A reduction class containing formulas with one monadic predicate and one binary function symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45-49
Author(s):  
Charles E. Hughes
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Conjunctive Normal Form ◽  
Function Symbol ◽  
Predicate Calculus ◽  
Binary Function ◽  
First Order ◽  
The Matrix ◽  
Reduction Class ◽  
Order Predicate Calculus

AbstractA new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

On simplifying the matrix of a wff

1968 ◽  
Vol 33 (2) ◽  
pp. 180-192 ◽  
Author(s):  
Peter Andrews
Keyword(s):  
Normal Form ◽  
Decision Procedure ◽  
Conjunctive Normal Form ◽  
Predicate Calculus ◽  
First Order ◽  
The Matrix ◽  
Order Predicate Calculus

In [3], [4], and [5] Joyce Friedman formulated and investigated certain rules which constitute a semi-decision procedure for wffs of first order predicate calculus in closed prenex normal form with prefixes of the form ∀x1 … ∀xκ∃y1 … ∃ym∀z1 … ∀zn. Given such a wff QM, where Q is the prefix and M is the matrix in conjunctive normal form, Friedman's rules can be used, in effect, to construct a matrix M* which is obtained from M by deleting certain conjuncts of M.

The decision problem for formulas in prenex conjunctive normal form with binary disjunctions

1970 ◽  
Vol 35 (2) ◽  
pp. 210-216 ◽  
Author(s):  
M. R. Krom
Keyword(s):  
Normal Form ◽  
Positive Solution ◽  
Decision Problem ◽  
Conjunctive Normal Form ◽  
Decision Problems ◽  
First Order ◽  
Reduction Class ◽  
Image Position ◽  
Positive Integers ◽  
Order Predicate Calculus

In [8] S. J. Maslov gives a positive solution to the decision problem for satisfiability of formulas of the formin any first-order predicate calculus without identity where h, k, m, n are positive integers, αi, βi are signed atomic formulas (atomic formulas or negations of atomic formulas), and ∧, ∨ are conjunction and disjunction symbols, respectively (cf. [6] for a related solvable class). In this paper we show that the decision problem is unsolvable for formulas that are like those considered by Maslov except that they have prefixes of the form ∀x∃y1 … ∃yk∀z. This settles the decision problems for all prefix classes of formulas for formulas that are in prenex conjunctive normal form in which all disjunctions are binary (have just two terms). In our concluding section we report results on decision problems for related classes of formulas including classes of formulas in languages with identity and we describe some special properties of formulas in which all disjunctions are binary including a property that implies that any proof of our result, that a class of formulas is a reduction class for satisfiability, is necessarily indirect. Our proof is based on an unsolvable combinatorial tag problem.

A normal form theorem for first order formulas and its application to Gaifman's splitting theorem

1984 ◽  
Vol 49 (4) ◽  
pp. 1262-1267
Author(s):  
Nobuyoshi Motohashi
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Atomic Formula ◽  
Splitting Theorem ◽  
First Order ◽  
Usual Manner ◽  
Binary Predicate ◽  
Normal Form Theorem ◽  
Image Position ◽  
Order Predicate Calculus

Let L be a first order predicate calculus with equality which has a fixed binary predicate symbol <. In this paper, we shall deal with quantifiers Cx, ∀x ≦ y, ∃x ≦ y defined as follows: CxA(x) is ∀y∃x(y ≦ x ∧ A(x)), ∀x ≦ yA{x) is ∀x(x ≦ y ⊃ A(x)), and ∃x ≦ yA(x) is ∃x(x ≦ y ∧ 4(x)). The expressions x̄, ȳ, … will be used to denote sequences of variables. In particular, x̄ stands for 〈x1, …, xn〉 and ȳ stands for 〈y1,…, ym〉 for some n, m. Also ∃x̄, ∀x̄ ≦ ȳ, … will be used to denote ∃x1 ∃x2 … ∃xn, ∀x1 ≦ y1 ∀x2 ≦ y2 … ∀xn ≦ yn, …, respectively. Let X be a set of formulas in L such that X contains every atomic formula and is closed under substitution of free variables and applications of propositional connectives ¬(not), ∧(and), ∨(or). Then, ∑(X) is the set of formulas of the form ∃x̄B(x̄), where B ∈ X, and Φ(X) is the set of formulas of the form.Since X is closed under ∧, ∨, the two sets Σ(X) and Φ(X) are closed under ∧, ∨ in the following sense: for any formulas A and B in Σ(X) [Φ(X)], there are formulas in Σ(X)[ Φ(X)] which are obtained from A ∧ B and A ∨ B by bringing some quantifiers forward in the usual manner.

Two interpolation theorems for a predicate calculus

1971 ◽  
Vol 36 (2) ◽  
pp. 262-270
Author(s):  
Shoji Maehara ◽  
Gaisi Takeuti
Keyword(s):  
Normal Form ◽  
Completeness Theorem ◽  
Interpolation Theorem ◽  
Second Order ◽  
Predicate Calculus ◽  
Cut Elimination ◽  
First Order ◽  
Image Position ◽  
Order Predicate Calculus

A second order formula is called Π1 if, in its prenex normal form, all second order quantifiers are universal. A sequent F1, … Fm → G1 …, Gn is called Π1 if a formulais Π1If we consider only Π1 sequents, then we can easily generalize the completeness theorem for the cut-free first order predicate calculus to a cut-free Π1 predicate calculus.In this paper, we shall prove two interpolation theorems on the Π1 sequent, and show that Chang's theorem in [2] is a corollary of our theorem. This further supports our belief that any form of the interpolation theorem is a corollary of a cut-elimination theorem. We shall also show how to generalize our results for an infinitary language. Our method is proof-theoretic and an extension of a method introduced in Maehara [5]. The latter has been used frequently to prove the several forms of the interpolation theorem.

Compactness and transfer for a fragment of L2

1977 ◽  
Vol 42 (2) ◽  
pp. 261-268
Author(s):  
M. Magidor ◽  
J. Malitz
Keyword(s):  
Function Symbol ◽  
Predicate Calculus ◽  
Infinite Cardinal ◽  
Small Fragment ◽  
First Order ◽  
Open Questions ◽  
Higher Power ◽  
Countably Compact ◽  
Order Predicate Calculus

The language Ln is obtained from the first order predicate calculus by adjoining the quantifier Qn which binds n variables. The formula Qnυ1 … υnΨ is given a κ-interpretation for each infinite cardinal κ, namely, “there is a set X of power κ such that Ψx1 … xn holds for all distinct x1 … xn ϵ X”. L<ω is the result of adjoining all the Qn quantifiers for each n ϵ ω to the first order predicate calculus.In [4] we showed that under the assumption (cf. [3]) L<ω is countably compact under the ω1-interpretation, and that any sentence σ ϵ L<ω that has a model in some κ-interpretation where κ is a regular infinite cardinal has a model in the ω1 interpretation. However, compactness for L<ω in the κ-interpretation for κ an infinite successor cardinal other than ω1 and the transfer of satisfiability from ω1 to any higher power remain open questions under any set theoretic assumptions.Here we restrict our attention to a small fragment L2− of L2 consisting of universal first order formulas along with formulas of the kind Q2υ1υ2∀υ3 … υnΨ and ¬Q2υ1υ2∀υ3 … υnφ where Ψ and φ are open and no function symbol of arity > 1 occurs in any formula. Assuming the existence of a κ-Souslin tree, this language is λ compact in the κ-interpretation when λ < κ.

Some forms of completeness

1962 ◽  
Vol 27 (3) ◽  
pp. 344-352 ◽  
Author(s):  
P. C. Gilmore
Keyword(s):  
Predicate Symbol ◽  
Constant Term ◽  
Predicate Calculus ◽  
First Order ◽  
Individual Variables ◽  
The Individual ◽  
And Function ◽  
Order Predicate Calculus

By a theory is meant an applied first-order predicate calculus with at least one predicate symbol and perhaps some individual constants and function symbols and a specified set of axioms. In addition to the terms defined by means of the individual variables, constants, and function symbols a theory may also include among its terms those constructed by means of operators such as the epsilon or iota operators; that is, expressions like (εχΡ) or (οχΡ), where P is a well formed formula (wff) of the theory, may also be terms. A constant term of a theory F is then a term in which no variable occurs free. We are interested only in theories which have at least one constant term so that if a theory doesn't have any individual constants it must necessarily admit as terms expressions constructed by means of operators. A sentence of a theory F is a closed wff.

scholarly journals On a Propositional Calculus Whose Decision Problem is Recursively Unsolvable

1970 ◽  
Vol 38 ◽  
pp. 145-152
Author(s):  
Akira Nakamura
Keyword(s):  
Decision Problem ◽  
Propositional Calculus ◽  
Predicate Calculus ◽  
Reduction Theorem ◽  
List Type ◽  
First Order ◽  
Order Predicate Calculus

The purpose of this paper is to present a propositional calculus whose decision problem is recursively unsolvable. The paper is based on the following ideas: (1) Using Löwenheim-Skolem’s Theorem and Surányi’s Reduction Theorem, we will construct an infinitely many-valued propositional calculus corresponding to the first-order predicate calculus.(2) It is well known that the decision problem of the first-order predicate calculus is recursively unsolvable.(3) Thus it will be shown that the decision problem of the infinitely many-valued propositional calculus is recursively unsolvable.

Infinitary logic and admissible sets

1969 ◽  
Vol 34 (2) ◽  
pp. 226-252 ◽  
Author(s):  
Jon Barwise
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Predicate Calculus ◽  
Generalized Quantifiers ◽  
Infinitary Logic ◽  
First Order ◽  
Admissible Sets ◽  
Classical Language ◽  
Second Order Logic ◽  
Order Predicate Calculus

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is(I) strong enough to express interesting properties not expressible by the classical language, but(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

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