unification algorithms cannot be combined in polynomial time we establish that there is no polynomial time general combination algorithm for unification in finitary equational theories unless the complexity class p of counting problems is contained in the class fp of function problems solvable in polynomial time the prevalent view in complexity theory is that such a collapse is extremely unlikely for a number of reasons including the fact that the containment of p in fp implies that p np our main result is obtained by establishing the intractability of the counting problem for general ag unification where ag is the equational theory of abelian groups more specifically we show that computing the cardinality of a minimal complete set of unifiers for general ag unification is a p hard problem in contrast ag unification with constants is known to be solvable in polynomial time since an algorithm for general ag unification can be obtained as a combination of a polynomial time algorithm for ag unification with constants and a polynomial time algorithm for syntactic unification it follows that no polynomial time general combination algorithm exists unless p is contained in fp this implication of our main results holds not only for the combination of unification algorithms but also for the combination of constraint solvers as well we also show that the counting problem for boolean ring unification is p hard this gives a lower bound on the performance of all algorithms for general br unification

a heuristic method to rectify intransitive judgments in pairwise comparison matrices abstract this paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a monte carlo simulation which confirms that relatively high percentage of comparison matrices satisfying saaty s cr criterion are ordinally inconsistent it is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process in contrast with cardinal inconsistency a heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices the proposed algorithm generates near optimal solutions and outperforms other tested approaches with respect to computation time

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

the ziegler spectrum for derived discrete algebras let be a derived discrete algebra we show that the krull gabriel dimension of the homotopy category of projective modules and therefore the cantor bendixson rank of its ziegler spectrum is 2 thus extending a result of bobinski and krause 8 we also describe all the indecomposable pure injective complexes and hence the ziegler spectrum for derived discrete algebras extending a result of z han 17 using this we are able to prove that all indecomposable complexes in the homotopy category of projective modules are pure injective so obtaining a class of algebras for which every indecomposable complex is pure injective but which are not derived pure semisimple

on solving equations and disequations we are interested in the problem of solving a system sl tl 1 i n pj qj 1 j m of equations and disequations also known as disunification solutions to disunification problems are substitutions for the variables of the problem that make the two terms of each equation equal but leave those of the disequations different we investigate this in both algebraic and logical contexts where equality is defined by an equational theory and more generally by a definitive clause equality theory e we show how e disunification can be reduced to e unification that is solving equations only and give a disunification algorithm for theories given a unification algorithm in fact this result shows that for theories in which the solutions of all unification problems can also be represented finitely we sketch how disunification can be applied to handle negation in logic programming with equality in a similar style to colmerauer s logic programming with rational trees and to represent many solutions to ac unification problems by a few solutions to aci disunification problems

on the complexity of recognizing the hilbert basis of a linear diophantine system the problem of computing the hilbert basis of a homogeneous linear diophantine system over nonnegative integers is often considered in automated deduction and integer programming in automated deduction the hilbert basis of a corresponding system serves to compute the minimal complete set of associative commutative unifiers whereas in integer programming the hilbert bases are tightly connected to integer polyhedra and to the notion of total dual integrality in this paper we sharpen the previously known result that the problem asking whether a given solution belongs to the hilbert basis of a given system is conp complete we show that the problem has a pseudopolynomial algorithm if the number of equations in the system is fixed but it is conp complete in the strong sense if the given system is unbounded this result is important in the scope of automated deduction where the input is given in unary and therefore the previously known conp completeness result was unusable moreover we show that from the complexity standpoint it is not important to know the underlying homogeneous linear diophantine system when we ask whether a given set of vectors constitutes a hilbert basis

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

combination of constraint solvers for free and quasi free structures abstract when combining languages for symbolic constraints one is typically faced with the problem of how to treat mixed constraints the two main problems are 1 how to define a combined solution structure over which these constraints are to be solved and 2 how to combine the constraint solving methods for pure constraints into one for mixed constraints the paper introduces the notion of a free amalgamated product as a possible solution to the first problem we define so called quasi free structures called strong simply combinable structures in a previous publication as a generalization of free structures for quasi free structures over disjoint signatures we describe a canonical amalgamation construction that yields the free amalgamated product the combination techniques known from unification theory can be used to combine constraint solvers for quasi free structures over disjoint signatures into a solver for their free amalgamated product in addition to term algebras modulo equational theories i e free algebras the class of quasi free structures contains many solution structures that are of interest in constraint logic programming such as the algebra of rational trees feature structures and domains consisting of hereditarily finite well founded or non well founded nested sets and lists

a geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices abstract the usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria in a multiple objective decision problem through binary comparison matrices or pairwise judgment matrices is receiving considerable attention the methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices the eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale a difficulty that is often experienced in using this approach in practice is the inconsistency of the binary evaluations this paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix where inconsistency is a problem it is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix the procedure is illustrated by means of examples

unification in commutative idempotent monoids abstract embedding sets as a datastructure into resolution based deduction requires a unification algorithm respecting associativity commutativity and idempotence of the constructor function union we shall present such an algorithm and prove its correctness and termination although this algorithm is very simple the enormous proliferation of unifiers restricts its value to theoretical considerations for practical purposes the more special unitary theory of boolean expressions seem to be better suited

alternating two way ac tree automata we explore the notion of alternating two way tree automata modulo the theory of finitely many associative commutative ac symbols this was prompted by questions arising in cryptographic protocol verification in particular in modeling group key agreement schemes based on diffie hellman like functions where the emptiness question for intersections of such automata is fundamental this also has independent interest we show that the use of general push clauses or of alternation leads to undecidability already in the case of one ac symbol with only functions of arity zero on the other hand emptiness is decidable in the general case of several function symbols including several ac symbols provided push clauses are unconditional and intersection clauses are final this class of automata is also shown to be closed under intersection

smashing subcategories and the telescope conjecture an algebraic approach we prove a modified version of ravenel s telescope conjecture it is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra this result is based on a new characterization of smashing subcategories which leads in addition to a classification of these subcategories in terms of the category of finite spectra the approach presented here is purely algebraic it is based on an analysis of pure injective objects in a compactly generated triangulated category and covers therefore also situations arising in algebraic geometry and representation theory

on the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices abstract several measures have been proposed in the literature to verify the rationality and consistency of judgements expressed by means of pairwise comparisons however some of these indices lack a meaningful interpretation because of the absence of thresholds associated with them in this paper we first introduce the consistency approximated thresholds for the salo hamalainen index by deriving an inconsistency tolerance level which is analogous to that proposed for the consistency ratio by saaty for the geometric consistency index by aguaron moreno jimenez and for the consistency measure by koczkodaj then via a simulation we propose a transitivity threshold for n n matrices which may give meaningful information about the degree of misclassification and the reliability of preferences while avoiding the need to revise the judgements on varying n we derive the transitivity thresholds for salo hamalainen and for the other consistency indices proposed in literature

computing random consistency indices and assessing priority vectors reliability abstract the paper deals with two crucial steps in multi criteria decision analysis that are consistency of the judgments and priority vectors for alternatives criteria from a side several consistency indices are proposed for measuring the consistency of a pairwise comparison matrix from another one conditions weaker than consistency such as transitivity and weak consistency are proposed for representing further levels of coherence of a decision maker when he she expresses his her preferences by means of reciprocal pairwise comparison matrix firstly in this paper a simulation is performed in order to establish a relation between random consistency index and coherence level then since weak consistency ensures reliability to priority vectors proposed in literature a second simulation is performed in order to measure in case of no weak consistency the reliability of these priority vectors

decision making with the ahp why is the principal eigenvector necessary in this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix a aij when a is a small perturbation of a consistent matrix when providing numerical judgments an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios near consistent matrices are essential because when dealing with intangibles human judgment is of necessity inconsistent and if with new information one is able to improve inconsistency to near consistency then that could improve the validity of the priorities of a decision in addition judgment is much more sensitive and responsive to large rather than to small perturbations and hence once near consistency is attained it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one if such perturbations were forced they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision

derived categories of nodal algebras abstract in this article we classify indecomposable objects of the derived categories of finitely generated modules over certain infinite dimensional algebras the considered class of algebras which we call nodal algebras contains such well known algebras as the complete ring of a double nodal point k x y xy and the completed path algebra of the gelfand quiver as a corollary we obtain a description of the derived category of harish chandra modules over sl 2 r we also give an algorithm which allows to construct projective resolutions of indecomposable complexes in the appendix we prove the krull schmidt theorem for homotopy categories

type dependencies for logic programs using aci unification abstract this paper presents a new notion of typing for logic programs which generalizes the notion of directional types the generation of type dependencies for a logic program is fully automatic with respect to a given domain of types the analysis method is based on a novel combination of program abstraction and aci unification which is shown to be correct and optimal type dependencies are obtained by abstracting programs replacing concrete terms by their types and evaluating the meaning of the abstract programs using a standard semantics for logic programs enhanced by aci unification this approach is generic and can be used with any standard semantics the method is both theoretically clean and easy to implement using general purpose tools the proposed domain of types is condensing which means that analyses can be carried out in both top down or bottom up frameworks with no loss of precision for goal independent analyses the proposed method has been fully implemented within a bottom up approach and the experimental results are promising

