Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479--506. Springer, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is too weak: confluence is a property of the reflexive transitive closure of , but the former is in general not first order definable in 0 . Hence, in order to express confluence, one has to use FO logic over or, alternatively, monadic second order logic (MSO logic) over 3 [26] (which allows to define the reflexive transitive closure) Known (un)decidability results for the above mentioned logics in the case of term rewriting systems and semi Thue systems, respectively, are collected in Table 1. Hence, most theories are undecidable. Note that the undecidability of the ....

Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479--506. Springer, 1985.

....to the rst order theory of its factors. Composition theorems for theories of orderings were rst explored by L auchli [20] and subsequently developed by Shelah [23] The technique was used in a series of papers by Gurevich and Shelah [9,11,12,14] and outlined in a survey exposition by Gurevich [10]. Thomas [24] provides an overview on using composition theorems where he suggests that, despite their success, such techniques are still largely ignored by the theoretical computer science community in favour of the well established automata theoretic techniques. He emphasizes the importance of ....

Y. Gurevich (1985). Monadic second-order theories. In Model-Theoretic Logics, (J. Barwise and S. Feferman, eds.), pp479-506, Springer-Verlag.

....with tree automatic presentation strictly includes the class of automatic structures (see [5] 2) The structure (N, stands for having no common divisor is automatic. 7 Composition of structures The composition method developed by Feferman and Vaught [21] and by Shelah [39] see also [26, 41]) considers compositions (products and sums) of structures according to some index structure and allows one to compute depending on the type of composition the first order or monadic second order theory of the whole structure from the respective theories of its components and the monadic ....

Y. Gurevich, Monadic second-order theories, in Model-Theoretic Logics, J. Barwise and S. Feferman, eds., Springer, 1985, pp. 479--506.

....more encyclopedic. In the area of (modal and) temporal logic good background texts are e.g. 30, 56, 21] For surveys Stirling s paper [51] is highly recommendable. Recommendable too are [15, 39] Finally, on the topic of classical second order theories, a useful reference is Gurevich s chapter [22]. Acknowledgements Colin Stirling helped initiate the writing of these notes. Thomas s handbook chapter [55] provided lots of inspiration. Thanks to David Turner for valuable suggestions, in particular concerning the expressive completeness proof for linear time temporal logic with until. s 2 ....

....connectives may be prohibitively costly. 4. Results similar to the above for the second order monadic theory of the integers (Z; can be obtained using automata theoretic techniques. The technique we used for proving expressive completeness for F(fl; U) has its roots with Shelah [49] see also [22]) The methods introduced by Shelah have been used to prove decidability and undecidability for a variety of orderings and, most notably: Theorem 30.1 (Shelah 1975) The monadic theory of (R; is undecidable. 2 31 Branching Time Logics A new dimension is added when formulas are allowed to talk ....

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Y. Gurevich. Monadic second-order theories. In Model-Theoretic Logics, J. Barwise and S. Feferman (eds), Springer-Verlag, pages 479--506, 1985.

....For distributive lattices (L 1 ; and (L 2 ; one has J(L 1 L 2 ; J(L 1 ; J(L 2 ; Thus, we have to show that the monadic theory of f(P 1 ; P 2 ; j (P i ; 2 J(L i )g is decidable. This follows from the composition theorem from Shelah [She75, Theorem 2. 4] cf. [Gur85] for the proof of this result) since MTh(J(L i ) is decidable. Note that in the corollary above we assumed from the very beginning that MTh(L i ) is decidable for i = 1; 2. Actually, the niteness of L 1 or L 2 follows without this assumption from the decidability of MTh(L) We nish this ....

Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479-506. Springer, 1985.

....second order theory of the whole structure from the respective theories of its components and the monadic theory of the index structure. The characterisation given in the previous section can be used to prove closure of automatic structures under such compositions of finitely many structures (see [23, 13, 16]) A generalised product as it is defined below is a generalisation of a direct product, a disjoint union, and an ordered sum. We will prove that given a finite sequence (A i ) i of structures which belong to some class K containing a complete structure, all their generalised products are ....

Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479--506. Springer, 1985.

....If C = fSg where S is in nite, then the L theory of C may be undecidable. However the study of the theories of in nite structures and especially of their monadic (second order) theories is a very rich topic that we cannot even touch here. We refer the reader to survey papers like those of Gurevich [44], Courcelle [10] Thomas [62] Proposition 1.1 (Trakhtenbrot [64] The rst order theory of the class of nite graphs is undecidable. So are a fortiori its monadic and its second order theories. The following result is a basic tool for obtaining undecidability results. It concerns square ....

