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.

The theory of all substructures of a structure: Characterisation and decision problems

1979 ◽  
Vol 44 (4) ◽  
pp. 583-598 ◽  
Author(s):  
Kenneth L. Manders
Keyword(s):  
Philosophy Of Science ◽  
Decision Problem ◽  
Decision Procedure ◽  
Relational Structure ◽  
Decision Problems ◽  
Partial Reduction ◽  
First Order ◽  
Entire Theory

AbstractAn infinitary characterisation of the first-order sentences true in all substructures of a structure M is used to obtain partial reduction of the decision problem for such sentences to that for Th(M). For the relational structure 〈R, ≤, + 〉 this gives a decision procedure for the ∃x∀y-part of the theory of all substructures, yet we show that the ∃x1x2∀y-part, and the entire theory, is Π11-complete. The theory of all ordered subsemigroups of 〈R, ≤, + 〉 is also shown Π11-complete. Applications in the philosophy of science are mentioned.

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.

Some undecidability results in strong algebraic languages

1984 ◽  
Vol 49 (3) ◽  
pp. 951-954
Author(s):  
Cornelia Kalfa
Keyword(s):  
Decision Problem ◽  
Decision Problems ◽  
Turing Machines ◽  
Constant Symbol ◽  
Finite Sets ◽  
Operation Symbol ◽  
Halting Problem ◽  
First Order ◽  
Order Language ◽  
Image Position

The recursively unsolvable halting problem for Turing machines is reduced to the problem of the existence or not of an algorithm for deciding whether a field is finite. The latter problem is further reduced to the decision problem of each of propertiesfor recursive sets Σ of equations of strong algebraic languages with infinitely many operation symbols.Decision problems concerning properties of sets of equations were first raised by Tarski [9] and subsequently examined by Perkins [6], McKenzie [4], McNulty [5] and Pigozzi [7]. Perkins is the only one who studied recursive sets; the others investigated finite sets. Since the undecidability of properties Pi for recursive sets of equations does not imply any answer to the corresponding decision problems for finite sets, the latter problems remain open.The work presented here is part of my Ph.D. thesis [2]. I thank Wilfrid Hodges, who supervised it.An algebraic language is a first-order language with equality but without relation symbols. It is here denoted by , where Qi is an operation symbol and cj, is a constant symbol.

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.

scholarly journals ON THE FINITENESS PROBLEM FOR AUTOMATON (SEMI)GROUPS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250052 ◽  
Author(s):  
ALI AKHAVI ◽  
INES KLIMANN ◽  
SYLVAIN LOMBARDY ◽  
JEAN MAIRESSE ◽  
MATTHIEU PICANTIN
Keyword(s):  
Decision Problem ◽  
Decision Procedure ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mealy Automata ◽  
Necessary And Sufficient ◽  
New Criteria

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskiĭ, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.

Church’s theorem and the decision problem

2018 ◽  
Author(s):  
Rohit Parikh
Keyword(s):  
Decision Problem ◽  
Combinatorial Problem ◽  
Order Logic ◽  
First Order Logic ◽  
Negative Solution ◽  
Great Contribution ◽  
Solvable Problem ◽  
First Order ◽  
Mathematical Definition ◽  
Definition Of

Church’s theorem, published in 1936, states that the set of valid formulas of first-order logic is not effectively decidable: there is no method or algorithm for deciding which formulas of first-order logic are valid. Church’s paper exhibited an undecidable combinatorial problem P and showed that P was representable in first-order logic. If first-order logic were decidable, P would also be decidable. Since P is undecidable, first-order logic must also be undecidable. Church’s theorem is a negative solution to the decision problem (Entscheidungsproblem), the problem of finding a method for deciding whether a given formula of first-order logic is valid, or satisfiable, or neither. The great contribution of Church (and, independently, Turing) was not merely to prove that there is no method but also to propose a mathematical definition of the notion of ‘effectively solvable problem’, that is, a problem solvable by means of a method or algorithm.

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.

Two variable first-order logic over ordered domains

2001 ◽  
Vol 66 (2) ◽  
pp. 685-702 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Order Logic ◽  
Decision Problems ◽  
First Order Logic ◽  
Satisfiability Problem ◽  
Exponential Time ◽  
Infinite Domains ◽  
First Order ◽  
Computation Tree ◽  
Common Extension ◽  
Undecidability Result

AbstractThe satisfiability problem for the two-variable fragment of first-order logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations.It is shown that FO2 over ordered, respectively wellordered. domains or in the presence of one well-founded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems is essentially the same as for plain unconstrained FO2. namely non-deterministic exponential time.In contrast FO2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings. wellorderings. or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO2 and computation tree logic CTL.

Sign in / Sign up

Export Citation Format

Share Document