On the reduction of the decision problem. First paper. Ackermann prefix, a single binary predicate

1939 ◽  
Vol 4 (1) ◽  
pp. 1-9 ◽  
Author(s):  
László Kalmár
Keyword(s):  
Decision Problem ◽  
Reduction Process ◽  
The Other ◽  
First Order ◽  
Other Hand ◽  
Special Cases ◽  
Binary Predicate ◽  
Ternary Variables ◽  
Order Predicate Calculus ◽  
Predicate Variable

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form(1) (Ex1)(x2)(Ex3)(x4)…(xm).On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove theTheorem. 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.2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

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.

On the reduction of the decision problem

1947 ◽  
Vol 12 (3) ◽  
pp. 65-73 ◽  
Author(s):  
László Kalmár ◽  
János Surányi
Keyword(s):  
Special Form ◽  
Decision Problem ◽  
Predicate Calculus ◽  
Reduction Theorem ◽  
First Order ◽  
Title One ◽  
Order Predicate Calculus ◽  
Predicate Variable

In the first paper of the above main title, one of us has proved that any formula of the first order predicate calculus is equivalent (as to being satisfiable or not) to some binary first order formula having a prefix of the form (Ex1)(x2)(Ex3) … (xn) and containing a single predicate variable. This result is an improvement of a theorem of Ackermann stating that any first order formula is equivalent to another with a prefix of the above form but saying nothing about the number of predicate variables appearing therein. Hence the question arises if other theorems reducing the decision problem to the satisfiability question of the first order formulas with a prefix of a special form can be improved in like manner. In the present paper we shall answer this question concerning Gödel's reduction theorem stating that any first order formula is equivalent to another the prefix of which has the form

Postnikov “Invariants” in 2004

2005 ◽  
Vol 12 (1) ◽  
pp. 139-155
Author(s):  
Julio Rubio ◽  
Francis Sergeraert
Keyword(s):  
Decision Problem ◽  
The Other ◽  
Topological Spaces ◽  
Exact Nature ◽  
Other Hand

Abstract The very nature of the so-called Postnikov invariants is carefully studied. Two functors, precisely defined, explain the exact nature of the connection between the category of topological spaces and the category of Postnikov towers. On one hand, these functors are in particular effective and lead to concrete machine computations through the general machine program Kenzo. On the other hand, the Postnikov “invariants” will be actual invariants only when an arithmetical decision problem – currently open – will be solved; it is even possible this problem is undecidable.

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.

scholarly journals Deontic Logics as Axiomatic Extensions of First-Order Predicate Logic: An Approach Inspired by Wolniewicz’s Formal Ontology of Situations

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 109 ◽  
Author(s):  
Malec
Keyword(s):  
Deontic Logic ◽  
Predicate Logic ◽  
The Other ◽  
Choice Situation ◽  
Formal Ontology ◽  
Legal Rules ◽  
First Order ◽  
The Third ◽  
Other Hand ◽  
First Order Predicate Logic

The aim of this article is to present a method of creating deontic logics as axiomatic theories built on first-order predicate logic with identity. In the article, these theories are constructed as theories of legal events or as theories of acts. Legal events are understood as sequences (strings) of elementary situations in Wolniewicz′s sense. On the other hand, acts are understood as two-element legal events: the first element of a sequence is a choice situation (a situation that will be changed by an act), and the second element of this sequence is a chosen situation (a situation that arises as a result of that act). In this approach, legal rules (i.e., orders, bans, permits) are treated as sets of legal events. The article presents four deontic systems for legal events: AEP, AEPF, AEPOF, AEPOFI. In the first system, all legal events are permitted; in the second, they are permitted or forbidden; in the third, they are permitted, ordered or forbidden; and in the fourth, they are permitted, ordered, forbidden or irrelevant. Then, we present a deontic logic for acts (AAPOF), in which every act is permitted, ordered or forbidden. The theorems of this logic reflect deontic relations between acts as well as between acts and their parts. The direct inspiration to develop the approach presented in the article was the book Ontology of Situations by Boguslaw Wolniewicz, and indirectly, Wittgenstein’s Tractatus Logico-Philosophicus.

scholarly journals A Survey of Conceptualizations of Politeness

2015 ◽  
Vol 5 (6) ◽  
pp. 115
Author(s):  
Lei Qiu
Keyword(s):  
Second Order ◽  
The Other ◽  
Theoretical Construct ◽  
First Order ◽  
Relational Work ◽  
Other Hand ◽  
Uniform Definition

