On fixed-point logic with counting

2000 ◽  
Vol 65 (2) ◽  
pp. 777-787 ◽  
Author(s):  
Jörg Flum ◽  
Martin Grohe
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Planar Graphs ◽  
Expressive Power ◽  
Descriptive Complexity ◽  
Main Motivation ◽  
First Order ◽  
Finite Structures ◽  
Tree Width

One of the fundamental results of descriptive complexity theory, due to Immerman [13] and Vardi [18], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered finite structures by a logic in a similar way.The most obvious shortcoming of fixed-point logic itself on unordered structures is that it cannot count. Immerman [14] responded to this by adding counting constructs to fixed-point logic. Although it has been proved by Cai, Fürer, and Immerman [1] that the resulting fixed-point logic with counting, denoted by IFP+C, still does not capture all of polynomial time, it does capture polynomial time on several important classes of structures (on trees, planar graphs, structures of bounded tree-width [15, 9, 10]).The main motivation for such capturing results is that they may give a better understanding of polynomial time. But of course this requires that the logical side is well understood. We hope that our analysis of IFP+C-formulas will help to clarify the expressive power of IFP+C; in particular, we derive a normal form. Moreover, we obtain a problem complete for IFP+C under first-order reductions.

scholarly journals Isomorphism, canonization, and definability for graphs of bounded rank width

2021 ◽  
Vol 64 (5) ◽  
pp. 98-105
Author(s):  
Martin Grohe ◽  
Daniel Neuen
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Descriptive Complexity ◽  
Graph Isomorphism Problem ◽  
Structural Graph ◽  
Graph Classes ◽  
Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

Finite model theory

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Model Theory ◽  
Classical Model ◽  
Expressive Power ◽  
Order Logic ◽  
Finite Model ◽  
Descriptive Complexity ◽  
Finite Model Theory ◽  
First Order ◽  
Finite Models ◽  
Finite Structures

This final chapter unites ideas from both model theory and complexity theory. Finite model theory is the part of model theory that disregards infinite structures. Examples of finite structures naturally arise in computer science in the form of databases, models of computations, and graphs. Instead of satisfiability and validity, finite model theory considers the following finite versions of these properties. • A first-order sentence is finitely satisfiable if it has a finite model. • A first-order sentence is finitely valid if every finite structure is a model. Finite model theory developed separately from the “classical” model theory of previous chapters. Distinct methods and logics are used to analyze finite structures. In Section 10.1, we consider various finite-variable logics that serve as useful languages for finite model theory. We define variations of the pebble games introduced in Section 9.2 to analyze the expressive power of these logics. Pebble games are one of the few tools from classical model theory that is useful for investigating finite structures. In Section 10.2, it is shown that many of the theorems from Chapter 4 are no longer true when restricted to finite models. There is no analog for the Completeness and Compactness theorems in finite model theory. Moreover, we prove Trakhtenbrot’s theorem which states that the set of finitely valid first-order sentences is not recursively enumerable. Descriptive complexity is the subject of 10.3. This subject describes the complexity classes discussed in Chapter 7 in terms of the logics introduced in Chapter 9. We prove Fagin’s theorem relating the class NP to existentional second-order logic. We prove the Cook–Levin theorem as a consequence of Fagin’s Theorem. This theorem states that the Satisfiability Problem for Propositional Logic is NP-complete. We conclude this chapter (and this book) with a section describing the close connection between logic and the P = NP problem. In this section, we discuss appropriate logics for the study of finite models. First-order logic, since it describes each finite model up to isomorphism, is too strong. For this reason, we must weaken the logic. It may seem counter-intuitive that we should gain knowledge by weakening our language.

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.

