Two theories with axioms built by means of pleonasms

1957 ◽  
Vol 22 (1) ◽  
pp. 36-38 ◽  
Author(s):  
Andrzej Ehrenfeucht
Keyword(s):  
Finite Number ◽  
Logical Constants ◽  
Recursive Functions ◽  
Undecidable Theory ◽  
Binary Predicate ◽  
The Common ◽  
Image Position ◽  
Infinite Power ◽  
Recursive Set

This paper contains examples T1 and T2 of theories which answer the following questions:(1) Does there exist an essentially undecidable theory with a finite number of non-logical constants which contains a decidable, finitely axiomatizable subtheory?(2) Does there exist an undecidable theory categorical in an infinite power which has a recursive set of axioms? (Cf. [2] and [3].)The theory T1 represents a modification of a theory described by Myhill [7]. The common feature of theories T1 and T2 is that in both of them pleonasms are essential in the construction of the axioms.Let T1 be a theory with identity = which contains one binary predicate R(x, y) and is based on the axioms A1, A2, A3, B1, B2, B3, B4, Cnm which follow.A1: x = x. A2: x = y ⊃ y = x. A3: x = y ∧ y = z ⊃ x = z.(Axioms of identity.)B1: R(x, x). B2: R(x, y) ⊃ R(y, x). B3: R (x, y) ∧ R(y, z) ⊃ R(x, z).(Axioms of equivalence.)B4: x = y ⊃ [R(z, x) ≡ R(z,y)].Let φn be the formulawhich express that there is an abstraction class of the relation R which has exactly n elements.Let f(n) and g(n) be two recursive functions which enumerate two recursively inseparable sets [5], and call these sets X1 and X2.We now specify the axioms Cmm.It is obvious that the set composed of the formulas A1−A3, B1−B4, Cnm (n,m = 1,2, …) is recursive.The theory T1 is essentially undecidable; for if there were a complete and decidable extension T′1 (of it, then the recursive sets Z = {n: φn is provable in T′1} and Z′ = {n: ∼φn is provable in T′1} would separate the sets X1 and X2.

Optimal selection of stochastic intervals under a sum constraint

1987 ◽  
Vol 19 (2) ◽  
pp. 454-473 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
R. R. Weber
Keyword(s):  
Decision Rule ◽  
Selection Process ◽  
Random Variables ◽  
Optimal Threshold ◽  
Optimal Selection ◽  
Expected Number ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Selection Of

We model a selection process arising in certain storage problems. A sequence (X1, · ··, Xn) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

Proof that there are infinitely many modalities in Lewis's system S2

1940 ◽  
Vol 5 (3) ◽  
pp. 110-112 ◽  
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Finite Number ◽  
Infinite Matrix ◽  
Image Position

In this note I show, by means of an infinite matrix M, that the number of irreducible modalities in Lewis's system S2 is infinite. The result is of some interest in view of the fact that Parry has recently shown that there are but a finite number of modalities in the system S2 (which is the next stronger system than S2 discussed by Lewis).I begin by introducing a function θ which is defined over the class of sets of signed integers, and which assumes sets of signed integers as values. If A is any set of signed integers, then θ(A) is the set of all signed integers whose immediate predecessors are in A; i.e., , so that n ϵ θ(A) is true if and only if n − 1 ϵ A is true.Thus, for example, θ({−10, −1, 0, 3, 14}) = {−9, 0, 1, 4, 15}. In particular we notice that θ(V) = V and θ(Λ) = Λ, where V is the set of all signed integers, and Λ is the empty set of signed integers.It is clear that, if A and B are sets of signed integers, then θ(A+B) = θ(A)+θ(B).It is also easily proved that, for any set A of signed integers we have . For, if n is any signed integer, then

Permutation endomorphisms and refinement of a theorem of Birkhoff

1960 ◽  
Vol 56 (4) ◽  
pp. 322-328 ◽  
Author(s):  
H. K. Farahat ◽  
L. Mirsky
Keyword(s):  
Abelian Group ◽  
Finite Number ◽  
Linear Combination ◽  
Finite Linear Combination ◽  
Usual Manner ◽  
Restricted Permutation ◽  
Image Position ◽  
Ring Of Endomorphisms ◽  
Finite Linear

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equationsA permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (01) ◽  
pp. 5-7
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I 1, I 2, ···, I n where I j is the strategy ‘play value m j ′ and m 1 < m 2 < ··· < m n . The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[I i , I j ] is the expected payoff for playing I i against I j . We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

III.—The Application of Operators to the Solution of the Algebraic Equation

1918 ◽  
Vol 37 ◽  
pp. 18-49
Author(s):  
James Littlejohn
Keyword(s):  
Algebraic Equation ◽  
The Common ◽  
Image Position

