W. Walkoe. Finite partially-ordered quantification. J. Symbolic Logic, 35:535-- 555, 1970. 16  Home/Search   Document Not in Database   Summary   Related Articles   Check

....supported by the Austrian Science Fund Project N Z29 INF, and the Max Kade Foundation. Most of this research has been carried out while the first author was with the University of Gieen, Germany, and the third author was with TU Wien. in [27] and subsequently studied in numerous papers, e.g. [18, 64, 5, 9, 33, 50]. Unifying the two approaches, the most prominent work from this area is the seminal paper of Lindstrom [34] whose formalization of generalized quantifiers is mostly used to date. After diminished attention for some time, generalized quantifiers gained in the 90 s increasing interest in ....

W. Walkoe. Finite Partially-Ordered Quantification. Journal of Symbolic Logic, 35:535--555, 1970.

.... Delta 9Q expressible queries (where Q is Pi n ) Pi n 1 is the set of all 8 Delta Delta Delta 8Q expressible queries (where Q is Sigma n ) and so on. Chandra and Harel [7] proved that on the class of all finite digraphs, these expressibility classes are distinct. Meanwhile, Walkoe [37] constructed a class of infinite structures such that if Q and Q 0 are distinct prefixes of the same length, then there exists a Q expressible query that is not Q 0 expressible in that class. Keisler and Walkoe [25] then proved that for each N , over the class of finite structures with one ....

W. Walkoe, Finite partially-ordered quantification, J. Sym. Logic 35 (1970), 535--555.

....controller is given, the singlestep control problem becomes considerably harder. This is because a (static or dynamic) type may specify partially ordered input output dependencies. These partially ordered dependencies correspond to boolean formulas with partially ordered (Henkin) quantifiers [Hen61,Wal70,BG86], whose complexity class for satisfiability is NE (a weak form of nondeterministic exponential time) GLV95] The solution of control problems in presence of types gives rise to additional surprising phenomena. For example, with static or syntactically composed dynamic types, two states s and t ....

....of as a set ff 1 ; f k g of Skolem functions: for 1 i k, the Skolem function f i provides a next value for y i , and has as arguments the variables in XM [fx 0 2 X 0o M j y i N xg. This set of Skolem functions corresponds to the following boolean formula with Henkin quantifiers [Hen61,Wal70]: H(N ; 0 (8 fx 0 2 X 0o M j y 1 N xg) 9y 0 1 ) Delta Delta Delta (8 fx 0 2 X 0o M j y k N xg) 9y 0 n ) 1 A ( M 0 ) The fixed type single step static control problem can be solved as follows. Lemma 2. Given a statically typed module (M; M ) a static ....

W. Walkoe. Finite partially-ordered quantification. J. Symbolic Logic, 35:535-- 555, 1970. 16

....if is equivalent to p , then p can be embedded in the prefix of . This strengthens a theorem of Walkoe. In this paper we address the following question. Given two first order prefixes p and q, is there a sentence with prefix p that is not equivalent to any sentence with prefix q Walkoe [4] proved that if p and q are different prefixes of length n, then there is such a sentence, containing a single n ary relation symbol. Keisler and Walkoe [3] then strengthened this result by showing that it also holds over the class of finite structures. Our main theorem improves on Walkoe s ....

W. Walkoe, Jr. Finite partially-ordered quantification. Journal of Symbolic Logic, 35:535--555, 1970.

....H Sxyzw (see Henkin (1961) as follows. 8x9y(8z=xy) 9w=xy) Sxyzw j 8x 9y 8z 9w Sxyzw: Further, each complex Henkin quantifier prefix H can be defined by a quantifier prefix H of the following bifarious form (2) see Krynicki (1993) qualifying that of Blass Gurevich (1986) and Walkoe (1970)) 8x 1 : x n 9y 8z 1 : z m 9w (for some n; m 2 ) 2) We call sentences whose prefixes are put into this form Krynicki normal form for Henkin quantifiers. The formulas for the Krynicki normal forms can be defined in L as follows: 8x 1 : x n 9y(8z 1 : z m =x 1 : x n ....

....can be defined using Sigma 1 1 sentences only. The expressive strength flows from the well known results that every Henkin quantifier can be defined as a Sigma 1 1 sentence (cf. Henkin (1961) and that every Sigma 1 1 sentence can be defined in a language of all Henkin quantifiers (cf. Walkoe (1970)) The usual way of formulating a truth definition for the language of first order Peano Arithmetic is to take the truth defining formula Tr(x) to have the form 9X(L(X) X(x) with X a second order variable, and L(x) a formula which runs through all the recursive clauses of the truth definition. ....

Walkoe, W.: (1970) Finite partially ordered quantification, Journal of Symbolic Logic 35, pp. 535--550.

....Henkin quantifier H Sxyzw not reducible to first order logic as follows. 8x9y(8z=x; y) 9w=x; y) Sxyzw j 8x 9y 8z 9w Sxyzw: Further, each complex Henkin quantifier prefix H can be defined by a quantifier prefix H of the following bifarious form (3) see [18] qualifying that of [6] and [25]) 8x 1 : x n 9y 8z 1 : z m 9w (for some n; m 2 ) 3) Sentences whose prefixes are put into this form can be called Krynicki normal form for Henkin quantifiers. The formulas for the Krynicki normal forms for Henkin quantifiers can be defined in L as follows: 8x 1 : x n 9y(8z ....

Walkoe, W.: (1970) Finite partially ordered quantification, Journal of Symbolic Logic 35, 535--550. School of Cognitive and Computing Sciences, University of Sussex, Falmer, Brighton BN1 9QH, U.K. E-mail address: ahtivp@cogs.susx.ac.uk

....; f 1 (x 1 ) f k (x k ) with the obvious meaning evaluates to true over A. Note that with Henkin quantifiers it is not necessary to list the variables of OE to be bound, given that these variables already occur explicitly in the quantifier. Henkin quantifiers were extensively studied in [9, 38, 2]. In particular, Walkoe [38] showed that the fragment H has the same expressive power as the fragment of existential second order formulas over arbitrary structures. As Blass and Gurevich [2] point out, from this and from Fagin s theorem [10] it immediately follows that H captures NP over ....

....with the obvious meaning evaluates to true over A. Note that with Henkin quantifiers it is not necessary to list the variables of OE to be bound, given that these variables already occur explicitly in the quantifier. Henkin quantifiers were extensively studied in [9, 38, 2] In particular, Walkoe [38] showed that the fragment H has the same expressive power as the fragment of existential second order formulas over arbitrary structures. As Blass and Gurevich [2] point out, from this and from Fagin s theorem [10] it immediately follows that H captures NP over finite structures. ....

Wilbur Walkoe. Finite Partially-Ordered Quantification. Journal of Symbolic Logic, vol. 35, pp.535-555, 1970.

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W. Walkoe. Finite partially-ordered quantification. J. Symbolic Logic, 35:535-- 555, 1970. 16

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W. J. Walkoe, Jr, Finite partially-ordered quantification. J. Symbolic Logic 35 535-- 555 (1970). 15

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Walkoe, W. Jr., Finite partially ordered quantification, The Journal of Symbolic Logic,

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