Some Aspects of Model Theory and Finite Structures

2002 ◽  
Vol 8 (3) ◽  
pp. 380-403 ◽  
Author(s):  
Eric Rosen
Keyword(s):  
Computer Science ◽  
Complexity Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Finite Model ◽  
Finite Model Theory ◽  
First Order ◽  
Single Sentence ◽  
Finite Structures

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

Boundary Classes of Planar Graphs

2008 ◽  
Vol 17 (2) ◽  
pp. 287-295 ◽  
Author(s):  
VADIM LOZIN
Keyword(s):  
Polynomial Time ◽  
Planar Graphs ◽  
Dominating Set ◽  
Helpful Tool ◽  
Tree Width

We analyse classes of planar graphs with respect to various properties such as polynomial-time solvability of thedominating setproblem or boundedness of the tree-width. A helpful tool to address this question is the notion of boundary classes. The main result of the paper is that for many important properties there are exactly two boundary classes of planar graphs.

scholarly journals Descriptive complexity of finite structures: Saving the quantifier rank

2005 ◽  
Vol 70 (2) ◽  
pp. 419-450 ◽  
Author(s):  
Oleg Pikhurko ◽  
Oleg Verbitsky
Keyword(s):  
Automorphism Group ◽  
Descriptive Complexity ◽  
First Order ◽  
Finite Structures ◽  
Universal Quantifiers ◽  
Quantifier Alternation

AbstractWe say that a first order formula Φ distinguishes a structure M over a vocabulary L from another structure M′ over the same vocabulary if Φ is true on M but false on M′. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M′. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M′.We prove that every n-element structure M is identifiable by a formula with quantifier rank less than and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition of two elements, the same result holds for definability rather than identification.The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than quantifiers suffice to identify M and. as long as we keep the number of universal quantifiers bounded by a constant, at total quantifiers are necessary.

scholarly journals Fifty years of the spectrum problem: survey and new results

2012 ◽  
Vol 18 (4) ◽  
pp. 505-553 ◽  
Author(s):  
Arnaud Durand ◽  
Neil D. Jones ◽  
Johann A. Makowsky ◽  
Malika More
Keyword(s):  
Complexity Theory ◽  
Recursion Theory ◽  
Finite Model ◽  
Chronological Order ◽  
Descriptive Complexity ◽  
First Order ◽  
Structural Graph ◽  
Theoretic Problem ◽  
Bounded Complexity

AbstractIn 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open.

Epsilon-logic is more expressive than first-order logic over finite structures

2000 ◽  
Vol 65 (4) ◽  
pp. 1749-1757 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Database Theory ◽  
Finite Structures

AbstractThere are properties of finite structures that are expressible with the use of Hilbert's ∈-operator in a manner that does not depend on the actual interpretation for ∈-terms. but not expressible in plain first-order. This observation strengthens a corresponding result of Gurevich, concerning the invariant use of an auxiliary ordering in first-order logic over finite structures. The present result also implies that certain non-deterministic choice constructs, which have been considered in database theory, properly enhance the expressive power of first-order logic even as far as deterministic queries are concerned, thereby answering a question raised by Abiteboul and Vianu.

scholarly journals Descriptive Complexity, Computational Tractability, and the Logical and Cognitive Foundations of Mathematics

2020 ◽  
Author(s):  
Markus Pantsar
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Theoretical Computer Science ◽  
Human Cognition ◽  
Complexity Classes ◽  
Foundations Of Mathematics ◽  
Descriptive Complexity ◽  
Cognitive Capacities ◽  
Computational Complexity Theory ◽  
First Order

Abstract In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes.

scholarly journals On the Descriptive Complexity of Color Coding

Algorithms ◽  
2021 ◽  
Vol 14 (3) ◽  
pp. 96
Author(s):  
Max Bannach ◽  
Till Tantau
Keyword(s):  
Complexity Theory ◽  
Parameterized Complexity ◽  
Syntactic Structure ◽  
Color Pattern ◽  
Fixed Number ◽  
Color Coding ◽  
Descriptive Complexity ◽  
First Order ◽  
Algorithmic Technique ◽  
Syntactic Properties

Color coding is an algorithmic technique used in parameterized complexity theory to detect “small” structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing—purely in terms of the syntactic structure of describing formulas—when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in FPT just because of the way they are commonly described using logical formulas.

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