completion of a set of rules modulo a set of equations abstract church rosser properties are first presented depending on an arbitrary relation r an equivalence relation e and a reduction relation r e used to compute normal forms of r modulo e terminating rewriting systems operating on equational congruence classes of terms of a free algebra are then considered in this framework the church rosser property is proved decidable for a very general reduction relation which may take into account the left linearity of rules for efficiency reasons under the only assumption of existence of a complete and finite unification algorithm for the underlying equational theory whose congruence classes are assumed to be finite this extends previous results by lankford and ballantyne peterson and stickel huet jouannaud a general completion procedure for mixed sets of rules and equations is then presented that generalizes and improves peterson and stickel s one in addition to computing a church rosser set of rules when it terminates it yields a semi decision pro

incremental constraint satisfaction for equational logic programming abstract in this paper we are concerned with an instance of the constraint logic programming clp scheme specialized in solving equations with respect to a horn equational theory e the intended structure h e is given by the finest partition induced by e on the herbrand universe h over a finite one sorted alphabet this work deals with the description of an incremental constraint solver as the kernel of an operational semantics for the language clp h e the primary issues are how to verify the solvability of constraints in the structure h e by using some sound and complete semantic unification procedure such as narrowing how to simplify constraints in a computation sequence how to achieve incrementality in the computation process and how to profit from finitely failed derivations as a heuristic for optimizing the algorithms

conditional rewrite rules abstract the notion of conditional term rewriting systems that allows to simulate theories specified by conditional equations is presented the paper considers general conditional rules as opposed to related works usually dealing with restricted cases means for the investigation of conditional rewriting i e iterative constructs are proposed evaluation by rewriting is shown to be correct with respect to the underlying algebraic theory a general evaluation procedure is given intractability results about one step rewriting even for canonical systems and computation of the normal form function are established

on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

rewriting systems over similarity and generalized pseudometric spaces and their properties we present a study of confluence and related properties of fuzzy relations defined over similarity spaces the ordinary confluence is an essential property of relations connected to the idea of rewriting and substituting which appears in abstract rewriting systems this paper is a continuation of our previous paper where we have introduced analogous notions related to substitutability in graded setting using residuated lattices as structures of truth degrees leaving the ordinary notions a particular case when the underlying structure is the two valued boolean algebra in this paper we further extend our previous results by developing the notions of confluence and related properties respecting a given similarity relation we also present definitions of confluence and related properties of relations on a generalized pseudometric spaces using the well known link between generalized pseudometrics and similarities we describe the connection of the notions defined on generalized pseudometric spaces to the corresponding notions on similarity spaces the introduced notions are further illustrated by example of rewriting based on graded if then rules

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

positivity of the bondi energy we give a new proof that the bondi momentum is time like and future directed this is done by extending a theorem showing the existence of solutions of witten s equation on a complete asymptotically flat surface to the case where the surface is asyptotically hyperbolic and spans a cut s of i by choosing appropriate boundary conditions the positivity of the bondi energy is also shown for the case where the surface is no longer free of inner boundaries but is internally bounded by the union of a finite number of marginally trapped surfaces this argument also applies to the case of asymptotically flat surfaces

on termination and confluence properties of disjoint and constructor sharing conditional rewrite systems abstract we investigate the modularity behavior of termination and confluence properties of join conditional term rewriting systems we give counterexamples showing that the properties weak termination weak innermost termination and strong innermost termination are not preserved in general under signature extensions then we develop sufficient conditions for the preservation of these properties under signature extensions and more generally for their modularity in the disjoint union case this leads to new criteria for modularity of termination and completeness generalizing known results for unconditional systems finally combining our analysis with recent related results on the preservation of semi completeness we show how to cover the non disjoint case of combined conditional rewrite systems with shared constructors too

