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

....(U; I; where U is a set, I is a mapping that maps p 2 r (r 0) and a 2 to an r ary relation over U , a binary associative function over U and an element of U , respectively, and is a variable assignment to U . Then, the satisfaction relation j= is de ned in a standard manner (cf. [14, 30]) A model of an atom A or a clause C over S is an interpretation I over S such that I j= A and I j= C, respectively. We assume that any variable in a clause is universally quanti ed. A model of an EFS H over S is a model of every clause in H over S. For an EFS H and a clause C over S, we say ....

H.- D. Ebbinghaus, J. Flum, W. Thomas, Mathematical logic (2nd Ed.) (SpringerVerlag, 1994).

....systems or more classically as the equivalence induced by the in nite Ehrenfeucht Fra ss e game whose moves capture the relativised pattern of modal quanti cation. 1.1. Ehrenfeucht Fra ss e games. Recall the classical EhrenfeuchtFra ss e characterisation of elementary equivalence (see [7, 6, 20] for textbook treatments) The game is played by two players over the two structures, A and B that are to be compared. In each round of the game the rst player marks an element of one of the structures (his choice) and the second player responds by selecting an element in the opposite structure. ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical logic, Springer, 1994.

....where D R n , nd its global minimum. Minimax Optimization: Given a function f : D 1 Dn R, where D 1 Dn R n , nd min x12D1 max x22D2 : min xn 1 2Dn 1 max xn2Dn f(x 1 ; xn ) Quanti ed Constraint Solving: Given a rst order formula over the reals [11], that is, an expression that contains quanti ers (9, 8) connectives ( predicate symbols (e.g. function symbols (e.g. sin, exp) rational constants and variables ranging over the reals) nd its solution set (or truth value in the case of a closed formula) A ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer Verlag, 1984.

....sets. We show that, for constraints containing only inequality predicates, stability under perturbation is independent of whether we use information about the continuity of constraints and equality of variables. The paper is self contained except for some basic knowledge of mathematical logic [10]. Its structure is as follows: In Section 2, we give a formalization of the problem. In Section 3, we show that for a certain class of quanti ed constraints their solution set is stable under perturbation except for one sign computation. In Section 4, we present criteria that are useful in ....

....font. The symbol will function as a meta variable over constraints, and the symbol will function as a meta variable over terms. For any syntactic entity s (i.e. constraint, term, predicate symbol, function symbol) we denote its meaning by [ s] Note that in contrast to mathematical logic [10] we assign meaning independent of a certain variable assignment. As a result, the meaning of a constraint is a solution set, the meaning of a term is a function, and the meaning [ x] of a variable x is the identity function (see [27] for details) Furthermore, the solution set of a closed ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer Verlag, 1984.

....solver for rst order constraints on this narrowing algorithm. Section 7 discusses the relation of the results to classical decision algorithms, and Section 8 concludes the paper. 2 Preliminaries We x a set V of variables. A rst order constraint is a formula in the rstorder predicate language [12] over the reals with predicate and function symbols interpreted as suitable real relations and functions, and with variables in V . In this paper we restrict ourselves to the predicate symbols , and , and assume that equalities are expressed by inequalities on the residual (i.e. f = 0 as ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer Verlag, 1984.

....we present the formal problem speci cation and give a basic algorithm for ful lling the speci cation. This algorithm mainly serves for illustration, and does not yet take into account eciency considerations. As already mentioned, a quanti ed constraint is a rst order formula over the reals [13]. So it contains quanti ers (9, 8) connectives ( predicate symbols (e.g. function symbols (e.g. sin, exp) rational constants and variables ranging over real numbers. A simple example is the quanti ed constraint 9y [x 2 y 2 1 y 0] The semantics of quanti ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer Verlag, 1984.

....that interpretation. But it is this type of informality, interpretation, and reasoning about the underlying model that leads to the claim that false is not provable, by de nition. A proof of G odel s second incompleteness theorem can be found in textbooks on mathematical logic (e.g. 20] 25] [15], and [16] A study of formal provability via modal logic is conducted by Boolos [0] where on page xvii, Boolos makes the point that although for every theorem S of a suciently strong formal system F, there is also a theorem of F to the e ect that S is a theorem of F, for no non theorem S 0 of ....

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

....structures of the Lindenbaum Tarski algebras of these logics, which are themselves the free algebras in the varieties that the logics define. The discovery of Fine [12] that an elementary class of Kripke frames determines a logic validated by its canonical frames generalises to the result [14, 15] that the powerset al..gebras of an ultraproduct closed class of structures generate a variety of BAO s closed under canonical extensions. The question of whether a logic is complete with respect to some class of Kripke frames corresponds to the question of whether a variety of algebras is ....

