Prefix classes of Krom formulas

1973 ◽  
Vol 38 (4) ◽  
pp. 628-642 ◽  
Author(s):  
Stål O. Aanderaa ◽  
Harry R. Lewis
Keyword(s):  
Decision Problem ◽  
Decision Problems ◽  
Quantification Theory

In this paper we consider decision problems for subclasses of Kr, the class of those formulas of pure quantification theory whose matrices are conjunctions of binary disjunctions of signed atomic formulas. If each of Q1, …, Qn is an ∀ or an ∃, then let Q1 … Qn be the class of those closed prenex formulas with prefixes of the form (Q1x1)… (Qnxn). Our results may then be stated as follows:Theorem 1. The decision problem for satisfiability is solvable for the class ∀∃∀ ∩ Kr.Theorem 2. The classes ∀∃∀∀ ∩ Kr and ∀∀∃∀ ∩ Kr are reduction classes for satisfiability.Maslov [11] showed that the class ∃…∃∀…∀∃…∃ ∩ Kr is solvable, while the first author [1, Corollary 4] showed ∃∀∃∀ ∩ Kr and ∀∃∃∀ ∩ Kr to be reduction classes. Thus the only prefix subclass of Kr for which the decision problem remains open is ∀∃∀∃…∃∩ Kr.The class ∀∃∀ ∩ Kr, though solvable, contains formulas whose only models are infinite (e.g., (∀x)(∃u)(∀y)[(Pxy ∨ Pyx) ∧ (¬ Pxy ∨ ¬Pyu)], which can be satisfied over the integers by taking P to be ≥). This is not the case for Maslov's class ∃…∃∀…∀∃…∃ ∩ Kr, which contains no formula whose only models are infinite ([2] [5]).Theorem 1 was announced in [1, Theorem 4], but the proof sketched there is defective: Lemma 4 (p. 17) is incorrectly stated. Theorem 2 was announced in [9].

Logical Decision Problems and Complexity of Logic Programs

1987 ◽  
Vol 10 (1) ◽  
pp. 1-33
Author(s):  
Egon Börger ◽  
Ulrich Löwen
Keyword(s):  
Fixed Point ◽  
Computational Complexity ◽  
Decision Problem ◽  
Decision Problems ◽  
Complexity Classes ◽  
Logic Programs ◽  
Finite Structures ◽  
Syntactic Restrictions

We survey and give new results on logical characterizations of complexity classes in terms of the computational complexity of decision problems of various classes of logical formulas. There are two main approaches to obtain such results: The first approach yields logical descriptions of complexity classes by semantic restrictions (to e.g. finite structures) together with syntactic enrichment of logic by new expressive means (like e.g. fixed point operators). The second approach characterizes complexity classes by (the decision problem of) classes of formulas determined by purely syntactic restrictions on the formation of formulas.

scholarly journals COVID-19 Vaccination Priority Evaluation

2021 ◽  
Author(s):  
Jozo J Dujmovic ◽  
Daniel Tomasevich
Keyword(s):  
Decision Problem ◽  
Evaluation Method ◽  
Social Conditions ◽  
Decision Problems ◽  
Organ Transplants ◽  
Quantitative Criterion

Computing the COVID-19 vaccination priority is an urgent and ubiquitous decision problem. In this paper we propose a solution of this problem using the LSP evaluation method. Our goal is to develop a justifiable and explainable quantitative criterion for computing a vaccination priority degree for each individual in a population. Performing vaccination in the order of the decreasing vaccination priority produces maximum positive medical, social, and ethical effects for the whole population. The presented method can be expanded and refined using additional medical and social conditions. In addition, the same methodology is suitable for solving other similar medical priority decision problems, such as priorities for organ transplants.

