H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, second edition, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(a tree language) T is recognizable 13 if there exists a tree automaton M, such that T = L(M) 3.1.2 Monadic second order logic Let r be the maximal arity of function symbols in the ranked alphabet . A tree t can be naturally viewed as a nite structure (in the sense of mathematical logic [8, 9]) over the binary relation symbols fS 1 ; S r g and the unary relation symbols fO f j f 2 g. We denote the vocabulary associated with by . The domain of t, viewed as a structure, equals the set of nodes of t. The relation S i in t equals the set of pairs (n; n ) such that n is ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, second edition, 1994.

....(a tree language) T is recognizable 13 if there exists a tree automaton M, such that T = L(M) 3.1.2 Monadic second order logic Let r be the maximal arity of function symbols in the ranked alphabet . A tree t can be naturally viewed as a nite structure (in the sense of mathematical logic [8, 9]) over the binary relation symbols fS 1 ; S r g and the unary relation symbols fO f j f 2 g. We denote the vocabulary associated with by . The domain of t, viewed as a structure, equals the set of nodes of t. The relation S i in t equals the set of pairs (n; n 0 ) such that n 0 ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, second edition, 1994.

....investigate their interrelationship and relationship with other logics in Section 3. In Section 4, we examine the nesting of for loops in both BQL and FO(FOR) We end with a discussion in Section 5. 2 Preliminaries Throughout the paper we use the terminology and notation of mathematical logic [EFT94]. For background on database theory we refer to Abiteboul, Hull, and Vianu [AHV95] and for nite model theory to Ebbinghaus and Flum [EF95] Immerman [Imm98] and Otto [Ott97] A relational vocabulary is what in the eld of databases is known as a relational schema; a structure over is what ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, second edition, 1994.

....2, we de ne BQL and FO(FOR) and investigate their interrelationship and relationship with other logics in Section 3. In Section 4, we examine the nesting of for loops in both BQL and FO(FOR) 2 Preliminaries Throughout the paper we will use the terminology and notation of mathematical logic [EFT94]. For background on database theory we refer to Abiteboul, Hull, and Vianu [AHV95] and for nite model theory to Ebbinghaus and Flum [EF95] Immerman [Imm98] and Otto [Ott97] A relational vocabulary is what in the eld of databases is known as a relational schema; a structure over is what ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, second edition, 1994. 51

....( 1 ) Phi ) 4. The function Phi # : FOL over S FOL over R is defined by Phi # ( Phi . Fig. 1. Translation scheme and its components Phi R instance Gamma S instance Phi R formulae Gamma S formulae Phi # The following fundamental theorem is easily verified, cf. [EFT94]. Its origins go back at least to the early years of modern logic, cf. HB70, page 277 ff] Theorem 4. Let Phi = hOE; 1 ; m i be a k R S translation scheme, I(R) be a R instance and be a FOL formula over S. Then I(R) j= Phi # ( iff Phi (I(R) j= Note that we now ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic, 2nd edition. Undergraduate Texts in Mathematics. Springer-Verlag, 1994.

....approaches: FOL[ATC;Q] will be different from IFPL[Q] when Q is a generalized quantifier treated as a variable. Once spelled out, this is not surprising, but illuminating. Outline of the paper We assume the reader is familiar with the basics of generalized quantifiers and logics as described in [EFT80, MP93, BF85] and with logics capturing complexity classes, as in [Imm87, Imm88, Imm89] In section 2 we introduce our notion of uniformly capturing of relativized complexity classes. In this section we give also a detailed outline of our results. In section 3 we discuss various interpretations of our results ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, 1980.

