Relativised quantification: Some canonical varieties of sequence-set algebras

1998 ◽  
Vol 63 (1) ◽  
pp. 163-184 ◽  
Author(s):  
Hajnal Andréka ◽  
Robert Goldblatt ◽  
István Németi
Keyword(s):  
Existential Quantifier ◽  
Cylindric Algebras ◽  
Order Model ◽  
Assigned Value ◽  
First Order ◽  
Cartesian Space ◽  
A Value ◽  
Quantificational Logic ◽  
Image Position ◽  
Existential Quantification

This paper explores algebraic aspects of two modifications of the usual account of first-order quantifiers.Standard first-order quantificational logic is modelled algebraically by cylindric algebras. Prime examples of these are algebras whose members are sets of sequences: given a first-order model U for a language that is based on the set {υκ: κ < α} of variables, each formula φ is represented by the setof all those α-length sequences x = 〈xκ: κ < α〉 that satisfy φ in U. Such a sequence provides a value-assignment to the variables (υκ is assigned value xκ), but it may also be viewed geometrically as a point in the α-dimensional Cartesian spaceαU of all α-length sequences whose terms come from the underlying set U of U. Then existential quantification is represented by the operation of cylindrification. To explain this, define a binary relation Tκ on sequences by putting xTκy if and only if x and y differ at most at their κth coordinate, i.e.,Then for any set X ⊆ αU, the setis the “cylinder” generated by translation of X parallel to the κth coordinate axis in αU. Given the standard semantics for the existential quantifier ∃υκ asit is evident that

A selection theorem

1983 ◽  
Vol 48 (3) ◽  
pp. 585-594
Author(s):  
Lefteris Miltiades Kirousis
Keyword(s):  
Partial Function ◽  
Existential Quantifier ◽  
Selection Theorem ◽  
Unbounded Subset ◽  
Image Position ◽  
Existential Quantification

In [1978] Harrington and MacQueen proved that if B is an (A, E)-semirecursive subset of A, such that the functions in BA can be coded as elements of A in an (A, E)-recursive way, then ENV(A, E) is closed under the existential quantifier ∃T ∈ B.Later Moschovakis showed that if ENV(Vκ, ∈, E) is closed under the quantifier ∃t ∈ λ, where λ is the p-cofinality of κ, thenthe p-cofinality of κ is the least ordinal λ for which there exists a (κ, <, E)-recursive partial function ƒ into κ, such that ƒ∣λ is total from λ onto an unbounded subset of κ.In this paper we prove that for any infinite ordinal κ if p-card(κ) = κ, then ENV(κ, <, E) is closed under ∃t ∈ μ, for μ < p-cf(κ); p-cf(κ) is the “boldface” analog of p-cf((κ) and p-card(κ) is defined similarly.From this follows that for any infinite ordinal κ the following two statements are equivalent.(i) ENV(κ, <, E) is closed under bounded existential quantification.(ii) ENV(κ, <, E) = ENV(κ, <, E#) or p-cf(κ) = κ.We also show that we cannot omit any of the hypotheses in the above theorem.We follow mainly the notation of Kechris and Moschovakis [1977].

On the reduction of the decision problem. Third paper. Pepis prefix, a single binary predicate

1950 ◽  
Vol 15 (3) ◽  
pp. 161-173 ◽  
Author(s):  
László Kalmár ◽  
János Surányi
Keyword(s):  
Decision Problem ◽  
Predicate Calculus ◽  
Existential Quantifier ◽  
First Order ◽  
Binary Predicate ◽  
Image Position ◽  
Analogous Theorem ◽  
Order Predicate Calculus ◽  
Predicate Variable

It has been proved by Pepis that any formula of the first-order predicate calculus is equivalent (in respect of being satisfiable) to another with a prefix of the formcontaining a single existential quantifier. In this paper, we shall improve this theorem in the like manner as the Ackermann and the Gödel reduction theorems have been improved in the preceding papers of the same main title. More explicitly, we shall prove theTheorem 1. To any given first-order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.An analogous theorem, but producing a prefix of the formhas been proved in the meantime by Surányi; some modifications in the proof, suggested by Kalmár, led to the above form.

