regular graphs and the spectra of two variable logic with counting the spectrum of a first order logic sentence is the set of natural numbers that are cardinalities of its finite models in this paper we show that when restricted to using only two variables but allowing counting quantifiers the class of spectra of first order logic sentences is exactly the class of semilinear sets and hence closed under complement at the heart of our proof are semilinear characterizations for the existence of regular and biregular graphs the class of graphs in which there are a priori bounds on the degrees of the vertices our proof also provides a simple characterization of models of two variable logic with counting that is up to renaming and extending the relation names they are simply a collection of regular and biregular graphs

on the efficiency of subsumption algorithms the costs of subsumption algorithms are analyzed by an estimation of the maximal number of unification attempts worst case unification complexity made for deciding whether a clause c subsumes a clause d for this purpose the clauses c and d are characterized by the following parameters number of variables in c number of literals in c number of literals in d and maximal length of the literals the worst case unification complexity immediately yields a lower bound for the worst case time complexity first two well known algorithms chang lee stillman are investigated both algorithms are shown to have a very high worst case time complexity then a new subsumption algorithm is defined which is based on an analysis of the connection between variables and predicates in c an upper bound for the worst case unification complexity of this algorithm which is much lower than the lower bounds for the two other algorithms is derived examples in which exponential costs are reduced to polynomial costs are discussed finally the asymptotic growth of the worst case complexity for all discussed algorithms is shown in a table for several combinations of the parameters

data complexity of the two variable fragment with counting quantifiers the data complexity of both satisfiability and finite satisfiability for the two variable fragment with counting quantifiers is np complete the data complexity of both query answering and finite query answering for the two variable guarded fragment with counting quantifiers is co np complete

removing redundancy from a clause abstract this paper deals with the problem of removing redundant literals from a given clause we first consider condensing a weak type of redundancy elimination a clause is condensed if it does not subsume any proper subset of itself it is often useful and sometimes necessary to replace a non condensed clause c by a condensation i e by a condensed subset of c which is subsumed by c after studying the complexity of an existing clause condensing algorithm we present a more efficient algorithm and provide arguments for the optimality of the new method we prove that testing whether a given clause is condensed is co np complete and show that several problems related to clause condensing belong to complexity classes that are probably slightly harder than np we also consider a stronger version of redundancy elimination a clause c is strongly condensed iff it does not contain any proper subset c such that c logically implies c we show that the problem of testing whether a clause is strongly condensed is undecidable

on a result of ozawa concerning uniqueness of meromorphic functions we prove a uniqueness theorem for meromorphic functions sharing three values with some finite weight which improves a recent result of the author

meromorphic functions sharing three weighted values in this paper we deal with the relation between the characteristics of two meromorphic functions that share three weighted values which improve the results of q c zhang and through the results two conjectures which proposed by mues and osgood yang are solved in a weaker condition

on meromorphic functions that share three values of finite weights abstract a uniqueness theorem for two distinct non constant meromorphic functions that share three values of finite weights is proved which generalizes two previous results by h x yi and x m li and h x yi as applications of it many known results by h x yi and p li etc could be improved furthermore with the concept of finite weight sharing extensions on osgood yang s conjecture and mues conjecture and a generalization of some prevenient results by m ozawa and h ueda ect could be obtained

the second main theorem for small functions and related problems we shall establish the following three results in more general forms 1 the second main theorem for small functions let f be a meromorphic function on the complex plane c let a1 aq be distinct meromorphic functions on c assume that ai are small with respect to f i e t r a 0 corollary 1 here as usual in nevanlinna theory the terms t r f and n r ai f denote for the characteristic function and the truncated counting function respectively 2 application to functional equations let kc be the field of meromorphic functions on c for a function r 0 r put k c a kc t r a o r which is a subfield of kc then the following holds let f x y k c x y be a polynomial in two variables over k c assume that the curve f x y 0 over k c has genus greater than one if 1 2 kc satisfy the functional equation f 1 2 0 then both 1 and 2 are contained in k c corollary 2 3 height inequality for curves over function fields let k be a function field of one variable over c let x be a smooth projective curve over k let d x be a reduced divisor let l be an ample line bundle on x and let e 0 then we have hk kx d p n 1 k s d p dk p ehk l p oe 1 for all p x k d theorem 5 here the notations are introduced in v1 v3 see also section 9 our proof uses ahlfors theory of covering surfaces and the geometry of the moduli space of q pointed stable curves of genus 0