Skolem reduction classes

1975 ◽  
Vol 40 (1) ◽  
pp. 62-68 ◽  
Author(s):  
Warren D. Goldfarb ◽  
Harry R. Lewis
Keyword(s):  
Decision Problem ◽  
Atomic Formula ◽  
Complex Situation ◽  
Quantification Theory ◽  
Reduction Class

Among the earliest and best-known theorems on the decision problem is Skolem's result [7] that the class of all closed formulas with prefixes of the form ∀···∀∃···∃ is a reduction class for satisfiability for the whole of quantification theory. This result can be refined in various ways. If the Skolem prefix alone is considered, the best result [8] is that the ∀∀∀∃ class is a reduction class, for Gödel [3], Kalmár [4], and Schütte [6] showed the ∀∀∃···∃ class to be solvable. The purpose of this paper is to describe the more complex situation that arises when (Skolem) formulas are restricted with respect to the arguments of their atomic subformulas. Before stating our theorem, we must introduce some notation.Let x, y1, y2, be distinct variables (we shall use v1, v2, ··· and w1, w2, ··· as metavariables ranging over these variables), and for each i ≥ 1 let Y(i) be the set {y1, ···, yi}. An atomic formula Pv1 ··· vk will be said to be {v1, ···, vk}-based. For any n ≥ 1, p ≥ 1, and any subsets Y1, ··· Yp of Y(n), let C(n, Y1, ···, Yp) be the class of all those closed formulas with prefix ∀y1 ··· ∀yn∃x such that each atomic subformula not containing the variable x is Yi-based for some i, 1 ≤ i ≤ p.

scholarly journals Integrating imperfection of information into the promethee multicriteria decision aid methods: a general framework

2012 ◽  
Vol 37 (1) ◽  
pp. 9-23 ◽  
Author(s):  
Sarah Ben Amor ◽  
Bertrand Mareschal
Keyword(s):  
Decision Problem ◽  
Decision Aid ◽  
General Framework ◽  
Imperfect Information ◽  
Possibility Theory ◽  
Decision Problems ◽  
Multicriteria Decision Aid ◽  
Multicriteria Decision ◽  
Information Models ◽  
Multicriteria Methods

Abstract.Multicriteria decision aid methods are used to analyze decision problems including a series of alternative decisions evaluated on several criteria. They most often assume that perfect information is available with respect to the evaluation of the alternative decisions. However, in practice, imprecision, uncertainty or indetermination are often present at least for some criteria. This is a limit of most multicriteria methods. In particular the PROMETHEE methods do not allow directly for taking into account this kind of imperfection of information. We show how a general framework can be adapted to PROMETHEE and can be used in order to integrate different imperfect information models such as a.o. probabilities, fuzzy logic or possibility theory. An important characteristic of the proposed approach is that it makes it possible to use different models for different criteria in the same decision problem.

Solving Decision Problems with Mathematical Expectation

2011 ◽  
Vol 128-129 ◽  
pp. 293-296
Author(s):  
Mei Zhang ◽  
Jing Hua Wen
Keyword(s):  
Mathematical Expectation ◽  
Decision Problem ◽  
Random Variables ◽  
Decision Problems ◽  
Employment Decision ◽  
Decision Variables ◽  
Project Contract

Mathematical expectation is one of the important digital characters of the random variables. Many decision variables in decision problem are random variables; it is difficult to define their concrete distribution, while we can use their mathematical expectation to resolve decision questions. The applications of mathematical expectation in decision question were analyzed by example from two aspects of project contract decision and obtain employment decision.

Organizationally Intractable Decision Problems and the Intellectual Virtues of Heuristics

2017 ◽  
Vol 43 (8) ◽  
pp. 2620-2637 ◽  
Author(s):  
Richard A. Bettis
Keyword(s):  
Artificial Intelligence ◽  
Strategic Management ◽  
Decision Problem ◽  
Decision Problems ◽  
Allocation Decision ◽  
Mba Programs ◽  
Powerful Approach ◽  
Rules Of Thumb ◽  
Intractable Problems ◽  
Computational Intractability