scholarly journals THE NEAT EMBEDDING PROBLEM FOR ALGEBRAS OTHER THAN CYLINDRIC ALGEBRAS AND FOR INFINITE DIMENSIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 208-222 ◽  
Author(s):  
ROBIN HIRSCH ◽  
TAREK SAYED AHMED
Keyword(s):  
Embedding Problem ◽  
Infinite Dimensions ◽  
Cylindric Algebras ◽  
First Order ◽  
Polyadic Algebras ◽  
Finite Set ◽  
Image Position ◽  
Substitution Algebras

Abstract Hirsch and Hodkinson proved, for $3 \le m &lt; \omega $ and any $k &lt; \omega $ , that the class $SNr_m {\bf{CA}}_{m + k + 1} $ is strictly contained in $SNr_m {\bf{CA}}_{m + k} $ and if $k \ge 1$ then the former class cannot be defined by any finite set of first-order formulas, within the latter class. We generalize this result to the following algebras of m-ary relations for which the neat reduct operator $_m $ is meaningful: polyadic algebras with or without equality and substitution algebras. We also generalize this result to allow the case where m is an infinite ordinal, using quasipolyadic algebras in place of polyadic algebras (with or without equality).

Cylindric modal logic

1995 ◽  
Vol 60 (2) ◽  
pp. 591-623 ◽  
Author(s):  
Yde Venema
Keyword(s):  
Algebraic Logic ◽  
Equational Theory ◽  
Completeness Theorem ◽  
Cylindric Algebras ◽  
First Order ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Existential Quantification ◽  
Cylindric Set Algebras

AbstractTreating the existential quantification ∃νi as a diamond ♢i and the identity νi = νj as a constant δij, we study restricted versions of first order logic as if they were modal formalisms. This approach is closely related to algebraic logic, as the Kripke frames of our system have the type of the atom structures of cylindric algebras; the full cylindric set algebras are the complex algebras of the intended multidimensional frames called cubes.The main contribution of the paper is a characterization of these cube frames for the finite-dimensional case and, as a consequence of the special form of this characterization, a completeness theorem for this class. These results lead to finite, though unorthodox, derivation systems for several related formalisms, e.g. for the valid n-variable first order formulas, for type-free valid formulas and for the equational theory of representable cylindric algebras. The result for type-free valid formulas indicates a positive solution to Problem 4.16 of Henkin, Monk and Tarski [16].

A first-order model of thermal lensing of laser propagation in the eye and implications for laser safety

2005 ◽  
Author(s):  
Robert J. Thomas ◽  
Rebecca L. Vincelette ◽  
Gavin D. Buffington ◽  
Amber D. Strunk ◽  
Michael A. Edwards ◽  
...  
Keyword(s):  
Thermal Lensing ◽  
Order Model ◽  
Laser Safety ◽  
First Order ◽  
Laser Propagation

Application of two approaches to model chlorine residuals in Severn Trent Water Ltd (STW) distribution systems

1997 ◽  
Vol 36 (5) ◽  
pp. 317-324 ◽  
Author(s):  
M.J. Rodriguez ◽  
J.R. West ◽  
J. Powell ◽  
J.B. Sérodes
Keyword(s):  
Distribution System ◽  
Distribution Systems ◽  
Treatment Plant ◽  
Residual Chlorine ◽  
Ann Model ◽  
Order Model ◽  
First Order ◽  
Chlorine Decay ◽  
Artificial Neural Network Ann ◽  
Key Parameter

Increasingly, those who work in the field of drinking water have demonstrated an interest in developing models for evolution of water quality from the treatment plant to the consumer's tap. To date, most of the modelling efforts have been focused on residual chlorine as a key parameter of quality within distribution systems. This paper presents the application of a conventional approach, the first order model, and the application of an emergent modelling approach, an artificial neural network (ANN) model, to simulate residual chlorine in a Severn Trent Water Ltd (U.K.) distribution system. The application of the first order model depends on the adequate estimation of the chlorine decay coefficient and the travel time within the system. The success of an ANN model depends on the use of representative data about factors which affect chlorine evolution in the system. Results demonstrate that ANN has a promising capacity for learning the dynamics of chlorine decay. The development of an ANN appears to be justifiable for disinfection control purposes, in cases when parameter estimation within the first order model is imprecise or difficult to obtain.

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

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