The solution of the common quadraticwhich is usually writtenis, when expanded,This expansion will be found to be the work of the operator each term being derived from its predecessor by this operator's agency, so that the whole solution may be written

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

Distinct Values of a Polynomial in Subsets of a Finite Field

1969 ◽  
Vol 21 ◽  
pp. 1483-1488
Author(s):  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Number ◽  
Image Position

If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm] for the finite field with pm elements and [pm]′ for [pm] – {0}. We consider in this paper only subsets H of [pm] for which |H| = h satisfies1.1If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p0(d), for some prime p0, depending only on d, which is greater than d.

A remark concerning decidability of complete theories

1950 ◽  
Vol 15 (4) ◽  
pp. 277-279 ◽  
Author(s):  
Antoni Janiczak
Keyword(s):  
Finite Number ◽  
Decision Procedure ◽  
Effective Procedure ◽  
Undecidable Theory ◽  
Consistent Extension ◽  
Complete Theories ◽  
General Conditions

A formalized theory is called complete if for each sentence expressible in this theory either the sentence itself or its negation is provable.A theory is called deciddble if there exists an effective procedure (called decision-procedure) which enables one to decide of each sentence, in a finite number of steps, whether or not it is provable in the theory.It is known that there exist complete but undecidable theories. There exist, namely, the so called essentially undecidable theories, i.e. theories which are undecidable and remain so after an arbitrary consistent extension of the set of axioms. Using the well-known method of Lindenbaum we can therefore obtain from each such theory a complete and undecidable theory.The aim of this paper is to prove a theorem which shows that complete theories satisfying certain very general conditions are always decidable. In somewhat loose formulation these conditions are: There exist four effective methods M1, M2, M3, M4, such that(a) M1 enables us to decide in each case whether or not any given formula is a sentence of the theory;(b) M2 gives an enumeration of all axioms of the theory;(c) the rules of inference can be arranged in a sequence R1, R2, … such that if p1, … pk, r are arbitrary sentences of the theory, we can decide by M3 whether or not r results from p1, … pk, by the n-th rule;(d) M4 enables us to construct effectively the negation of each effectively given sentence.In order to express these conditions more precisely we shall make use of an arithmetization of the considered theory .

The arithmetic of certain semigroups of positive operators

1968 ◽  
Vol 64 (1) ◽  
pp. 161-166 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Finite Number ◽  
Positive Definite ◽  
Positive Operators ◽  
Linear Operators ◽  
Real Numbers ◽  
Ultraspherical Polynomials ◽  
Real Polynomials ◽  
Fixed Parameter ◽  
Summation Sign ◽  
Image Position

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measureand which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided thatHere the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that

A definition of existence in terms of abstraction and disjunction

1957 ◽  
Vol 22 (4) ◽  
pp. 343-344
Author(s):  
Frederic B. Fitch
Keyword(s):  
Finite Number ◽  
Infinite Number ◽  
Recursive Relation ◽  
Basic Logic ◽  
Class A ◽  
Definition Of ◽  
Image Position

Greater economy can be effected in the primitive rules for the system K of basic logic by defining the existence operator ‘E’ in terms of two-place abstraction and the disjunction operator ‘V’. This amounts to defining ‘E’ in terms of ‘ε’, ‘έ’, ‘o, ‘ό’, ‘W’ and ‘V’, since the first five of these six operators are used for defining two-place abstraction.We assume that the class Y of atomic U-expressions has only a single member ‘σ’. Similar methods can be used if Y had some other finite number of members, or even an infinite number of members provided that they are ordered into a sequence by a recursive relation represented in K. In order to define ‘E’ we begin by defining an operator ‘D’ such thatHere ‘a’ may be thought of as an existence operator that provides existence quantification over some finite class of entities denoted by a class A of U-expressions. In other words, suppose that ‘a’ is such that ‘ab’ is in K if and only if, for some ‘e’ in A, ‘be’ is in K. Then ‘Dab’ is in K if and only if, for some ‘e and ‘f’ in A, ‘be’ or ‘b(ef)’ is in K; and ‘a’, ‘Da’, ‘D(Da)’, and so on, can be regarded as existence operators that provide for existence quantification over successively wider and wider finite classes. In particular, if ‘a’ is ‘εσ’, then A would be the class Y having ‘σ’ as its only member, and we can define the unrestricted existence operator ‘E’ in such a way that ‘Eb’ is in K if and only if some one of ‘εσb’, ‘D(εσ)b’, ‘D(D(εσ))b’, and so on, is in K.

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