completion for unification abstract syntactic theories have the nice property that a unification algorithm may be computed directly from the form of the axioms of a specific presentation called resolvent of the theory in this work we present and prove a completion algorithm that for a given presentation returns a resolvent set of axioms whenever it terminates

semantic confluence tests and completion methods we present semantic methods for showing that a term rewriting system is confluent we also present methods for completing a given term rewriting system to obtain an equivalent confluent system these methods differ from the well known and widely studied knuth bendix method in that they emphasize semantics rather than syntax also they often require more user interaction than the purely syntactic knuth bendix method the concept of ground confluence is discussed methods for demonstrating ground confluence are also given we give decision procedures for some sub problems that arise in this method

refutational theorem proving using term rewriting systems abstract in this paper we propose a new approach to theorem proving in first order logic based on the term rewriting method first for propositional calculus we introduce a canonical term rewriting system for boolean algebra this system enables us to transform the first order predicate calculus into a form of equational logic and to develop several complete strategies both clausal and nonclausal for first order theories based on the knuth bendix completion procedure more importantly our strategies can deal with predicate logic and built in equational theories in a uniform and effective way we also describe an implementation and comparisons with some other first order theorem proving methods

towards a foundation of completion procedures as semidecision procedures completion procedures originated from the seminal work of knuth and bendix are well known as procedures for generating confluent rewrite systems i e decision procedures for equational theories in this paper we present a new abstract framework for the utilization of completion procedures as semidecision procedures for theorem proving the key idea in our approach in that a semidecision process should be target oriented i e keep into account the target theorem to be proved for the inference rules of a completion procedure we present target oriented schemes of contraction inference rules i e inference rules that delete sentences which are redundant for proving the target for the search plan we give a target oriented definition offairness according to which not all critical pairs need to be considered we prove that our notion of fairness together with the refutational completeness of the inference rules is sufficient for a completion procedure to be a semidecision procedure by relaxing the requirement of considering all critical pairs our target oriented framework should be more suitable for designing efficient procedures for theorem proving the generation of decision procedures is included as a special side effect and all the results of the classical approach to completion are re obtained in our framework the application of completion to disprove inductive conjectures i e the so called inductionless induction method is also covered as a semidecision process finally we present according to our framework some equational completion procedures based on unfailing knuth bendix completion

the homotopy category of complexes of projective modules abstract the homotopy category of complexes of projective left modules over any reasonably nice ring is proved to be a compactly generated triangulated category and a duality is given between its subcategory of compact objects and the finite derived category of right modules

computing with rewrite systems term rewriting systems that is sets of directed equations provide a paradigm of computation with particularly simple syntax and semantics rewrite systems may be used for straightforward computation by simplifying terms we show how in addition restricted forms of the knuth bendix completion procedure may be used to interpret logic programs written as a set of equivalence preserving rewrite rules we discuss verification issues and also illustrate the use of the full completion procedure to synthesize rewrite programs from specifications

ramification and causality abstract in formal systems for reasoning about actions the ramification problem denotes the problem of handling indirect effects these effects are not explicitly represented in action specifications but follow from general laws describing dependencies among components of the world state an adequate treatment of indirect effects requires a suitably weakened version of the general law of persistence it also requires a method to avoid unintuitive changes suggested by the aforementioned dependency laws we propose a solution to the ramification problem that uses directed relations between two single effects stating the circumstances under which the occurrence of the first causes the second we argue for the necessity of an approach based on causality by elaborating the limitations of common paradigms employed to handle ramifications the principle of categorization and the policy of minimal change our abstract solution is realized on the basis of a particular action calculus namely the fluent calculus

a superposition oriented theorem prover abstract this paper describes a theorem proving procedure which combines the approach of resolution with that of rewriting the basic theoretical result is the completeness of a strong restriction of paramodulation for locking resolution procedures its oriented character suggests to consider the restricted paramodulation as a form of superposition the rewriting operation this is achieved by means of a new formalism of clauses named equational clauses in which each literal is converted into an equation thereby superposition on equational clauses is shown to embody not only paramodulation but also binary resolution so clausal superposition will build up our major rule of inference in addition term simplification is incorporated in our procedure as well as subsumption experimental results and potential applications for our theorem prover are lastly reported