on results of h ueda and g brosch concerning the unicity of meromorphic functions by emplying the main ideas from g brosch two results of r nevanlinna s four value theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values and a fourth value or a pair of values weakly are proved which extend and generalize previous results of r nevanlinna g g gundersen h ueda and g brosch

a guide to completeness and complexity for modal logics of knowledge and belief abstract we review and re examine possible worlds semantics for propositional logics of knowledge and belief with three particular points of emphasis 1 we show how general techniques for finding decision procedures and complete axiomatizations apply to models for knowledge and belief 2 we show how sensitive the difficulty of the decision procedure is to such issues as the choice of modal operators and the axiom system and 3 we discuss how notions of common knowledge and distributed knowledge among a group of agents fit into the possible worlds framework as far as complexity is concerned we show among other results that while the problem of deciding satisfiability of an s5 formula with one agent is np complete the problem for many agents is pspace complete adding a distributed knowledge operator does not change the complexity but once a common knowledge operator is added to the language the problem becomes complete for exponential time

weak and strong fuzzy interval orders abstract the purpose of this paper is to introduce and investigate the fuzzification of the classical interval order one of the most interesting classical preference structures without incomparability in this study we consider fuzzy preference structures as defined by de baets et al fuzzy preference structures without incomparability receive special attention their fuzzy preference and fuzzy large preference relations share certain types of the t n ferrers property two special types of the j n ferrers property are introduced the weak ferrers and strong ferrers properties the classical interval order is briefly reviewed t fuzzy interval orders are introduced and it is shown that their fuzzy preference relation is sup t transitive two special types of t fuzzy interval orders are considered weak and strong fuzzy interval orders corresponding to transforms of w and to m the particular intermediate role of the weak ferrers property of the fuzzy preference relation of a fps without incomparability is demonstrated on the one hand it is a necessary condition for the fps to be a strong fuzzy interval order while on the other hand it is a sufficient condition for this structure to be a weak fuzzy interval order finally the concept of an cut of a fps is introduced leading to an interesting characterization of the strong ferrers property of a fps without incomparability

on the refutational completeness of signed binary resolution and hyperresolution signed binary resolution and hyperresolution belong to the basic resolution proof methods both the resolution methods are refutation sound and under some finitary restrictions refutation complete our aim is to investigate their refutational completeness in a more general case we shall assume that clausal theories may be countable sets and the set of truth values an arbitrary one there are unsatisfiable countable clausal theories for which there do not exist refutations by signed binary resolution and hyperresolution we propose a criterion on the existence of a refutation of an unsatisfiable countable clausal theory by the investigated resolution methods two important applications of the achieved results to automated deduction in signed logic herbrand s theorem and a signed davis putnam logemann loveland dpll procedure will be discussed

modular proof systems for partial functions with evans equality the paper presents a modular superposition calculus for the combination of first order theories involving both total and partial functions the modularity of the calculus is a consequence of the fact that all the inferences are pure only involving clauses over the alphabet of either one but not both of the theories when refuting goals represented by sets of pure formulae the calculus is shown to be complete provided that functions that are not in the intersection of the component signatures are declared as partial this result also means that if the unsatisfiability of a goal modulo the combined theory does not depend on the totality of the functions in the extensions the inconsistency will be effectively found moreover we consider a constraint superposition calculus for the case of hierarchical theories and show that it has a related modularity property finally we identify cases where the partial models can always be made total so that modular superposition is also complete with respect to the standard total function semantics of the theories