.... Sigma 2 then Phi # ( Sigma 1 ) Phi # ( Sigma 2 ) Fig. 1. Translation scheme and its components Phi R instance Gamma S instance Phi R formula Gamma S formula Phi # 2. 5 The fundamental property of translation schemes The following fundamental theorem is easily verified, cf. [EFT94]. Its origins go back at least to the early years of modern logic, cf. HB70, page 277 ff] Theorem 5. Let Phi = hOE; OE 1 ; OE m i be a k R S translation scheme, I(R) be a R instance and be a FOL formula over S. Then I(R) j= Phi # ( iff Phi (I(R) j= Note that we ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic, 2nd edition. Undergraduate Texts in Mathematics. Springer-Verlag, 1994.

....extended the deadline so as to allow substantial revision of this paper. 2 The General Framework We assume the reader is familiar with the basics of complexity theory as presented in [HU80, GJ79] or the excellent surveys [Sto87, Joh90] and with the basics of abstract model theory as presented in [EFT80, CK90] or in [Ebb85] of [BF85] We start by introducing a notion of relational regular complexity classes, similar in spirit to both Lindstrom s abstract definition of logics and the computable queries of A. Chandra and D. Harel [CH80, CH82] Independently A. Dawar in [Daw94] introduced basically the ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, 1980.

....5. i) Phi # ( 2 FOL (SOL) provided 2 FOL (SOL) even for vectorized Phi (k 2) ii) Phi # ( 2 MSOL provided 2 MSOL, but only for non vectorized Phi (k = 1) In the sequel, we assume that Phi is not vectorized, unless stated otherwise. The following facts hold (for proof see [EFT94]) Proposition6. Let Phi = hOE; 1 ; m i be a k oe translation scheme, A a structure and a L(oe) formula. Then A j= Phi # ( iff Phi (A) j= Proposition7. Let Phi = hOE; 1 ; m i be a k oe translation scheme, A a structure. In this case if ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic, 3rd edition. Undergraduate Texts in Mathematics. Springer-Verlag, 1994.

....Forth through Transitive Closure Operators We provide here the concepts that we need in order to formulate the Ehrenfeucht Fraiss e style characterization of our logics. We give are definitions in terms of systems of partial isomorphisms rather than games. The two approaches are equivalent, cf. [EFT80]. Our approach makes the winning strategies of the games explicit and is more suitable for written presentation, whereas the formulation with games is more intuitive as long as the wining strategies are not explicitly formulated. Note that through this section we deal with arbitrary structures, ....

.... l n 2 and for every two l tuples c; d over Im(f) l and for every relation R B 2l such that ( c; d) 62 TC(R) There exist a relation R 0 A 2l such that 1 Since our vocabularies always contain equality this condition is superfluous, however we maintained it for consistency with [EFT80] 2 By definition of partial isomorphism this implies I B (c) 2 Im(f ) i) f Gamma1 ( c) f Gamma1 ( d) 62 TC(R 0 ) ii) for every two l tuples ( a; b) 62 R 0 there exist a g 2 I n Gamma2l such that: ii.a) g extends f (ii.b) c i 2 Dom(g) d i 2 Dom(g) for 1 i l (ii.c) ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, 1980.

....and Sigma 2 be a set of S sentences. i) If K 1 K 2 then Phi (K 1 ) Phi (K 2 ) ii) If Sigma 1 j= Sigma 2 then Phi # ( Sigma 1 ) Phi # ( Sigma 2 ) 2. 4 The fundamental property of translation schemes The following fundamental theorem is easily verified, cf. [EFT94]. Its origins go back at least to the early years of modern logic, cf. HB70, page 277 ff] Theorem5. Let Phi = hOE; 1 ; m i be a k R S translation scheme, I(R) be a R instance and be a FOL formula over S. Then I(R) j= Phi # ( iff Phi (I(R) j= Note that we now ....

....from a class DEP Let us fix a class of dependencies DEP . For a set of first order sentences Sigma we say that Sigma eq DEP if there is a set of dependencies D DEP which is equivalent to Sigma . In general, checking whether Sigma eq DEP is undecidable, by Trakhtenbrot s theorem, cf. [EFT94], but this is not the point here. We want to study dependency preserving transformations where the dependencies are restricted to DEP . 7.1 Many options Our general situation is now as follows: Assumption] R; Sigma R and S; Sigma S are database schemes. Phi is a R Gamma ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic, 2nd edition. Undergraduate Texts in Mathematics. Springer-Verlag, 1994.

....genuine difficulty of the problems. It also indicates a line of attack, which has been vindicated recently for only slightly simpler problems in [AF90] In section 5, finally, we discuss some open problems. We assume the reader is familiar with the basics of abstract model theory as presented in [EFT80, CK90] or in the first chapters of [BF85] We also assume the reader is familiar with the basics of complexity theory as presented in [HU80, GJ79] or the excellent surveys [Sto87, Joh90] Occasionally, we refer also to the parallel complexity classes NC and bfNC 1 . NC consists of those ....

....Quantifiers We provide here the concepts that we need in order to formulate the Ehrenfeucht Fraiss e style characterization of our logics. We basically follow [Cai80] but give our definitions in terms of systems of partial isomorphisms rather than games. The two approaches are equivalent, cf. EFT80] Our approach makes the winning strategies of the games explicit and is more suitable for written presentation, whereas the formulation with games is more intuitive as long as the wining strategies are not explicitly formulated. Note that through this section we deal with arbitrary structures, ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, 1980.

....We leave these last two questions open for further research. 3 The Logics INV (L) and Delta(L) In this section we deal with weakly regular logics L on finite structures and prove some general facts concerning invariant definability. Background on logics and logical reductions can be found in [EFT94, EF95, MP95]. The proofs of all the statements in this section are easy and left to the reader. We first introduce our notion of parametrically definable classes of structures (or global relations) It will serve as a tool to obtain more invariantly definable classes once we have some at our disposal. ....

H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic, 2nd edition. Undergraduate Texts in Mathematics. Springer-Verlag, 1994.

....that of having the same expressive power through embeddings is indeed an equivalence relation. Also, it seems to capture the notion of everything that can be said in L 1 can be said in L 2 and vice versa. It is related to the standard definition of equivalence of logic systems as presented in [7] on the one hand, and to the notion of equivalence of equational theories in equational logic [19] on the other. But there are important differences between Baader s definition and the other two. Unlike Baader s definition, the definition stemming from equivalence of logic systems applies to ....

H. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, 1984.

....state estimate set characterised by the set of x i , 1 i N , such that CSE k (x i ) holds. We now examine the evolution of full and fragment theories, and define the relationship between H m k Gamma1 and H m k . We give below a well known lemma for later use: Lemma 5. 1 Coincidence Lemma [EFT84] Let H 0 = I 0 ; V 0 and H 00 = I 00 ; V 00 be structures for the languages L 0 and L 00 respectively, both with the same domain D. Let L = L 0 L 00 . a) For any t ffl T erm(L) if H 0 and H 00 agree on the symbols of S(L) occurring in t, then I 0 (t) I 00 ....

....h(I(c) I 0 (h(c) 2) For every f ffl Func(L) and t 1 ; Delta Delta Delta ; t l ffl T erm(L) h(I(f( t) I 0 (f) h( t) 3) For every P ffl P re(L) and t 1 ; Delta Delta Delta ; t l ffl T erm(L) I(P ( t) iff I 0 (P )I 0 ( h( t) Lemma 5. 3 Isomorphism Lemma [EFT84] If I and I 0 are isomorphic, then for any F ffl WFF (L) we have H j= F ( H 0 j= F: The following theorem is the result of application of Theorem 2.1. Theorem 5.1 ( CW90,W91] For M together with the consistent observation sequence o k 1 , k 0, M Sigma k is consistent. MARKOVIAN ....

H.D. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic. Undergraduate Texts in Mathematics, Springer, New York, 1984.

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H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Undergraduate Texts in Mathematics. Springer-Verlag, second edition, 1994.

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H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic, 2nd edition. Undergraduate Texts in Mathematics. Springer-Verlag, 1994.

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