Tailoring recursion for complexity

1995 ◽  
Vol 60 (3) ◽  
pp. 952-969 ◽  
Author(s):  
Erich Grädel ◽  
Yuri Gurevich
Keyword(s):  
Polynomial Time ◽  
Recursion Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Complexity Classes ◽  
First Order ◽  
Logarithmic Space ◽  
Functional Analog

AbstractWe design functional algebras that characterize various complexity classes of global functions. For this purpose, classical schemata from recursion theory are tailored for capturing complexity. In particular we present a functional analog of first-order logic and describe algebras of the functions computable in nondeterministic logarithmic space, deterministic and nondeterministic polynomial time, and for the functions computable by AC1 -circuits.

Strong extension axioms and Shelah's zero-one law for choiceless polynomial time

2003 ◽  
Vol 68 (1) ◽  
pp. 65-131 ◽  
Author(s):  
Andreas Blass ◽  
Yuri Gurevich
Keyword(s):  
Fixed Point ◽  
Polynomial Time ◽  
Order Logic ◽  
Complexity Class ◽  
First Order Logic ◽  
Infinitary Logic ◽  
Extensive Discussion ◽  
First Order ◽  
Detailed Proof ◽  
Strong Extension

AbstractThis paper developed from Shelah's proof of a zero-one law for the complexity class “choiceless polynomial time,” defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general.

Finite Variable Logics in Descriptive Complexity Theory

1998 ◽  
Vol 4 (4) ◽  
pp. 345-398 ◽  
Author(s):  
Martin Grohe
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Complexity Classes ◽  
Finite Model ◽  
Polynomial Space ◽  
Finite Model Theory ◽  
First Order ◽  
First Order Theory ◽  
Finite Structures

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider infinitary variants and extensions by so-called counting quantifiers.Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time (PTIME) or polynomial space (PSPACE) can be characterized as collections of all classes of (ordered) finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth.

Universal first-order logic is superfluous in the second level of the polynomial-time hierarchy

2019 ◽  
Vol 27 (6) ◽  
pp. 895-909
Author(s):  
Nerio Borges ◽  
Edwin Pin
Keyword(s):  
Polynomial Time ◽  
Numerical Data ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Syntactic Approach ◽  
Polynomial Time Hierarchy

Abstract In this paper we prove that $\forall \textrm{FO}$, the universal fragment of first-order logic, is superfluous in $\varSigma _2^p$ and $\varPi _2^p$. As an example, we show that this yields a syntactic proof of the $\varSigma _2^p$-completeness of value-cost satisfiability. The superfluity method is interesting since it gives a way to prove completeness of problems involving numerical data such as lengths, weights and costs and it also adds to the programme started by Immerman and Medina about the syntactic approach in the study of completeness.

Recursion theory and formal deducibility

1970 ◽  
Vol 35 (4) ◽  
pp. 556-558
Author(s):  
E. M. Kleinberg
Keyword(s):  
Recursion Theory ◽  
Order Logic ◽  
Negative Integer ◽  
Predicate Calculus ◽  
First Order Logic ◽  
Recursively Enumerable Set ◽  
First Order ◽  
Recursively Enumerable ◽  
Order Predicate Calculus

The enumeration, given a first-order sentence , of all sentences deducible from in the first-order predicate calculus, and the enumeration, given a non-negative integer n, of the recursively enumerable set Wn, are two well-known examples of effective processes. But are these processes really distinct? Indeed, might there not exist a Gödel numbering of the sentences of first-order logic such that for each n, if n is the number assigned to the sentence , then Wn is the set of numbers assigned to all sentences deducible from ? If this were the case, the first sort of enumeration would just be a particular instance of the second.

Computability and complexity

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Polynomial Time ◽  
Time Complexity ◽  
Finite Graph ◽  
Computability Theory ◽  
Theoretical Computer Science ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Mathematical Problems ◽  
Time Required

In this chapter we study two related areas of theoretical computer science: computability theory and computational complexity. Each of these subjects take mathematical problems as objects of study. The aim is not to solve these problems, but rather to classify them by level of difficulty. Time complexity classifies a given problem according to the length of time required for a computer to solve the problem. The polynomial-time problems P and the nondeterministic polynomial-time problems NP are the two most prominent classes of time complexity. Some problems cannot be solved by the algorithmic process of a computer. We refer to problems as decidable or undecidable according to whether or not there exists an algorithm that solves the problem. Computability theory considers undecidable problems and the brink between the undecidable and the decidable. There are only countably many algorithms and uncountably many problems to solve. From this fact we deduce that most problems are not decidable. To proceed beyond this fact, we must state precisely what we mean by an “algorithm” and a “problem.” One of the aims of this chapter is to provide a formal definition for the notion of an algorithm. The types of problems we shall consider are represented by the following examples. • The even problem: Given an n ∈ ℕ, determine whether or not n is even. • The 10-clique problem: Given finite graph, determine whether or not there exists a subgraph that is isomorphic to the 10-clique. • The satisfiability problem for first-order logic: Given a sentence of first-order logic, determine whether or not it is satisfiable. The first problem is quite easy. To determine whether a given number is even, we simply check whether the last digit of the number is 0, 2, 4, 6 or 8. The second problem is harder. If the given graph is large and does contain a 10-clique as a subgraph, then we may have to check many subsets of the graph before we find it. Time complexity gives precise meaning to the ostensibly subjective idea of one problem being “harder” than another. The third problem is the most difficult of the three problems.

Linear time computable problems and first-order descriptions

1996 ◽  
Vol 6 (6) ◽  
pp. 505-526 ◽  
Author(s):  
Detlef Seese
Keyword(s):  
Mathematical Logic ◽  
Polynomial Time ◽  
Linear Time ◽  
Order Logic ◽  
First Order Logic ◽  
Algorithmic Problem ◽  
Bounded Degree ◽  
First Order ◽  
Relational Structures ◽  
Formal Theories

It is well known that every algorithmic problem definable by a formula of first-order logic can be solved in polynomial time, since all these problems are inL(see Aho and Ullman (1979) and Immerman (1987)). Using an old technique of Hanf (Hanf 1965) and other techniques developed to prove the decidability of formal theories in mathematical logic, it is shown that an arbitraryFO-problem over relational structures of bounded degree can be solved in linear time.

Relativized logspace and generalized quantifiers over finite ordered structures

1997 ◽  
Vol 62 (2) ◽  
pp. 545-574 ◽  
Author(s):  
Georg Gottlob
Keyword(s):  
Sufficient Conditions ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
General Point ◽  
Complexity Classes ◽  
Generalized Quantifiers ◽  
Ordered Structures ◽  
First Order ◽  
Henkin Quantifiers

AbstractWe here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures LC, i.e., logarithmic space relativized to an oracle in C. We show that this is not always true. However, after studying the problem from a general point of view, we derive sufficient conditions on C such that FO(Q) captures LC. These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by NP. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures LNP. This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many families Q of generalized quantifiers (including the family of Henkin quantifiers), each FO(Q)-formula can be replaced by an equivalent FO(Q)-formula with only two occurrences of generalized quantifiers. This generalizes and extends an earlier normal-form result by I. A. Stewart [Fundamenta Inform, vol. 18, 1993].

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

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