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.

A NORMAL FORM FOR FIRST-ORDER LOGIC OVER DOUBLY-LINKED DATA STRUCTURES

2008 ◽  
Vol 19 (01) ◽  
pp. 205-217 ◽  
Author(s):  
STEVEN LINDELL
Keyword(s):  
Normal Form ◽  
Data Structures ◽  
Linked Data ◽  
Linear Time ◽  
Order Logic ◽  
Information Storage ◽  
First Order Logic ◽  
Total Size ◽  
Bounded Degree ◽  
First Order

We use singulary vocabularies to analyze first-order definability over doubly-linked data structures. Singulary vocabularies contain only monadic predicate and monadic function symbols. A class of mathematical structures in any vocabulary can be elementarily interpreted in a singulary vocabulary, while preserving notions of total size and degree. Doubly-linked data structures are a special case of bounded-degree finite structures in which there are reciprocal connections between elements, corresponding closely with physically feasible models of information storage. They can be associated with logical models involving unary relations and bijective functions in what we call an invertible singulary vocabulary. Over classes of these models, there is a normal form for first-order logic which eliminates all quantification of dependent variables. The paper provides a syntactically based proof using counting quantifiers. It also makes precise the notion of implicit calculability for arbitrary arity first-order formulas. Linear-time evaluation of first-order logic over doubly-linked data structures becomes a direct corollary. Included is a discussion of why these special data structures are appropriate for physically realizable models of information.

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.

Notions of locality and their logical characterizations over finite models

1999 ◽  
Vol 64 (4) ◽  
pp. 1751-1773 ◽  
Author(s):  
Lauri Hella ◽  
Leonid Libkin ◽  
Juha Nurmonen
Keyword(s):  
Small Degree ◽  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
Finite Models ◽  
Order Case ◽  
The Relationship

AbstractMany known tools for proving expressibility bounds for first-ordér logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree.

scholarly journals Learning Relational Representations with Auto-encoding Logic Programs

2019 ◽  
Author(s):  
Sebastijan Dumancic ◽  
Tias Guns ◽  
Wannes Meert ◽  
Hendrik Blockeel
Keyword(s):  
Deep Learning ◽  
Relational Learning ◽  
Order Logic ◽  
First Order Logic ◽  
Relational Data ◽  
Logic Programs ◽  
Learning Methods ◽  
First Order ◽  
Relational Structures ◽  
Representation Language

Deep learning methods capable of handling relational data have proliferated over the past years. In contrast to traditional relational learning methods that leverage first-order logic for representing such data, these methods aim at re-representing symbolic relational data in Euclidean space. They offer better scalability, but can only approximate rich relational structures and are less flexible in terms of reasoning. This paper introduces a novel framework for relational representation learning that combines the best of both worlds. This framework, inspired by the auto-encoding principle, uses first-order logic as a data representation language, and the mapping between the the original and latent representation is done by means of logic programs instead of neural networks. We show how learning can be cast as a constraint optimisation problem for which existing solvers can be used. The use of logic as a representation language makes the proposed framework more accurate (as the representation is exact, rather than approximate), more flexible, and more interpretable than deep learning methods. We experimentally show that these latent representations are indeed beneficial in relational learning tasks.

scholarly journals Local Normal Forms for First-Order Logic with Applications to Games and Automata

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Thomas Schwentick ◽  
Klaus Barthelmann
Keyword(s):  
Normal Forms ◽  
Winning Strategy ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Relational Structures ◽  
Second Order Logic ◽  
Monadic Second Order Logic

International audience Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

scholarly journals Successor-Invariant First-Order Logic on Classes of Bounded Degree (Extended Abstract)

2021 ◽  
Author(s):  
Julien Grange
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
Successor Relation

We study the expressive power of successor-invariant first-order logic, which is an extension of first-order logic where the usage of a successor relation on the vertices of the graph is allowed, as long as the validity of formulas is independent on the choice of a particular successor. We show that when the degree is bounded, successor-invariant first-order logic is no more expressive than first-order logic.

Logical Methods for Self-Configuration of Network Devices

2011 ◽  
pp. 189-216 ◽  
Author(s):  
Sylvain Hallé ◽  
Roger Villemaire ◽  
Omar Cherkaoui
Keyword(s):  
Mathematical Logic ◽  
Order Logic ◽  
Point Of View ◽  
First Order Logic ◽  
Declarative Languages ◽  
First Order ◽  
User Intervention

The goal of self-configuration consists of providing appropriate values for parameters that modulate the behaviour of a device. In this chapter, self-configuration is studied from a mathematical logic point of view. In contrast with imperative means of generating configurations, characterized by scripts and templates, the use of declarative languages such as propositional or first-order logic is argued. In that setting, device configurations become models of particular logical formulæ, which can be generated using constraint solvers without any rigid scripting or user intervention.

Infinitary logics

2018 ◽  
Author(s):  
Bernd Buldt
Keyword(s):  
Mathematical Logic ◽  
Order Logic ◽  
Logical Analysis ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Essential Tool

An infinitary logic arises from ordinary first-order logic when one or more of its finitary properties is allowed to become infinite, for example, by admitting infinitely long formulas or infinitely long or infinitely branched proof figures. The need to extend first-order logic became pressing in the late 1950s when it was realized that many of the fundamental notions of mathematics cannot be expressed in first-order logic in a way that would allow for their logical analysis. Because infinitary logics often do not suffer the same limitation, they have become an essential tool in mathematical logic.

Minimal predicates, fixed-points, and definability

2005 ◽  
Vol 70 (3) ◽  
pp. 696-712 ◽  
Author(s):  
Johan Van Benthem
Keyword(s):  
Mathematical Logic ◽  
Fixed Points ◽  
Expressive Power ◽  
Correspondence Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Technical Result ◽  
Unique Existence ◽  
First Order ◽  
Preservation Theorem

AbstractMinimal predicates P satisfying a given first-order description ϕ(P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘PIA conditions’ ϕ(P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal in expressive power to LFP(FO), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond.

Sign in / Sign up

Export Citation Format

Share Document