combining deduction and model checking into tableaux and algorithms for converse pdl this paper presents a prefixed tableaux calculus for propositional dynamic logic with converse based on a combination of different techniques such as prefixed tableaux for modal logics and model checkers for calculus we prove the correctness and completeness of the calculus and illustrate its features we also discuss the transformation of the tableaux method naively nexptime into an exptime algorithm

indicators of fuzzy relations in this paper first we give the definitions of various indicators of fuzzy relations and their basic properties then we investigate the relationships between these indicators particularly between those of t transitivity negative s transitivity t s semitransitivity and t s ferrers property the investigation is divided into three parts results in the general case results under a condition proposed by peng et al and results under special t norms and t conorms on account of our research certain results on fuzzy relations in the literature are extended to the degree description

on the characteristics of meromorphic functions that share three values cm in this paper by studying the counting functions of the common 1 points of meromorphic functions a more precise relation between the characteristics of meromorphic functions that share three values cm has been obtained as applications of this many known results can be improved

ordered chaining calculi for first order theories of transitive relations we propose inference systems for binary relations that satisfy composition laws such as transitivity our inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods as they are used in the context of resolution type theorem proving we establish the refutational completeness of these calculi and prove that our methods are compatible with the usual simplification techniques employed in refutational theorem provers such as subsumption or tautology deletion various optimizations of the basic chaining calculus will be discussed for theories with equality and for total orderings a key to the practicality of chaining methods is the extent to which so called variable chaining can be avoided we demonstrate that rewrite techniques considerably restrict variable chaining and that further restrictions are possible if the transitive relation under consideration satisfies additional properties such as symmetry but we also show that variable chaining cannot be completely avoided in general

deciding expressive description logics in the framework of resolution we present a decision procedure for the description logic shiq based on the basic superposition calculus and show that it runs in exponential time for unary coding of numbers to derive our algorithm we extend basic superposition with a decomposition inference rule which transforms conclusions of certain inferences into equivalent but simpler clauses this rule can be used for general first order theorem proving with any resolution based calculus compatible with the standard notion of redundancy

fuzzy preference structures without incomparability abstract in this paper we establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability this study is carried out for the fuzzy preference structures introduced recently by de baets van de walle and kerre a set of remarkable theorems gives detailed insight in the relationships between the sup t transitivity of the fuzzy preference and indifference relations and the sup t transitivity of the fuzzy large preference relation several paths of thought involving t norms with or without zero divisors are explored and where required illustrative counterexamples confirm the falsity of certain implications finally we introduce the t n ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this ferrers property

on the complexity of qualitative spatial reasoning a maximal tractable fragment of the region connection calculus the computational properties of qualitative spatial reasoning have been investigated to some degree however the question for the boundary between polynomial and np hard reasoning problems has not been addressed yet in this paper we explore this boundary in the region connection calculus rcc 8 we extend bennett s encoding of rcc 8 in modal logic based on this encoding we prove that reasoning is np complete in general and identify a maximal tractable subset of the relations in rcc 8 that contains all base relations further we show that for this subset path consistency is sufficient for deciding consistency

some further results on weighted sharing three values and brosch s theorem using the notion of weighted sharing of values we prove two uniqueness theorems which improve the results proved by t c alzahary t c alzahary weighted sharing three values and brosch s theorem j math anal appl 323 2006 8 25 t c alzahary on a result of ueda concerning unicity of meromorphic functions kodai math j 30 2007 140 146 under a weaker hypothesis

reduction rules for resolution based systems abstract inference rules for resolution based systems can be classified into deduction rules which add new objects and reduction rules which remove objects traditional reduction rules like subsumption do not actively contribute to a solution but they help to avoid redundancies in the search space we present a number of advanced reduction rules which can cope with high degrees of redundancy and play a distinctly active part because they find trivial solutions on their own and thus relieve the control component for the deduction rules from low level tasks we describe how these reduction rules can be implemented with reasonable efficiency in a clause graph resolution system but they are not restricted to this particular representation

