p URL: http://www.elsevier. nl/loc ate/en tcs/vo lume2. html 14 pages Linear Time Computable Problems and Logical Descriptions Detlef Seese University Karlsruhe TH, AIFB Institute for Applied Computer Science and Formal Description Methods, D - 76128 Karlsruhe, Germany e-mail: [email protected] Abstract It is a general problem to investigate the trade o b etween the complexityofal- gorithmic problems, the structure of the input ob jects and the expressivepower of problem description languages. The article concentrates on linear time algorithms and on rst order logic FO as problem description language. One of the main results is a pro of that each FO-problem can b e solved in linear time for arbitrary relational structures of universally b ounded degree. Keywords: Linear Time Computability, Relational Structures, Descriptive Complexity Theory, First-Order Logic, Finite Mo del Theory. 1 Intro duction It could b e one of the ultimate goals of computer science to nd a general problem solver, i.e. a machinery which transforms each algorithmic problem P into an algorithm to solve P for arbitrary inputs. But unfortunately,itis well-known that there is no such algorithm whose input is a formal description ' of an arbitrary algorithmic problem P and whose output is an algorithm to solve P . Hence it is an interesting question to nd restrictions on ' and P under which the ab ove question could have an armative answer. A p ossible formalization is to nd a language L or more precise a logic, i.e. a language and its semantics and a class K of structures, such that each problem P which can b e describ ed by a formula ' in L can b e solved eciently for each structure G 2 K .Now it is the problem to nd L and K in suchaway that the expressivepower of L is strong, i.e. many algorithmic problems can b e expressed in it, the class K is large, i.e. it contains many structures interesting for applications, and all these problems can b e solved with low complexity, i.e. in p olynomial or b etter in linear time. Many approaches develop ed in connection with this problem concentrated on strong languages with high expressivepower, but had to pay the price c 1995 Elsevier Science B. V. Seese of very restricted classes of structures. The most successful ones investi- gated extensions of the strong monadic second order language MSO or re- lated algebraic-logical approaches in connection with structures of universally b ounded tree width, i.e. structures which are up to a certain parameter close related to trees [2,6,8,9,11,13,14,16,31,36,40,42]. In many of these cases one gets even linear time algorithms. However in case that one is interested in larger classes of structures it seems to b e necessary to lower the expressive power of the corresp onding language L. With resp ect to such classes of struc- tures almost all existing general results b elong to the theory of descriptive complexity. Historically, the investigation of connections b etween complexity classes on one hand and descriptions in logical languages on the other started with Ronald Fagin's seminal pap er [22], where he proved that NP coincides with the class of problems expressible in existential second order logic. Immerman [33] proved corresp onding results for P, NL and several other complexity classes using extensions of the classical rst-order calculus byvarious op erators and prompted thus the development of descriptive complexityasanown branch of complexity theory. Several of the articles in this area concentrated on the famos P-NP{ or the NP-Co-NP{Problem and investigated languages whose expressivepower is higher than that of rst-order logic, e.g. extensions of rst-o der logic by certain op erators, as e.g. a xp oint op erator LFP, for whichitwas proved that P=FO+ LFP if structures with an ordered universe are regarded see [33]. Pure rst order logic did not get so much attention, since its expressivepower is relatively weak and it was one of the early results of this area that FO is strictly contained in L and hence in P, where L denotes the class of deterministic logspace computable problems see [4,32]. The main result of this article is a pro of that each rst-order problem can b e solved in linear time if only relational structures of b ounded degree are regarded. The basic idea of the pro of is a lo calization technique which based on a metho d whichwas originally develop ed by Hanf [30] to show that two in nite structures agree on all rst-order sentences. Fagin, Sto ckmeyer and Vardi [23] develop ed a variant of this technique which is applicable in descriptive complexity theory to nite relational structures of universally b ounded degree. Variants of this result can b e found also in [25] see also [46]. The essential content of this result, which is denoted also as the Hanf-Sphere Lemma, is that two relational structures of b ounded degree ful ll the same rst-order sentences of a certain quanti er-rank, if b oth contain up to a certain number m the same numb er of isomorphism typ es of substructures of a b ounded radius r . The pap er is organized as follows: section 2 intro duces the basic terminol- ogy and the notion BIORAM which serves as basis of the linear time com- putability used in this pap er. Section 3 intro duces lo cal r -typ es and handles the case of structures of b ounded degree by reducing the general problem to a lo cal investigation of r -typ es. Some op en problems and remarks conclude the pap er in section 4. 2 Seese A Martin Kreidler prepared the L T X-version of this article. I wish to thank E him for his ecient and diligentwork. Furthermore, I thank him for his helpful suggestions and careful reading of several versions of the \ nal" text. Finally, Imust express my gratitude to Wolfgang Thomas and his colleagues in Kiel, where I presented and discussed the rst version of the main result. 2 De nitions and conventions This section is devoted to a brief intro duction of the basic terminology, Notions from logic or complexity theory not intro duced in this or the following sections are standard and the reader is referred e.g. to [10,12,20,37]. A nite signature S for relational structures is a nite set of relation sym- b ols R ;:::;R , each with a xed arity r > 0, and constant symb ols c ;:::;c , 1 s i 1 t G G G G G but without function symb ols. An S -structure G =A ;R ;:::;R ;c ;:::;c 1 1 s t G consists of a nite set A , the domain or universe, from whichwe assume that G G it is the set f1;:::;ng for a natural number n, relations R over A of arity i G G r for each i :1is and elements c of A for each j :1j t. The i j individuals of the domain of a structure are sometimes denoted as points or in analogy to graphs as vertices .For a structure G we will denote its domain by jG j. The numb er of the elements of an arbitrary set B will b e denoted also as j B j, but this will cause no diculties, since sets and structures can b e typ ographically distinguished. An S -structure G is called nite in case that its domain is. Unless otherwise stated, throughout the rest of this article we make the assumption that all structures that will b e considered are nite. If S is a signature, let ST RUC T S= fG : G is a nite S -structureg. G G For a subset B A , that contains all c with 1 j t, the induced j G G r G r G substructure GB is the structure B; R \ B ; :::; R \ B ;c ;:::;c . 1 1 1 s s t G G Individuals a and b from A are said to b e adjacent by R , if there are i G x ;:::;x , such that a = b or x ;:::;x 2 R and a = x ;b = x for some 1 r 1 r j k i i i G j; k r . We will sometimes denote x ;:::;x asR -edge connecting a and i 1 r i i G b. a and b are said to b e adjacent in G if there is a relation R such that a and b i G are adjacentbyR . In this case a is said to b e incident with the corresp onding i edge x ; :::; x . The degree of an individual a is the cardinality of the set of 1 r i individuals adjacenttoabut not equal to a. The degree of a structure is the maximum of the degrees of its individuals. A sequence x ;:::;x is called a 0 m G -path, or simply a path if it is clear which structure is used, if for every j<m,x and x are adjacent. m is denoted as the length of this path. j j +1 G The distance d a; bbetween a and b in G is the length of a shortest path G from a to b in G .For r 0 the r -neighbourhood of x in G , N a, consists r G G of all b 2 A with d a; b r .