: GUREVICH Y., Monadic second-order theories, in J. Barwise and S. Feferman eds., \Model theoretic logic", Springer, Berlin, 1985, pp. 479-506.

....second order theory of the whole structure from the respective theories of its components and the monadic theory of the index structure. The characterisation given in the previous section can be used to prove closure of automatic structures under such compositions of finitely many structures (see [23, 13, 16]) A generalised product as it is defined below is a generalisation of a direct product, a disjoint union, and an ordered sum. We will prove that given a finite sequence (A i ) i of structures which belong to some class K containing a complete structure, all their generalised products are ....

Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479--506. Springer, 1985.

....Theorem In the proof of Theorem 4 we shall use a version of the Feferman Vaught Theorem, FV59] adapted to MSOL. It is not clear who observed first that this adaptation to MSOL is true, but it is already in [Lau68, She75] and follows from [Fef57, Ehr61] For a good exposition, cf. [Gur79, Gur85]. We review some notation from [CM93] Definition34. Let A be a structure, let A be the domain of A and let be a MSOL( formula with free set variables X 1 ; Xn . We denote by sat(A; the set of n tuples of subsets of A for which holds in A. Formally: sat(A; f(D 1 ; ....

.... and have quantifier depth no larger than the quantifier depth of , and for every two p graphs G and H presented over 1;p such that V (G) V (H) sat(G Phi H; 1im sat(G; i ) Thetasat(H; i ) Proof: Immediate reformulation of the result by Feferman Vaught as discussed in [Gur85]. The result can also be proved directly using pebble games for MSOL. 2 A more sophisticated construction where the union is disjoint can be derived as in Lemma 2.4 of [CM93] but is not needed here. 4.4 The linear time algorithms The main ideas for proving Theorem 4 are as follows: i) If G is a ....

Y. Gurevich. Monadic second order theories. In Model-Theoretic Logics, Perspectives in Mathematical Logic, chapter 14. Springer Verlag, 1985.

....we have two successors, for left and right . Then, just as finite sets of positions over one successor are encoded by strings, sets of positions over two successors are encoded by binary trees. Working out the details yields the weak secondorder monadic theory of two successors (or WS2S) [9, 15, 17]. The syntax of WS2S is similar to WS1S except we use (instead of 0) to denote the root address in a tree and we now have two successor :0 (left child) and :1 (right child) Semantically, WS2S formulae are interpreted in the domain D = f0; 1g and denotes the empty string, 0 denotes ....

Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479--506. Springer, 1985.

.... as an alternative to the automata theoretic approach popularized by Buchi, were first explored by Lauchli [16] and subsequently developed greatly by Shelah [20] The technique was used in a series of papers by Gurevich and Shelah [7, 9, 10, 11] and outlined in a survey exposition by Gurevich [8]. Hafer and Thomas [12] provide a composition theorem for MPL over full binary trees, and in the present paper we prove a composition theorem for MPL over wide trees. Thomas [21] provides a recent overview on using composition theorems where he suggests that, despite their success, such techniques ....

Y. Gurevich (1985). Monadic second-order theories. In Model-Theoretic Logics, (J. Barwise and S. Feferman, eds.), pp479-506, Springer-Verlag.

....tool. We conclude by applying a constraint logic programming (CLP) extension resulting from the embedding of the compiler as its constraint solver to remaining engineering problems. We assume some basic familiarity with both the decidability proof for WS2S (an introductory presentation is in Gurevich 1985) and tree automata (an introduction can be found in G ecseg and Steinby 1997) 2 From Logic to Automata MSO logic is a straightforward extension of rst order logic to include variables that range over sets (i.e. monadic predicates) and quanti ers over these variables. Let Nn denote the ....

Gurevich, Y. (1985). Monadic second-order theories, in J. Barwise and S. Feferman (eds), Model-Theoretic Logics, Springer, Heidelberg, pp. 479-506.

.... used as an alternative to the automata theoretic approach popularized by Buchi, were first explored by Lauchli [15] and subsequently developed by Shelah [18] The technique was used in a series of papers by Gurevich and Shelah [6, 8, 9, 10] and outlined in a survey exposition by Gurevich [7]. Hafer and Thomas [11] provide a composition theorem for MPL over binary trees, and in the present paper we prove a composition theorem for MPL over wide trees. Thomas [19] provides a recent overview on using composition theorems where he suggests that, despite their success, such techniques are ....