exp time tableaux for alc abstract the last years have seen two major advances in knowledge representation and reasoning first many interesting problems ranging from semi structured data to linguistics were shown to be expressible in logics whose main deductive problems are exptime complete second experiments in automated reasoning have substantially broadened the meaning of practical tractability instances of realistic size for pspace complete problems are now within reach for implemented systems still there is a gap between the reasoning services needed by the expressive logics mentioned above and those provided by the current systems indeed the algorithms based on tree automata which are used to prove exptime completeness require exponential time and space even in simple cases on the other hand current algorithms based on tableau methods can take advantage of such cases but require double exponential time in the worst case we propose a tableau calculus for the description logic alc for checking the satisfiability of a concept with respect to a tbox with general axioms and transform it into the first simple tableau based decision procedure working in single exponential time to guarantee the ease of implementation we also discuss the effects that optimizations propositional backjumping simplification semantic branching etc might have on our complexity result and introduce a few optimizations ourselves

a new subsumption method in the connection graph proof procedure compared to most of the other resolution based proof procedures the connection graph proof procedure cgpp is superior due to its effective search space reduction rules a test for subsumption in cgpp is sufficiently efficient in comparison with the standard test but the subsumption with no restriction may cause logical inconsistency the restriction on subsumption to maintain the consistency has been an open problem since it was first noticed in 1981 we define a new subsumption which maintains the logical consistency and provides the graph transformation method which makes c subsume d with the new definition if c subsumes d with the original definition of sumsumption in the subsumption test in cgpp it is essential to find a set of pairwise strongly compatible matching substitutions between literals in two clauses this paper presents an algorithm which finds a set of pairwise strongly compatible matching substitutions efficiently based on both the reduction of possible matching substitutions and bit operations the worst case time complexity of this algorithm is analyzed in terms of the estimated maximal number of string comparisons and it is shown that our complexity is lower than the other algorithms based on the s link test based on the new definition of subsumption and the improved algorithm for the pairwise strongly compatible test an unrestricted subsumption algorithm and a restricted subsumption algorithm are suggested

fuzzy semi orders the case of t norms without zero divisors for crisp relations the concept of a semi order can be stated in a number of equivalent ways when trying to extend this concept to the fuzzy setting we observe that the generalized definitions fail to be equivalent in this contribution we discuss which is the most natural definition of a fuzzy semi order and study the hierarchy among the alternative definitions in particular when using a t norm without zero divisors

complexity results for first order two variable logic with counting let c 2 p denote the class of first order sentences with two variables and with additional quantifiers there exists exactly at most at least i for i leq p and let c 2 be the union of c 2 p taken over all integers p we prove that the satisfiability problem for c 2 1 sentences is nexptime complete this strengthens the results by e gradel ph kolaitis and m vardi bull symbolic logic 3 1997 pp 53 69 who showed that the satisfiability problem for the first order two variable logic l 2 is nexptime complete and by e gradel m otto and e rosen 12th annual ieee symposium on logic in computer science 1997 pp 306 317 who proved the decidability of c 2 our result easily implies that the satisfiability problem for c 2 is in nondeterministic doubly exponential time it is interesting that c 2 1 is in nexptime in spite of the fact that there are sentences whose minimal and only models are of doubly exponential size it is worth noticing that by a recent result of e gradel m otto and e rosen proceedings of 14th annual symposium on theoretical aspects of computer science lecture notes in comput sci 1200 springer verlag berlin 1997 extensions of two variable logic l 2 by a weak access to cardinalities through the hartig or equicardinality quantifier is undecidable the same is true for extensions of l 2 by very weak forms of recursion the satisfiability problem for logics with a bounded number of variables has applications in artificial intelligence notably in modal logics see e g w van der hoek and m de rijke j logic comput 5 1995 pp 325 345 where counting comes in the context of graded modalities and in description logics where counting can be used to express so called number restrictions see e g a borgida artificial intelligence 82 1996 pp 353 367