....an ideal of # that is disjoint from the principal filter generated by x j .Butthen# must contain an ultrafilter t j that includes x j and is disjoint from u j . This is enough to ensure that (i) and (ii) hold with j in place of i. The full details of this argument may be found in Theorem 2.2. 1 of [14], where the proof is shown to work for any normal operator on a distributive lattice. Now to each BAO # = ##,m # # # # # of type # we can associate the type # relational structure # st # = #S#,R m# # # # # which we call the canonical structure of #. Its complex algebra will be denoted ....

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H. D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, Berlin, 1984. vi+216 pp.

....We say that GTS propositions P and P 0 are syntactically equivalent if and only if P = P 0 . The reader should convince himself that syntactic equivalence implies logical equivalence, but the converse is not true, of course. 9 This is standard stu we include merely for completeness: see [EFT94] [I need to check to see if this is really in that book] 4 GTS PREDICATE CALCULUS 64 Lemma 45 (Proposition Equivalence Finite Generation) The syntactic equivalence of two GTS propositions can be proved by a nite number of applications of the equations in the Proposition Syntactic Equivalence ....

....P R for every proposition P 2 P. The notation [V 1 7 E 1 ; V 2 7 E 2 ; V n 7 E n ]P means the set of all propositions [V 1 7 E 1 ; V 2 7 E 2 ; V n 7 E n ]P for every proposition P 2 P. 13 See books on mathematical logic by Kleene[Kle67] Hamilton[Ham88] or Ebbinghaus[EFT94]. 5 GTS MODELS 86 Note that we are using sets of variables and propositions because in this section, where we are dealing with modeling and validity, the order of variables and propositions in their various lists does not matter. De nition 81 (Logical Equivalence) Two GTS propositions, P and P ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, 1994.

....3 recalls some characterizations of unstable rst order constraints, Section 4 proves the main theorem, and Section 5 concludes the paper. 1 2 A Stochastic Model for First Order Constraints A real rst order constraint (or short: constraint) is a formula in the rstorder predicate language [1] with variables in a set V , the predicate symbols and , and arbitrary function symbols. Such rst order constraints can express equalities of the form f = 0 via the two inequalities f 0 f 0. We assume that all atomic constraints are of the form f 0 or f 0. Furthermore we divide the ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer Verlag, 1984.

....# k, it holds that P (a i 1 , a i m ) ## P (b i 1 , b i m ) If Duplicator has a winning strategy in the k round game for N on two strings u and v, we write u # N k v. The fundamental use of the game comes from the fact that it characterises first order logic (c.f. e.g. [EFT94]) In our context, this can be formulated as follows: 2.1 Theorem (Ehrenfeucht, Frasse) A language L # A # is definable in FO[N ] iff there is a finite subset N # of N and a number k such that, for every u # L, v ## L, Spoiler has a winning strategy in the k round game for N # on u ....

H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, New York, 2nd edition, 1994.

....trees, and outline a proof of completeness based on ideas and results from the paper. The last section 5 discusses a research program arising from this study. 2 Preliminaries Here we summarize some basic topological facts that will be used further. For details on definitions and related results, Ebbinghaus et al., 94]and[Hodges, 93] 3 are general references on the necessary logical background, and [Engelking, 85] on topology. Let L be a first order language, SEN(L) be the set of sentences of L, and C(L) be the set of complete theories in L.TheStone topology S(L) is defined on the set C(L) by a base ....

.... introduced by using an appropriate metric (which need not be inducing the Stone topology) on C(L) The notion of quantifier rank of a formula is introduced as usual in languages with relational signatures, and appropriately modified for languages including constant and functional symbols, as in [Ebbinghaus et al., 94] p. 257. Let SEN (n) L)bethesetofL sentences of (modified) rank # n and for every # # SEN, # (n) ##SEN (n) L) 6 First, we define distance in C(L) as follows: d(T 1 ,T 2 ) # 0ifT 1 = T 2 , 1 n 1 if n is the least integer such that T (n) 1 #= T (n) 2 . Proposition 4.2 ....

Ebbinghaus H.-D., J. Flum, W. Thomas, Mathematical Logic, Springer-Verlag, 2nd ed., 1994.

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

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H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer Verlag, 2 edition, 1995. ISBN 3-540-60149-X.

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H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer Verlag, Berlin Heidelberg, 2nd edition, 1994.

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H.-D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, second ed. Springer-Verlag, 1994.

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H.-D. Ebbinghaus, J. Flum and W. Thomas, Mathematical Logic, Springer-Verlag, 1984.

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H. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, 1984.

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H. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, 1984.

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H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer Verlag, 1984.

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H.-D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, Springer, 1994.

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H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, 1984.

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H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, 1984. 35

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H. D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, Berlin, 1984. vi+216 pp.

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H.D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. SpringerVerlag, Berlin, 1984.

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