Y. Gurevich (1985). Monadic second-order theories. In Model-Theoretic Logics, (J. Barwise and S. Feferman, eds.), pp479-506, Springer-Verlag.

....it is necessary that ST outputs non Zeno signals. It can be shown that for every retrospective speed independent function G there exists a system of equations that defines G. 9.3. Monadic Second Order Theory of Order Recall that the language of monadic second order theory of order (see e.g. [5, 18]) has individual variables, monadic second order variables, a binary predicate , the usual propositional connectives and first and second order quantifiers. The atomic formulas are formulas of the form: t v and x(t) b, where t; v are individual variables and x is a monadic second order ....

Y. Gurevich. Monadic Second-Order Theories. In Handbook of Model-Theoretical Logics, Eds J. Barwise and S. Feferman, pp: 479-506, 1985, Springer Verlag.

....and [LuSh 435] are particular cases: the probability for fi; jg being an edge of M n for i; j 2 [n] is p n i . So in [ShSp 304] p n i = p n and in [LuSh 435] p n i = p i . We then note that we shall get the same theory as in case (A) above in the limit, while simplyfying the probabilistic arguments, if we change the context to: Second context for M n (graph on f1; ng) with probability of fi; jg being an edge is p n i = 1 n a 1 2 ji Gammajj . So the probability basically has two parts 1) 1 2 ji Gammajj ) depends only on the distance, but decays fast, so the ....

.... (x 0 ; x 1 ; x 2 ; a 0 ) 0 ; 1 ; 2 2 = 0 1 (x 0 ; x 1 ; x 2 ) b 0 ; b 1 ; b 2 ) M n ffl 3 (x 0 ; x 1 ; x 2 ; a 0 ) 0 ; 1 ; 2 2 = 0 1 (x 0 ; x 1 ; x 2 ) b 0 ; b 1 ; b 2 ) But it is not a priori clear whether our first order formulas distinguish large and small in such interpretation. Note: all this not why 0 1 law holds, just explain the situation, and show we cannot prove the theory is too nice on the one hand but that this is not sufficient for failure of 0 1 law. Still what we say applies to both contexts, which shows that ....

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Yuri Gurevich. Monadic Second--Order Theories. In J. Barwise and S. Feferman, editors, Model Theoretic Logics, Perspectives in Mathematical Logic, chapter XIII, pages 479--506. Springer-Verlag, New York Berlin Heidelberg Tokyo, 1985.

....path. Suppose, for contradiction, that C is defined by a m.SO(9) sentence . Then, over the class D, defines D 0 . Now, view each A 2 D as a word, in the sense of formal language theory, over the alphabet fm; og, which contains exactly one occurence of the letter m) By Buchi s theorem, see [12]) every m.SO sentence defines a regular language over word models , as above. But D 0 is not a regular language, so it cannot be defined by any m.SO sentence. One can also prove the result using a simple m.SO pebble game instead of Buchi s theorem. The next proposition separates the first two ....

Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479--506. Springer-Verlag, 1985.

....of the first order language of the binary tree by allowing quantification over arbitrary subsets as well as over elements. Afterwards we define tree automata and quote some facts linking these objects to monadic logic. For more about monadic languages and on their links to tree automata see [Sh] [Gu], GuSh] and [Th] Notation. We use upper case letters X;Y; P; Q; Q : to denote set variables and constants. x; y; oe; denote element variables and constants. Definition 4.1. Let L be the first order language whose non logical symbols are the binary relations left(y; x) and right(y; ....

....q) is successful if for every branch B Sigma there exists a pair (L i ; U i ) 2 F such that Phi j 2 B Sigma odd : r(j) 2 L i Psi is finite Phi j 2 B Sigma odd : r(j) 2 U i Psi is infinite. Note. This is an adjustment of the definition of Rabin tree automata (see [Gu] and [Th] In a Buchi tree automaton the accepting condition is a subset F S and in our case we will demand that a successful run satisfies: for every branch B Sigma, Phi j 2 B Sigma odd : r(j) 2 F Psi is infinite. Definition 4.5. Given a k tree automaton A let Sigma(A) denote ....

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Y. Gurevich, Monadic Second--order Theories, Model Theoretic Logics (J. Barwise and S. Feferman, eds.), Springer-Verlag, Berlin, 1985, pp. 479--506.

....and with Magidor in [GMS83] Other applications appeared in [Th80] Gu82] CFGS82] and [Ze94] This list is not claimed to be complete; in particular, there may be further work of Shelah on the topic which is not cited here. Although the subject was exposed in Gurevich s concise survey [Gu85], it did not attract much attention among theoretical computer scientists. Preference was (and still is) given to the automata theoretic method: by its connection with a computational model it looks more intuitive, it incorporates programs in the form of state transition systems, and it does not ....

....to (A ; P ) 2I . Clearly, these Q j define a partition of I. It will turn out that, for a suitable r, the r type of such an expansion suffices to determine T k ( P 2I (A ; P ) We present the result here at a more relaxed pace than in [Sh75] where it is just stated) and [Gu79] [Gu85] (where the proofs are rather condensed and given in the more abstract framework of arbitrary signatures and general Feferman Vaught type theorems) Our specific choice of signature leads to a certain simplification. Theorem 5. Composition Theorem, Sh75] Gu79] From a sequence k = k 1 ; ....

Y. Gurevich, Monadic second-order theories, in: Model-Theoretic Logics (J. Barwise, S. Feferman, Eds.), Springer-Verlag, Berlin-Heidelberg-New York 1985, pp. 479-506.

.... (since the graphs of all primitive recursive functions are definable) but the monadic second order theory of even two successors (i.e. the monadic second order theory of binary trees) is decidable [ Rabin, 1969 ] A discussion and compendium of monadic second order theories can be found in [ Gurevich, 1985 ] 6.2 Restricted expressiveness 2: Fixpoint logics Because inductive definitions play particularly important roles in various applications of higher order logic, it is natural to isolate them, and to consider extensions of first order logic with fixpoint constructs. 88 The study of proof ....

Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretical Logics, pages 479--506. Springer-Verlag, Berlin, 1985.

....Feferman Vaught Theorem In the proof of Theorem 4 we shall use a version of the Feferman Vaught Theorem, FV59] adapted to MSOL. It is not clear who observed first that this adaptation to MSOL is true, but it is already in [Lau68, She75] and follows from [Fef, Ehr61] For a good exposition, cf. [Gur79, Gur85]. We review some notation from [CM93] Definition22. Let A be a structure, let A be the domain of A and let be a MSOL( formula with free set variables X 1 ; Xn . We denote by sat(A; the set of subsets of A n for which holds in A. Formally: sat(A; f(D 1 ; Dn ) ....

.... and have quantifier depth no larger than the quantifier depth of , and for every two k graphs G and H represented over 1;L such that V (G) V (H) sat(G Phi H; 1im sat(G; i ) Thetasat(H; i ) Proof. Immediate reformulation of the result by Feferman Vaught as discussed in [Gur85]. The result can also be proved directly using pebble games for MSOL. 3.4 Linear algorithms for optimization problems on bounded clique width graphs Let G be a graph, let f 1 ; fm be m evaluation functions associating integer values to the vertices of G, let D 1 ; D l V (G) ....

Y. Gurevich. Monadic second order theories. In Model-Theoretic Logics, Perspectives in Mathematical Logic, chapter 14. Springer Verlag, 1985.

....For every monadic first order sentence that respects fin there exists a PDC formula D such that and D are equivalent. Remark (about the proof of Theorem 5.3. The proof uses model theoretical techniques developed by Shelah [21] for the decidability of the second order monadic logic (see [4] for a survey of these techniques) The proof will be given in the full version of the paper. Let us only mention that it provides an explicit translation from monadic sentences into PMVC and PDC. A syntactical proof of Theorem 5.3 can be based on the techniques from [16] Remark It is widely ....

Y. Gurevich. Monadic second order theories. In J. Barwise and S. Feferman eds., Model theoretical logics, pp. 479-506, Springer Verlag.

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Y. Gurevich. Monadic second order theories. In J. Barwise and S. Feferman eds., Model theoretical logics, pp. 479-506, Springer Verlag, 1986. 17

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Y. S. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-theoretic Logics. Springer Verlag, 1985.

