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.

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 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.

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 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.

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.

Conciliating States and Locations: Towards a More Comprehensive and In- Depth Account of the Spanish Copula Estar

2013 ◽  
Vol 6 (1) ◽  
Author(s):  
Ma. Eugenia Mangialavori
Keyword(s):  
Syntactic Structure ◽  
Integrative Analysis ◽  
Semantic Features ◽  
Selectional Restrictions ◽  
Complementary Distribution ◽  
Spatial Domains ◽  
Structural Analogy ◽  
Syntactic Properties ◽  
Copular Clauses

AbstractThe case of estar may reveal how different proposals of study have failed to grasp grammatically relevant semantic features shared by its occurrences. The results of this study indicate that an integrative analysis of estar clauses would account not only for the consistent lexical properties observed - comprising (a)analogous lexical-syntactic structure predicting possible copular complements, (b)analogous selectional restrictions and (c)interpretative effects -, but also for the complementary distribution of two aspectually nontrivial verbal alternations (ser / estar and estar / haber). Our proposal lays on the standard syntactic structure of copular clauses - assumed to embrace locative clauses, against what traditional Spanish grammar suggests - in combination with (i) the structural analogy between estar’s alternative complements (APs and PPs) and (ii) the understanding of states as abstract spatial domains (be at). Thus, the eventual differences between clauses like ‘estoy triste’ and ‘estoy en casa’ could be accounted for by virtue of the semantic / syntactic properties of the lexical head selected.

scholarly journals Machine-based methods in parameterized complexity theory

2005 ◽  
Vol 339 (2-3) ◽  
pp. 167-199 ◽  
Author(s):  
Yijia Chen ◽  
Jörg Flum ◽  
Martin Grohe
Keyword(s):  
Complexity Theory ◽  
Parameterized Complexity

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 The Parameterized Complexity of Motion Planning for Snake-Like Robots

2019 ◽  
Author(s):  
Siddharth Gupta ◽  
Guy Sa'ar ◽  
Meirav Zehavi
Keyword(s):  
Motion Planning ◽  
Parameterized Complexity ◽  
Initial Position ◽  
Polynomial Kernel ◽  
Planning Problem ◽  
Color Coding ◽  
Grid Graphs ◽  
Input Size ◽  
Motion Problems ◽  
General Graphs

We study a motion-planning problem inspired by the game Snake that models scenarios like the transportation of linked wagons towed by a locomotor to the movement of a group of agents that travel in an ``ant-like'' fashion. Given a ``snake-like'' robot with initial and final positions in an environment modeled by a graph, our goal is to decide whether the robot can reach the final position from the initial position without intersecting itself. Already on grid graphs, this problem is PSPACE-complete [Biasi and Ophelders, 2018]. Nevertheless, we prove that even on general graphs, it is solvable in time k^{O(k)}|I|^{O(1)} where k is the size of the robot, and |I| is the input size. Towards this, we give a novel application of color-coding to sparsify the configuration graph of the problem. We also show that the problem is unlikely to have a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion problems has been~largely~neglected, thus our work is pioneering in this regard.

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