order sorted algebra i equational deduction for multiple inheritance overloading exceptions and partial operations abstract this paper generalizes many sorted algebra msa to order sorted algebra osa by allowing a partial ordering relation on the set of sorts this supports abstract data types with multiple inheritance in roughly the sense of object oriented programming several forms of polymorphism and overloading partial operations as total on equationally defined subsorts exception handling and an operational semantics based on term rewriting we give the basic algebraic constructions for osa including quotient image product and term algebra and we prove their basic properties including quotient homomorphism and initiality theorems the paper s major mathematical results include a notion of osa deduction a completeness theorem for it and an osa birkhoff variety theorem we also develop conditional osa including initiality completeness and mckinsey malcev quasivariety theorems and we reduce osa to conditional msa which allows lifting many known msa results to osa retracts which intuitively are left inverses to subsort inclusions provide relatively inexpensive run time error handling we show that it is safe to add retracts to any osa signature in the sense that it gives rise to a conservative extension a final section compares and contrasts many different approaches to osa this paper also includes several examples demonstrating the flexibility and applicability of osa including some standard benchmarks like stack and list as well as a much more substantial example the number hierarchy from the naturals up to the quaternions

elan from a rewriting logic point of view elan implements computational systems a concept that combines two first class entities rewrite rules and rewriting strategies elan can be used either as a logical framework or to describe and execute deterministic as well as non deterministic rule based processes with the general goal to make precise a rewriting logic based semantics of elan this paper has three contributions a presentation of the concepts of rules and strategies available in elan an expression of rewrite rules with matching conditions in conditional rewriting logic and finally an enrichment mechanism of a rewrite theory into a strategy theory in conditional rewriting logic

nakayama automorphisms of frobenius algebras abstract we show that the nakayama automorphism of a frobenius algebra r over a field k is independent of the field theorem 4 consequently the k dual functor on left r modules and the bimodule isomorphism type of the k dual of r and hence the question of whether r is a symmetric k algebra are independent of k we give a purely ring theoretic condition that is necessary and sufficient for a finite dimensional algebra over an infinite field to be a symmetric algebra theorem 7

a mechanical solution of schubert s steamroller by many sorted resolution abstract we demonstrate the advantage of using a many sorted resolution calculus by a mechanical solution of a challenge problem this problem known as schubert s steamroller had been unsolved by automated theorem provers before our solution clearly demonstrates the power of a many sorted resolution calculus the proposed method is applicable to all resolution based inference systems

levels of undecidability in rewriting undecidability of various properties of first order term rewriting systems is well known an undecidable property can be classified by the complexity of the formula defining it this classification gives rise to a hierarchy of distinct levels of undecidability starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy where quantification over function variables is allowed in this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies we consider properties related to normalization strong normalization weak normalization and dependency problems and properties related to confluence confluence local confluence and the unique normal form property for all of these we distinguish between the single term version and the uniform version where appropriate we also distinguish between ground and open terms most uniform properties are p 2 0 complete the particular problem of local confluence turns out to be p 2 0 complete for ground terms but only p 2 0 complete and thereby recursively enumerable for open terms the most surprising result concerns dependency pair problems without minimality flag we prove this problem to be p 1 1 complete hence not in the arithmetical hierarchy but properly in the analytic hierarchy some of our results are new or have appeared in our earlier publications others are based on folklore constructions and are included for completeness as their precise classifications have hardly been noticed previously

deformations of the hemisphere that increase scalar curvature consider a compact riemannian manifold m of dimension n whose boundary m is totally geodesic and is isometric to the standard sphere s n 1 a natural conjecture of min oo asserts that if the scalar curvature of m is at least n n 1 then m is isometric to the hemisphere s n equipped with its standard metric this conjecture is inspired by the positive mass theorem in general relativity and has been verified in many special cases in this paper we construct counterexamples to min oo s conjecture in dimension n 3

equational rules for rewriting logic in addition to equations and rules we introduce equational rules that are oriented while having an equational interpretation correspondence between operational behavior and intended semantics is guaranteed by a property of coherence which can be checked by examination of critical pairs and linearity conditions we present applications of this theory to three examples where the rewrite relation is interpreted respectively as equality transition and deduction

strict coherence of conditional rewriting modulo axioms conditional rewriting modulo axioms with rich types makes specifications and declarative programs very expressive and succinct and is used in all well known rule based languages however the current foundations of rewriting modulo axioms have focused for the most part on the unconditional and untyped case the main purpose of this work is to generalize the foundations of rewriting modulo axioms to the conditional order sorted case a related goal is to simplify such foundations in particular even in the unconditional case the notion of strict coherence proposed here makes rewriting modulo axioms simpler and easier to understand properties of strictly coherent conditional theories like operational equi termination of the r b and r b relations and general conditions for the conditional churchrosser property modulo b are also studied

specification and proof in membership equational logic this paper is part of a long term effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic basis on which efficient execution by rewriting and powerful theorem proving tools can be based in particular our rewriting techniques provide semantic foundations for maude s functional sublanguage where they have been efficiently implemented

a definition of total energy momenta and the positive mass theorem on asymptotically hyperbolic 3 manifolds i two sets of asymptotically hyperbolic initial data are defined which correspond to the spatial infinity in asymptotically ads spacetimes and to the null infinity in asymptotically minkowski spacetimes respectively the positive mass theorem involving the total energy the total linear momentum and the total angular momentum is established for these initial data sets

semantic foundations for generalized rewrite theories rewriting logic rl is a logic of actions whose models are concurrent systems rewrite theories involve the specification of equational theories of data and state structures together with a set of rewrite rules that model the dynamics of concurrent systems since its introduction more than one decade ago rl has attracted the interest of both theorists and practitioners who have contributed in showing its generality as a semantic and logical framework and also as a programming paradigm the experimentation conducted in these years has suggested that some significant extensions to the original definition of the logic would be very useful in practice these extensions may develop along several dimensions like the choice of the underlying equational logic the kind of side conditions allowed in rewrite rules and operational concerns for the execution of certain rewrites in particular the maude system now supports subsorting and conditional sentences in the equational logic for data and also frozen arguments to block undesired nested rewrites moreover it allows equality and membership assertions in rule conditions in this paper we give a detailed presentation of the inference rules model theory and completeness of such generalized rewrite theories our results provide a mathematical semantics for maude and a foundation for formal reasoning about maude specifications

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

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

function definitions in term rewriting and applicative programming the frameworks of unconditional and conditional term rewriting and applicative systems are explored with the objective of using them for defining functions in particular a new operational semantics tue reduction is elaborated for conditional term rewriting systems for each framework the concept of evaluation of terms invoking defined functions is formalized we then discuss how it may be ensured that a function definition in each of these frameworks is meaningful by defining restrictions that may be imposed to guarantee termination unambiguity and completeness of definition the three frameworks are then compared studying when a definition may be translated from one formalism to another

bilinear forms on frobenius algebras abstract we analyze the homothety types of associative bilinear forms that can occur on a hopf algebra or on a local frobenius k algebra r with residue field k if r is symmetric then there exists a unique form on r up to homothety iff r is commutative if r is frobenius then we introduce a norm based on the nakayama automorphism of r we show that if two forms on r are homothetic then the norm of the unit separating them is central and we conjecture the converse we show that if the dimension of r is even then the determinant of a form on r taken in k k 2 is an invariant for r

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

real killing spinors and holonomy we give a description of all complete simply connected riemannian manifolds carrying real killing spinors furthermore we present a construction method for manifolds with the exceptional holonomy groupsg2 and spin 7

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

a family of asymptotically hyperbolic manifolds with arbitrary energy momentum vectors a family of non radial solutions to the yamabe equation modeled on the hyperbolic space is constructed using power series as a result we obtain a family of asymptotically hyperbolic metrics with spherical conformal infinity with scalar curvature greater than or equal to n n 1 but which are a priori not complete moreover any vector of rn 1 is performed by an energy momentun vector of one suitable metric of this family they can in particular provide counter examples to the positive energy momentum theorem when one removes the completeness assumption

scalar curvature rigidity of asymptotically hyperbolic spin manifolds in this paper we study the hyperbolic version of this result using witten s method and show that a spin manifold which is strongly asymptotically hyperbolic cannot have scalar curvature r n n 1 everywhere unless it is isometric to hyperbolic space