singular value decomposition in ahp abstract the analytic hierarchy process ahp has been accepted as a leading multiattribute decision aiding model both by practitioners and academics the foundation of the ahp is the saaty s eigenvector method em and associated inconsistency index that are based on the largest eigenvalue and associated eigenvector of an n n positive reciprocal matrix the elements of the matrix are the decision maker s dm numerical estimates of the preference of n alternatives with respect to a criterion when they are compared pairwise using the 1 9 ahp fundamental comparison scale the components of the normalized eigenvector provide approximations of the unknown weights of the criteria alternatives and the deviation of the largest eigenvector from n yields a measure of how inconsistent the dm is with respect to the pairwise comparisons singular value decomposition svd is an important tool of matrix algebra that has been applied to a number of areas e g principal component analysis canonical correlation in statistics the determination of the moore penrose generalized inverse and low rank approximation of matrices in this paper using the svd and the theory of low rank approximation of a pairwise comparison matrix we offer a new approach for determining the associated weights we prove that the rank one left and right singular vectors that is the vectors associated with the largest singular value yield theoretically justified weights we suggest that an inconsistency measure for these weights is the frobenius norm of the difference between the original pairwise comparison matrix and one formed by the svd determined weights how this measure can be applied in practice as a means of measuring the confidence the dm should place in the svd weights is still an open question we illustrate the svd approach and compare it to the em for some numerical examples

the algebras with discrete derived category abstract we classify the finite dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non isomorphic indecomposable complexes

the harmonic consistency index for the analytic hierarchy process a new consistency measure the harmonic consistency index is obtained for any positive reciprocal matrix in the analytic hierarchy process we show how this index varies with changes in any matrix element a tight upper bound is provided for this new consistency measure when the entries of matrix are at most 9 as is often recommended using simulation the harmonic consistency index is shown to give numerical values similar to the standard consistency index but it is easier to compute and interpret in addition new properties of the column sums of reciprocal matrices are obtained

contribution of individual judgments toward inconsistency in pairwise comparisons pairwise comparison pc is a well established method to assist decision makers in estimating their preferences in pcs the acquired judgments are used to construct a pc matrix pcm that is used to check whether the inconsistency in judgments is acceptable or requires revision the use of consistency ratio cr a widely used measure for inconsistency has been widely debated and the literature survey has identified a need for a more appropriate measure considering this need a new measure termed congruence is proposed in this paper the measure is shown to be useful in finding the contribution of individual judgments toward overall inconsistency of a pcm and therefore can be used to detect and correct cardinally inconsistent judgments the proposed measure is applicable to incomplete sets of pc judgments without modification unlike cr which requires a complete set of pc judgments to address ordinal inconsistency another measure termed dissonance is proposed as a supplement to the congruence measure the two measures appear useful in detecting both outliers and the phenomenon of consistency deadlock where all judgments equally contribute toward the overall inconsistency

the homotopy category of flat modules and grothendieck duality let r be a ring we prove that the homotopy category k r proj is always aleph 1 compactly generated and depending on the ring r it may or may not be compactly generated we use this to give a description of k r proj as a quotient of k r flat the remarkable fact is that this new description of k r proj generalizes to non affine schemes this will appear in murfet s thesis

on the parameterized complexity of associative and commutative unification this article studies the parameterized complexity of the unification problem with associative commutative or associative commutative functions with respect to the parameter number of variables it is shown that if every variable occurs only once then both of the associative and associative commutative unification problems can be solved in polynomial time but that in the general case both problems are w 1 hard even when one of the two input terms is variable free for commutative unification an algorithm whose time complexity depends exponentially on the number of variables is presented moreover if a certain conjecture is true then the special case where one input term is variable free belongs to fpt some related results are also derived for a natural generalization of the classic string and tree edit distance problems that allows variables

morphisms between indecomposable complexes in the bounded derived category of a gentle algebra in this article we provide a simple combinatorial description of morphisms between indecomposable complexes in the bounded derived category of a gentle algebra

a unification algorithm for associative commutative functions an important component of automated theorem proving systems are unification algorithms which fred most general substitutions which when apphed to two expressions make them equivalent functions which are associative and commutative such as the arithmetic addition and multiphcatton functions are often the subject of automated theorem proving an algorithm which unifies terms whose function is associative and commutauve is presented here termmaaon soundness and completeness of the algorithm have been proved for a subclass of the general case the algorithm is an efficient alternative to other methods of handling associative commutative functions

discrete derived categories i homomorphisms autoequivalences and t structures discrete derived categories were studied initially by vossieck j algebra 243 168 176 2001 and later by bobinski et al cent eur j math 2 19 49 2004 in this article we describe the homomorphism hammocks and autoequivalences on these categories we classify silting objects and bounded t structures

permutative rewriting and unification permutative rewriting provides a way of analyzing deduction modulo a theory defined by leaf permutative equations our analysis naturally leads to the definition of the class of unify stable axiom sets in order to enforce a simple reduction strategy we then give a uniform unification algorithm modulo theories e axiomatized this way we prove that it computes complete sets of unifiers of simply exponential cardinality and that the e unification decision problem belongs to np