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