on maximal surfaces in asymptotically flat space times existenc of maximal and almost maximal hypersurfaces in asymptotically flat space times is established under boundary conditions weaker than those considered previously we show in particular that every vacuum evolution of asymptotically flat data for the einstein equations can be foliated by slices maximal outside a spatially compact set and that every strictly stationary asymptotically flat space time can be foliated by maximal hypersurfaces amongst other uniqueness results we show that maximal hypersurfaces can be used to partially fix an asymptotic poincare group

conditional equational theories and complete sets of transformations abstract the idea of combining the advantages of function and logic programming has attracted many researchers their work ranges from the integration of existing languages over higher order logic to equational logic languages where logic programs are augmented with equational theories recently it has been proposed to handle those equational theories by complete sets of transformations these transformations are extensions of the rules introduced by herbrand and later used by martelli and montanari to compute the most general unifier of two expressions we generalize this idea to complete sets of transformations for arbitrary conditional equational theories the largest class of equational theories that admit a least herbrand model the completeness proof is based on the observation that each refutation with respect to linear paramodulation and reflection can be modelled by the transformations as certain conditions imposed on an equational theory restrict the search space generated by paramodulation and reflection we can easily refine our transformations due to the completeness proof if the conditional equational theory is ground confluent or canonical

on completeness of narrowing strategies abstract we address the problem of unification modulo a set of equations using the narrowing relation we propose some syntactical criteria on algebraic specifications that ensure the completeness of narrowing strategies we then prove a theorem relating narrowing and reduction relations the completeness of narrowing strategies is proved and conditions for the computation of a minimal ground complete set of e unifiers are given as well as an algorithm transforming specifications satisfying huet and hullot s principle of definition into specifications fulfilling the proposed criteria

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

a mass for alf manifolds we prove a positive mass theorem on alf manifolds i e complete noncompact manifolds that are asymptotic to a circle fibration over a euclidean base with fibers of asymptotically constant length

the first eigenvalue of the dirac operator on quaternionic kahler manifolds in 15 we proved a lower bound for the spectrum of the dirac operator on quaternionic kahler manifolds in the present article we show that the only manifolds in the limit case i e the only manifolds where the lower bound is attained as an eigenvalue are the quaternionic projective spaces we use the equivalent formulation in terms of the quaternionic killing equation introduced in 16 and show that a nontrivial solution defines a parallel spinor on the associated hyperkahler manifold

scalar curvature rigidity of hyperbolic product manifolds this paper presents a scalar curvature rigidity result of real hyperbolic product manifolds in analogy to m min oo s result in 14 in order to prove this we consider dirac bundles obtained from the spinor bundle and we derive killing equations trivializing these dirac bundles moreover an integrated bochner weitzenbock formula is shown which allows the usage of the non compact bochner technique

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

the dirac witten operator on pseudo riemannian manifolds we study the dirac witten operator for spacelike spin submanifolds in pseudo riemannian manifolds when the normal bundles are spin and odd dimensional we derive new eigenvalue lower bounds we also prove the generalized positive mass theorem using this operator

conditional rewriting logic as a unified model of concurrency abstract rewriting with conditional rewrite rules modulo a set e of structural axioms provides a general framework for unifying a wide variety of models of concurrency concurrent rewriting coincides with logical deduction in conditional rewriting logic a logic of actions whose models are concurrent systems this logic is sound and complete and has initial models in addition to general models interpreted as concurrent systems which provide a more operational style of semantics more restricted semantics with an incresingly denotational flavor such as preorder poset cpo and standard algebraic models appear as special cases of the model theory this permits dealing with operational and denotational issues within the same model theory and logic a programming language called maude whose modules are rewriting logic theories is defined and given denotational and operational semantics maude provides a simple unification of concurrent programming with functional and object oriented programming and supports high level declarative programming of concurrent systems

abstract data type systems abstract this paper is concerned with the foundations of an extension of pure type systems by abstract data types hence the name of abstract data type systems adts generalize inductive types as they are defined in the calculus of constructions by providing definitions of functions by pattern matching on the one hand and relations among constructors of the inductive type on the other it also generalizes the first order framework of abstract data types by providing function types and higher order equations the first half of the paper describes the framework of adts while the second half investigates cases where adts are strongly normalizing this is shown to be the case for the polymorphic lambda calculus with possibly subtypes enriched by higher order algebraic rules obeying a strong generalization of primitive recursion of higher type that we call the general schema this covers in particular the case of inductive types whose constructors do not have functional arguments we conjecture that this result holds true for all calculi of the so called barendregt s cube on the other hand the definition of a schema for the higher order rules allowing for more general inductive types is left open

scalar curvature and asymptotic symmetric spaces a well known fact is that a complete riemannian spin manifold which is strongly asymptotically flat and has nonnegative scalar curvature must be isometric to the euclidean space using witten s positive mass argument this paper proves the analogous rigidity result for certain asymptotic symmetric spaces

a rationale for conditional equational programming abstract conditional equations provide a paradigm of computation that combines the clean syntax and semantics of lisp like functional programming with prolog like logic programming in a uniform manner for functional programming equations are used as rules for left to right rewriting for logic programming the same rules are used for conditional narrowing together rewriting and narrowing provide increased expressive power we discuss some aspects of the theory of conditional rewriting and the reasons underlying certain choices in designing a language based on them the most important correctness property a conditional rewriting program may posses is ground confluence this ensures that at most one value can be computed from any given variable free input term we give criteria for confluence reasonable conditions for ensuring the completeness of narrowing as an operational mechanism for solving goals are provided these results are then extended to handle rewriting with existentially quantified conditions and built in predicates some termination issues are also considered including the case of rewriting with higher order terms

skew group algebras of jacobian algebras for a quiver with potential q w with an action of a finite cyclic group g we study the skew group algebra lambda g of the jacobian algebra lambda p q w by a result of reiten and riedtmann

chern s magic form and the gauss bonnet chern mass in this note we use chern s magic form phi k in his famous proof of the gauss bonnet theorem to define a mass for asymptotically flat manifolds it turns out that the new defined mass is equivalent to the one that we introduced recently by using the gauss bonnet chern curvature l k moreover this equivalence implies a simple proof of the equivalence between the adm mass and the intrinsically defined mass via the ricci tensor which was reconsidered by miao tam proc am math soc 144 753 761 2016 and herzlich ann henri poincare 17 12 3605 3617 2016 very recently

on the confluence of lambda calculus with conditional rewriting the confluence of untyped l calculus with unconditional rewriting is now well un derstood in this paper we investigate the confluence of l calculus with conditional rewriting and provide general results in two directions first when conditional rules are algebraic this extends results of muller and dougherty for unconditional rewriting two cases are considered whether b reduction is allowed or not in the evaluation of conditions moreover dougherty s result is improved from the assumption of strongly normalizing b reduction to weakly normalizing b reduction we also provide examples showing that outside these conditions modularity of confluence is difficult to achieve second we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules

logicality of conditional rewrite systems a conditional term rewriting system is called logical if it has the same logical strength as the underlying conditional equational system in this paper we summarize known logicality results and we present new sufficient conditions for logicality of the important class of oriented conditional term rewriting systems

metriques d einstein a cusps et equations de seiberg witten if m4 g0 is a finite volume quotient of the complex hyperbolic space we prove that any einstein complete bounded curvature metric g on m such that the diameter of horocycles goes to zero and the mean curvature of the horocycles is bounded from below by a positive constant is equal up to a diffeomorphism to the standard complex hyperbolic metric g0 this generalizes a theorem of lebrun in the compact case to prove the theorem we produce a solution of the seiberg witten equations on the noncompact manifold m g in fact m can be compactified as an orbifold m and we get the wanted solution as a limit of solutions for a sequence of metrics on m which approximate g

perturbative solutions of the extended constraint equations in general relativity the extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface mathcal z in an asymptotically simple space time satisfying the vacuum conformal einstein equations developed by h friedrich the extended constraint equations consist of a quasi linear system of partial differential equations for the induced metric the second fundamental form and two other tensorial quantities defined on mathcal z and are equivalent to the usual constraint equations that mathcal z satisfies as a space like hypersurface in a space time satisfying einstein s vacuum equation this article develops a method for finding perturbative asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on euclidean space this method is fundamentally different from the classical method of lichnerowicz and york that is used to solve the usual constraint equations