<p>Along with the general trends of research from traditional Gricean approach to postmodern approach, politeness has been conceptualized as facework, social indexing concept, relational work and interactional work. Based on examination of debates over East group-oriented and Western individual-oriented politeness, first-order and second-order politeness, as well as the universality and relativity of conceptualizations, this paper has roughly demonstrated that the tension between universality and relativity of politeness can help to explain the reason for lack of uniform definition and concept in this field. It is essential for researchers to seek a universal second-order culture-general theoretical construct on one hand, and to look at first-order culture-specific constructs on the other hand.</p>

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.

scholarly journals Human perception of shape from touch

2011 ◽  
Vol 366 (1581) ◽  
pp. 3106-3114 ◽  
Author(s):  
Astrid M. L. Kappers
Keyword(s):  
Human Perception ◽  
The Other ◽  
Dominant Factor ◽  
Stimulus Information ◽  
After Effects ◽  
First Order ◽  
Other Hand ◽  
After Effect ◽  
The Difference

In this paper, I focus on the role of active touch in three aspects of shape perception and discrimination studies. First an overview is given of curvature discrimination experiments. The most prominent result is that first-order stimulus information (that is, the difference in attitude or slope over the stimulus) is the dominant factor determining the curvature threshold. Secondly, I compare touch under bimanual and two-finger performance with unimanual and one-finger performance. Consistently, bimanual or two-finger performance turned out to be worse. The most likely explanation for the former finding is that a loss of accuracy during intermanual comparisons is owing to interhemispheric relay. Thirdly, I address the presence of strong after-effects after just briefly touching a shape. These after-effects have been measured and studied in various conditions (such as, static, dynamic, transfer to other hand or finger). Combination of the results of these studies leads to the insight that there are possibly different classes of after-effect: a strong after-effect, caused by immediate contact with the stimulus, that does only partially transfer to the other hand, and one much less strong after-effect, caused by moving over the stimulus for a certain period, which shows a full transfer to other fingers.

scholarly journals Personal Assessment Effects in Spreading the Quality of Culture Among Employees of the Special Care Organizations in Jordan

2016 ◽  
Vol 12 (22) ◽  
pp. 224
Author(s):  
Ihsan Gdaifan Ali Alsaree'
Keyword(s):  
Special Education ◽  
Reliability And Validity ◽  
Special Care ◽  
The Other ◽  
Organizational Assessment ◽  
First Order ◽  
Quality Culture ◽  
Other Hand ◽  
Personal Assessment

The aim of this study is to recognize the effect of self-organizational assessment in spreading quality culture among employees at special education organizations in Jordan. The study sample consisted of employees in 20 special education organizations that present services for persons with disability who were chosen randomly. The study instrument was designed. It consisted of 64 items distributed into seven domains. Also, their reliability and validity were calculated. The findings show that the level of selforganizational assessment in spreading quality culture among employees at special education organizations has an average which ranged from 2.89 to 2.61. Here, the domain of programs and services was in the first order with the highest average of 2.89. On the other hand, the domain of mixing was in the last order with an average of 2.61. In addition, the average of the effectiveness as whole was 2.73. Furthermore, the findings show that there were statistical differences (a= 0.05) due to the effect of the organization type. As a result, T value was 2.711 with a statistical significant of 0.007 in favor of the special organizations.

The power of locality domains in phonology

Phonology ◽  
2017 ◽  
Vol 34 (2) ◽  
pp. 385-405 ◽  
Author(s):  
Thomas Graf
Keyword(s):  
Computational Complexity ◽  
The Other ◽  
Adequate Model ◽  
First Order ◽  
Other Hand ◽  
Integral Role ◽  
Domain Restrictions ◽  
The One

Domains play an integral role in linguistic theories. This paper combines locality domains with current models of the computational complexity of phonology. The first result is that if a specific formalism – strictly piecewise grammars – is supplemented with a mechanism to enforce first-order definable domain restrictions, its power increases so much that it subsumes almost the full hierarchy of subregular languages. However, if domain restrictions are based on linguistically natural intervals, we instead obtain an empirically more adequate model. On the one hand, this model subsumes only those subregular classes that have been argued to be relevant for phonotactic generalisations. On the other hand, it excludes unnatural generalisations that involve counting or elaborate conditionals. It is also shown that strictly piecewise grammars with interval-based domains are theoretically learnable, unlike those with arbitrary, first-order domains.

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