the complexity of concept languages abstract a basic feature of terminological knowledge representation systems is to represent knowledge by means of taxonomies here called terminologies and to provide a specialized reasoning engine to do inferences on these structures the taxonomy is built through a representation language called a concept language or description logic which is given a well defined set theoretic semantics the efficiency of reasoning has often been advocated as a primary motivation for the use of such systems the main contributions of the paper are 1 a complexity analysis of concept satisfiability and subsumption for a wide class of concept languages 2 algorithms for these inferences that comply with the worst case complexity of the reasoning task they perform

hyperresolution for godel logic with truth constants abstract in the paper we generalise the well known hyperresolution principle to the general first order godel logic with explicit partial truth we propose a hyperresolution calculus suitable for automated deduction in a useful expansion of godel logic by intermediate truth constants and the equality strict order projection operators we solve the deduction problem of a formula from a countable theory in this expansion we expand godel logic by a countable set of intermediate truth constants of the form c c 0 1 our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory consisting of order clauses an order clause is a finite set of order literals of the form e 1 e 2 where e i is an atom or a truth constant or a quantified atom and is a connective equality or strict order and are interpreted by the equality and standard strict linear order on 0 1 respectively we shall investigate the canonical standard completeness where the semantics of godel logic is given by the standard g algebra and truth constants are interpreted by themselves the hyperresolution calculus is refutation sound and complete for a countable order clausal theory under a certain condition for the set of truth constants occurring in the theory as an interesting consequence we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in godel logic with truth constants and the equality strict order projection operators

proof theory for locally finite many valued logics semi projective logics we extend the methodology in baaz and fermuller 1999 5 to systematically construct analytic calculi for semi projective logics a large family of propositional locally finite many valued logics our calculi defined in the framework of sequents of relations are proof search oriented and can be used to settle the computational complexity of the formalized logics as a case study we derive sequent calculi of relations for nilpotent minimum logic and for hajek s basic logic extended with the n contraction axiom n 1 the introduced calculi are used to prove that the decidability problem in these logics is co np complete

a proper hierarchy of propositional sequent calculi abstract it is known that lk a system of propositional sequent calculus without a cut rule written in tree form does not p simulate lk with a full cut rule written in tree form 1 6 it is also known that lk with a full cut rule in tree form p simulates lk in dag form in this paper we show that lk in tree form with a cut rule where the complexity of cut formulas is bounded by k 1 has an exponential speed up over the one bounded by k we also show that lk in tree form with a bounded complexity cut rule does not polynomially simulate cut free lk in dag form

reasoning about temporal relations a maximal tractable subclass of allen s interval algebra we introduce a new subclass of allen s interval algebra we call ord horn subclass which is a strict superset of the pointisable subclass we prove that reasoning in the ord horn subclass is a polynomial time problem and show that the path consistency method is sufficient for deciding satisfiability further using an extensive machine generated case analysis we show that the ord horn subclass is a maximal tractable subclass of the full algebra assuming p np in fact it is the unique greatest tractable subclass amongst the subclasses that contain all basic relations

two theses of knowledge representation language restrictions taxonomic classification and the utility of representation services abstract levesque and brachman argue that in order to provide timely and correct responses in the most critical applications general purpose knowledge representation systems should restrict their languages by omitting constructs which require nonpolynomial worst case response times for sound and complete classification they also separate terminological and assertional knowledge and restrict classification to purely terminological information we demonstrate that restricting the terminological language and classifier in these ways limits these general purpose facilities so severely that they are no longer generally applicable we argue that logical soundness completeness and worst case complexity are inadequate measures for evaluating the utility of representation services and that this evaluation should employ the broader notions of utility and rationality found in decision theory we suggest that general purpose representation services should provide fully expressive languages classification over relevant contingent information approximate forms of classification involving defaults and rational management of inference tools

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem