Note on a problem of L. Henkin

1956 ◽  
Vol 21 (1) ◽  
pp. 33-35 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Normal Form ◽  
Predicate Calculus ◽  
Universal Quantifiers ◽  
Image Position ◽  
Necessary And Sufficient

A statement X in the lower predicate calculus is said to be persistent with respect to the set of statements K ([2], [3]), if whenever X holds in a model M of K then X holds also in all extensions of M which are models of K. If X is persistent with respect to the empty set, then it may be said to be absolutely persistent.A statement X is called existential, if it is in prenex normal form and does not contain any universal quantifiers. This includes the possibility that X does not contain any quantifiers at all.Let E be the class of all existential statements. Then it is not difficult to see that E is quasi-disjunctive. That is to say, given statements Y1, Y2 in E, there exists a statement Y in E such thatis provable.L. Henkin [1] has raised the question how to characterise the statements X which are persistent with respect to a given set K (e.g. a set of axioms for a field or a group) by a syntactical condition. He has shown that, in order that a statement X be absolutely persistent, it is necessary and sufficient that there exist a statement Y ϵ E such thatis provable.

scholarly journals The number of proofs for a BCK-formula

1993 ◽  
Vol 58 (2) ◽  
pp. 626-628 ◽  
Author(s):  
Yuichi Komori ◽  
Sachio Hirokawa
Keyword(s):  
Normal Form ◽  
Modus Ponens ◽  
Basic Notion ◽  
Type Assignment ◽  
Natural Deduction System ◽  
Cartesian Closed Categories ◽  
Image Position ◽  
Coherence Theorem ◽  
Cartesian Closed ◽  
Necessary And Sufficient

In this note, we give a necessary and sufficient condition for a BCK-formula to have the unique normal form proof.We call implicational propositional formulas formulas for short. BCK-formulas are the formulas which are derivable from axioms B = (a → b) → (c → a) → c → b, C = (a→b→c)→b→a→c, and K = a→b→a by substitution and modus ponens. It is known that the property of being a BCK-formula is decidable (Jaskowski [11, Theorem 6.5], Ben-Yelles [3, Chapter 3, Theorem 3.22], Komori [12, Corollary 6]). The set of BCK-formulas is identical to the set of provable formulas in the natural deduction system with the following two inference rules.Here γ occurs at most once in (→I). By the formulae-as-types correspondence [10], this set is identical to the set of type-schemes of closed BCK-λ-terms. (See [5].) A BCK-λ-term is a λ-term in which no variable occurs twice. Basic notion concerning the type assignment system can be found [4]. Uniqueness of normal form proofs has been known for balanced formulas. (See [2,14].) It is related to the coherence theorem in cartesian closed categories. A formula is balanced when no variable occurs more than twice in it. It was shown in [8] that the proofs of balanced formulas are BCK-proofs. Relevantly balanced formulas were defined in [9], and it was proved that such formulas have unique normal form proofs. Balanced formulas are included in the set of relevantly balanced formulas.

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.

Random models and solvable Skolem classes

1993 ◽  
Vol 58 (3) ◽  
pp. 908-914
Author(s):  
Warren Goldfarb
Keyword(s):  
Normal Form ◽  
Systematic Approach ◽  
Finite Model ◽  
Theoretic Property ◽  
Quantification Theory ◽  
Maslov Class ◽  
Random Models ◽  
Universal Quantifiers ◽  
Necessary And Sufficient ◽  
Made In

A Skolem class is a class of formulas of pure quantification theory in Skolem normal form: closed, prenex formulas with prefixes ∀…∀∃…∃. (Pure quantification theory contains quantifiers, truth-functions, and predicate letters, but neither the identity sign nor function letters.) The Gödel Class, in which the number of universal quantifiers is limited to two, was shown effectively solvable (for satisfiability) sixty years ago [G1]. Various solvable Skolem classes that extend the Gödel Class can be obtained by allowing more universal quantifiers but restricting the combinations of variables that may occur together in atomic subformulas [DG, Chapter 2]. The Gödel Class and these extensions are also finitely controllable, that is, every satisfiable formula in them has a finite model. In this paper we isolate a model-theoretic property that connects the usual solvability proofs for these classes and their finite controllability. For formulas in the solvable Skolem classes, the property is necessary and sufficient for satisfiability. The solvability proofs implicitly relied on this fact. Moreover, for any formula in Skolem normal form, the property implies the existence of a finite model.The proof of the latter implication uses the random models technique introduced in [GS] for the Gödel Class and modified and applied in [Go] to the Maslov Class. The proof thus substantiates the claim made in [Go] that random models can be adapted to the Skolem classes of [DG, Chapter 2]. As a whole, the results of this paper provide a more general, systematic approach to finite controllability than previous methods.

scholarly journals A Note on Decomposable and Reducible Integer Matrices

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1125
Author(s):  
Carlos Marijuán ◽  
Ignacio Ojeda ◽  
Alberto Vigneron-Tenorio
Keyword(s):  
Normal Form ◽  
Linear Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Integer Matrix ◽  
Hermite Normal Form ◽  
Necessary And Sufficient

We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficiently determined by elementary linear algebra techniques, and which completely determines the decomposability of the first one.

Resolution systems and their applications I

1980 ◽  
Vol 3 (2) ◽  
pp. 235-268
Author(s):  
Ewa Orłowska
Keyword(s):  
Theorem Proving ◽  
Sufficient Conditions ◽  
Predicate Calculus ◽  
Resolution Method ◽  
First Order ◽  
Deductive Systems ◽  
Resolution System ◽  
Resolution Principle ◽  
Necessary And Sufficient ◽  
Classical Predicate

The central method employed today for theorem-proving is the resolution method introduced by J. A. Robinson in 1965 for the classical predicate calculus. Since then many improvements of the resolution method have been made. On the other hand, treatment of automated theorem-proving techniques for non-classical logics has been started, in connection with applications of these logics in computer science. In this paper a generalization of a notion of the resolution principle is introduced and discussed. A certain class of first order logics is considered and deductive systems of these logics with a resolution principle as an inference rule are investigated. The necessary and sufficient conditions for the so-called resolution completeness of such systems are given. A generalized Herbrand property for a logic is defined and its connections with the resolution-completeness are presented. A class of binary resolution systems is investigated and a kind of a normal form for derivations in such systems is given. On the ground of the methods developed the resolution system for the classical predicate calculus is described and the resolution systems for some non-classical logics are outlined. A method of program synthesis based on the resolution system for the classical predicate calculus is presented. A notion of a resolution-interpretability of a logic L in another logic L ′ is introduced. The method of resolution-interpretability consists in establishing a relation between formulas of the logic L and some sets of formulas of the logic L ′ with the intention of using the resolution system for L ′ to prove theorems of L. It is shown how the method of resolution-interpretability can be used to prove decidability of sets of unsatisfiable formulas of a given logic.

Delay stabilization of periodic orbits in coupled oscillator systems

2010 ◽  
Vol 368 (1911) ◽  
pp. 319-341 ◽  
Author(s):  
B. Fiedler ◽  
V. Flunkert ◽  
P. Hövel ◽  
E. Schöll
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Periodic Orbits ◽  
Coupled Oscillators ◽  
Coupled Oscillator ◽  
Non Invasive ◽  
Coupling Parameters ◽  
Necessary And Sufficient ◽  
Time Delayed Feedback ◽  
Coupled Oscillator Systems

We study diffusively coupled oscillators in Hopf normal form. By introducing a non-invasive delay coupling, we are able to stabilize the inherently unstable anti-phase orbits. For the super- and subcritical cases, we state a condition on the oscillator’s nonlinearity that is necessary and sufficient to find coupling parameters for successful stabilization. We prove these conditions and review previous results on the stabilization of odd-number orbits by time-delayed feedback. Finally, we illustrate the results with numerical simulations.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

scholarly journals Existence of nonoscillatory solutions of first order nonlinear neutral equations

1990 ◽  
Vol 32 (2) ◽  
pp. 180-192 ◽  
Author(s):  
Lu Wudu
Keyword(s):  
Nonoscillatory Solution ◽  
Sufficient Condition ◽  
Neutral Equation ◽  
Necessary And Sufficient Condition ◽  
Nonoscillatory Solutions ◽  
Neutral Equations ◽  
First Order ◽  
Image Position ◽  
Necessary And Sufficient

AbstractConsider the nonlinear neutral equationwhere pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).

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.

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