....of the index structure with disjoint unary relations t where t(i) means that t is the (k 1 , k d ) theory of the i component. The authors used the method on numerous occasions. Article [5] lays a technical foundation for more advanced applications of the method. Section 3 of the survey [6] describes the composition method (calling it the modeltheoretic decidability technique) and Section 5 mentions various applications of the method. See also the dissertation [20] and exposition [19] A recent 8 sophisticated use of the composition method over finite structures is found in [16] ....

....Remark 2.15 One of the reviewers asked how does the composition method compare with the automata theoretic method. The application domains of the two methods intersect. For example, the decidability of S1S, established first by means of Buchi automata [2] has also a simple model theoretic proof [15, 6]. Rabin used automata to prove the decidability of the MSO theory, known as S2S, of the infinite binary tree [12, 1] It is not clear whether the composition theory can be used for the purpose. One of the consequences of Rabin s result is the decidability of the MSO theory of rational order. This ....

Yuri Gurevich, "Monadic second-order theories", In "ModelTheoretical Logics" (ed. J. Barwise and S. Feferman), Springer-Verlag, 1985, 479--506.

....makes sense already now. The (weak) monadic second order logic is the fragment of the full second order logic, which allows quantification over elements and (finite) monadic predicates only. A profound introduction into the contemporary world of monadic secondorder theories can be found in [18]. The large expressive power of the monadic (second order) logic makes results concerning decidable monadic second order theories interesting. But unfortunately (or fortunately with respect to the above question) the investigation of monadic theories with respect to decidability yields ....

....monadic (second order) logic makes results concerning decidable monadic second order theories interesting. But unfortunately (or fortunately with respect to the above question) the investigation of monadic theories with respect to decidability yields undecidability in very many cases (see e.g. [14, 18, 20, 22, 23, 24, 55, 56, 59, 63, 66]) The most powerful positive result is Rabins s proof [38] see also [19] that S2S, the monadic theory of two successor functions is decidable. A large variety of theories was proved to be decidable via interpretability into S2S (see e.g. 2, 18, 32, 37, 38, 55, 57, 59, 60, 66, 74] The main ....

[Article contains additional citation context not shown here]

Y. Gurevich, Monadic second-order theories, in: J. Barwise and S. Feferman, eds., Model- Theoretic Logics (Springer, New York, 1985) Chapter XIII, 479-506.

....problem of satisfiability over the natural numbers. Hence, the translation cannot be recursive. The proof of the if direction of Theorem 1 will be based on the composition theorem [6, 2] The composition theorem is a very powerful theorem for the proofs of decidability of various theories [6, 3]. Here is the first application of the composition theorem to definability. The method of interpretation is also used extensively in our proof. The rest of the paper is organized as follows. In Section 2 we fix notations and terminology, state some preliminary results and provide a formulation of ....

Y. Gurevich. Monadic second order theories. In J. Barwise and S. Feferman eds., Model theoretical logics, pp. 479-506, Springer Verlag, 1986.

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Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479--506. Springer, 1985.

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Y. Gurevich, Monadic second-order theories, in Model-Theoretic Logics (J. Barwise and S. Feferman, eds.), Springer-Verlag, Berlin, 1985, pp. 479--506.

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Y. Gurevich (1985). Monadic second-order theories. In Model-Theoretic Logics, (J. Barwise and S. Feferman, eds.), 479-506, Springer-Verlag.

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Y. Gurevich, Monadic second-order theories, in: J. Barwise, S. Feferman (Eds.), Model-Theoretic Logics, Springer-Verlag, Berlin, 1985, pp. 479--506.

No context found.

Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 479--506. Springer-Verlag, 1985.

No context found.

Y. Gurevich. Monadic Second-Order Theories. In Handbook of Model-Theoretical Logics, Eds J. Barwise and S. Feferman, pp: 479-506, 1985, Springer Verlag.

No context found.

Y. GUREVICH, Monadic second-order theories, in "Model-Theoretic Logics," Chap. XlII, pp. 479-506, (Barwise and Feferman, Eds.), Springer-Verlag, New York, 1985.

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Gu Y. Gurevich, Monadic Second-Order Theories, in Model-Theoretic Logics, ed. J. Barwise and S. Feferman, Springer-Verlag, New York 1985, pp 479--506.

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Y. Gurevich. Monadic second order theories. In Model-Theoretic Logics, Perspectives in Mathematical Logic, chapter 14. Springer Verlag, 1985.

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Y. Gurevich. Monadic second order theories. In Model-Theoretic Logics, Perspectives in Mathematical Logic, chapter 14. Springer Verlag, 1985.

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