There is no theory in strategic management and other related fields for identifying decision problems that cannot be solved by organizations using rational analytical technologies of the type typically taught in MBA programs. Furthermore, some and perhaps many scholars in strategic management believe that the alternative of heuristics or “rules of thumb” is little more than crude guesses for decision making when compared to rational analytical technologies. This is reflected in a paucity of research in strategic management on heuristics. I propose a theory of “organizational intractability” based roughly on the metaphor provided by “computational intractability” in computer science. I demonstrate organizational intractability for a common model of the joint strategic planning and resource allocation decision problem. This raises the possibility that heuristics are necessary for deciding many important decisions that are intractable for organizations. This possibility parallels the extensive use of heuristics in artificial intelligence for computationally intractable problems, where heuristics are often the most powerful approach possible. Some important managerial heuristics are documented from both the finance and strategic management literatures. Based on all of this, I discuss some directions for theory of and research on organizational intractability and heuristics in strategic management.

scholarly journals Reference Model of Project Prototyping Problem

2011 ◽  
Vol 3 (1) ◽  
pp. 33-46
Author(s):  
Marcin Relich ◽  
Zbigniew Banaszak
Keyword(s):  
Programming Languages ◽  
Decision Problem ◽  
Reference Model ◽  
The Other ◽  
Decision Problems ◽  
Open Structure ◽  
Decision Variables ◽  
The Hierarchical Structure ◽  
Constraints Satisfaction ◽  
Natural Way

Reference Model of Project Prototyping ProblemThe paper presents the idea of reference model of project prototyping problem for the projects that are at risk of failure. The hierarchical structure of declarative model connects two fields: functionalities of a typical service enterprise and management system of project execution in the enterprise. The functionalities as separate Constraints Satisfaction Problems (CSP) are described. CSP contains the sets of decision variables, their domains and constraints, which link these variables. The separated problems described as CSP, then in single main CSP are integrated. On the other hand, these problems can decompose into the subproblems concerning the functionalities of different fields. The open structure of model enables to solve the decision problems with different level of specificity. The decision problem can regard a query about the results of proposed decisions as well as the decisions guaranteeing the expected results. A declarative kind of proposed reference model in a natural way allows to implement its in constraint programming languages. The possibility of this approach illustrates an example.

scholarly journals The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines

2020 ◽  
Author(s):  
Matheus Santana Lima
Keyword(s):  
Communication Systems ◽  
Decision Problem ◽  
Communication Channel ◽  
Solution Space ◽  
Search Space ◽  
Decision Problems ◽  
Turing Machines ◽  
Bernoulli Process ◽  
General Process ◽  
New Interpretation

We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials problems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines.

Information Distribution Decisions Supported by Argumentation

2011 ◽  
pp. 489-495 ◽  
Author(s):  
Ramon Brena ◽  
Carlos Chesñevar
Keyword(s):  
Information Technology ◽  
Knowledge Management ◽  
Decision Problem ◽  
Decision Problems ◽  
Information Distribution ◽  
Huge Amount ◽  
Automatic Reasoning ◽  
Human Labor ◽  
Access Rights ◽  
Technology Tool

Information and knowledge (IK) are each day more valuable assets in modern organizations (Atkinson, Court, & Ward, 1998; Carrillo, 1998; Liebowitz & Beckman, 1998). Distributing IK is indeed one of the main processes in knowledge management (Liebowitz & Wilcox, 1997; Horibe, 1999). Now, deciding which piece of IK goes to which member of an organization is a decision problem not simple in real organizations. There are many aspects that should be considered, such as what are the responsibilities and tasks of each person, which access rights they have, what are their preferences, and so on. Taking into account all these aspects requires either a huge amount of human labor or the help of an information-technology tool. In this article we explore how a particular technology, automated argumentation, which is a kind of automatic reasoning, can be applied to solve decision problems related to information distribution in an organization.

Decision problems for multiple successor arithmetics

1966 ◽  
Vol 31 (2) ◽  
pp. 182-190 ◽  
Author(s):  
J. W. Thatcher
Keyword(s):  
Decision Problem ◽  
Decision Procedure ◽  
Decision Problems ◽  
First Order

Let Nk denote the set of words over the alphabet Σk = {1, …, k}. Nk contains the null word which is denoted λ. We consider decision problems for various first-order interpreted predicate languages in which the variables range over Nk (k ≧ 2). Our main result is that there is no decision procedure for truth in the interpreted language which has the subword relation as its only non-logical primitive. This, together with known results summarized in Section 4, settles the decision problem for any language constructed on the basis of the relations and functions listed below.

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