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

bottom up tree pushdown automata classification and connection with rewrite systems we define different types of bottom up tree pushdown automata and study their connections with rewrite systems along this line of research we complete and generalize the results of gallier book and salomaa we define the notion of a tail reduction free trf rewrite system using the decidability of ground reducibility we prove the decidability of the trf property monadic rewrite systems of book gallier and salomaa become a natural particular case of trf rewrite systems we associate a deterministic bottom up tree pushdown automaton with any left linear trf rewrite system finally we generalize monadic rewrite systems by introducing the notion of a semi monadic rewrite system and show that like a monadic rewrite system it preserves recognizability

a technique for establishing completeness results in theorem proving with equality it is proved that an automatic theorem proving system consisting of resolution paramodulation factoring equality reversal simplification and subsumption removal is complete in first order logic with equality when restricted to equality units the system is similar to the knuth bendix procedure for deriving consequences from equalities however our proofs of completeness are restricted to the case in which the ordering on words terms or atoms that is required in this type of process is order isomorphic to the positive integers the completeness of resolution and paramodulation without the functionally reflexive axioms is a simple corollary of our result the methods used are based upon the familiar ideas associated with semantic trees and should be helpful in showing that other theorem proving systems with equality are complete

numerical error bounds for fourth order boundary value problems simultaneous estimation ofu x andu x for certain nonlinear two point boundary value problems of the fourth order an estimation theory is developed which yields simultaneous estimates of the solution and its second derivative methods for computing numerical error bounds for approximate solutions are described and tested the theory provides also uniqueness and existence statements the results can be applied to many problems for which a corresponding theory on two sided bounds is not suitable

decidability of regularity and related properties of ground normal form languages we study language theoretical properties of the set of reducible ground terms and its complement the set of ground normal forms induced by a given rewriting system as a tool for our analysis we introduce the property of finite irreducibility of a term with respect to a variable and prove it to be decidable it turns out that this property generalizes numerous interesting properties of the language of ground normal forms in particular we show that testing regularity of this language can be reduced to verifying this property in this way we prove the decidability of the regularity of the set of ground normal forms the problem mentioned in the list of open problems in rewriting by dershowitz jouannaud and klop in proceedings 4th conference on rewriting techniques and applications r v book ed springer verlag berlin new york 1991 also the decidability of the existence of an equivalent ground term rewriting system and some other results are proved

estimation of the global discretization error in shooting methods for linear boundary value problems abstract in this paper we consider an error estimation procedure in the numerical solution of a linear boundary value problem for a system of m first order equations when the simple or parallel shooting method is used we show that the global discretization error can be estimated through the numerical solution of the principal error equation related to only one initial value problem rather than the m 1 involved by shooting four significant numerical experiments are presented

on ground confluence of term rewriting systems abstract it is undecidable whether or not a finite noetherian term rewriting system is ground confluent this undecidability result holds even when systems involving only unary function symbols and one constant are being considered or when left linear or right linear monadic systems are being considered on the other hand if all right hand sides of rules of a finite noetherian rewriting system r are either ground terms or proper subterms of the corresponding left hand sides then it is decidable whether or not r is ground confluent

automatic proofs by induction in theories without constructors abstract inductionless induction consists of using pure equational reasoning for proving the validity of an equation in the initial algebra of a set of equational axioms which would normally require some kind of induction under given hypotheses the equation is valid iff adding it to the set of axioms does not result in an inconsistency this inconsistency can be found by the knuth bendix completion algorithm provided that the signature of the algebra is split into free constructors and defined symbols which must be completely defined in terms of constructors this is the base of the so called inductive completion algorithm of huet and hullot two key concepts inductive reducibility and inductive co reducibility allow us to extend these techniques in various directions incomplete specifications nonfree constructors no constructors specified equational term rewriting systems the method is adapted for proving the consistency property of an enrichment of a specification by new operators and new equations in addition we get also a simple algorithm to exhibit a set of constructors of a specification finally inductive co reducibility is reduced to inductive reducibility and an algorithm for deciding inductive reducibility is given for left linear term rewriting systems

approximate solution of the system of nonlinear integral equation by newton kantorovich method the newton kantorovich method is developed for solving the system of nonlinear integral equations the existence and uniqueness of the solution are proved and the rate of convergence of the approximate solution is established finally numerical examples are provided to show the validity and the efficiency of the method presented

the undecidability of the first order theories of one step rewriting in linear canonical systems by reduction from the halting problem for minsky s two register machines we prove that there is no algorithm capable of deciding the theory of one step rewriting of an arbitrary finite linear confluent finitely terminating term rewriting system weak undecidability we also present a fixed such system with undecidable theory of one step rewriting strong undecidability this improves over all previously known results of the same kind

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

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

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

fuzzy term rewriting system abstract this paper provides an approach to treat fuzzy equality in first order logic specially in a theory in which only equality occurs as a predicate symbol namely equational theory fuzzy term rewriting systems frs that is the set of left to right directional equations are also discussed finally a fuzzy recognition algorithm is presented by applying frs which shows its potential use

proving weak properties of rewriting abstract in rule based programming properties of programs such as termination are in general considered in their strong acceptance i e on every computation branch but in practice they may hold in their weak acceptance only i e on at least one computation branch moreover weak properties are often enough to ensure that programs give the expected result there are very few results to handle weak properties of rewriting we address here two of them termination and reducibility to a constructor form in a unified framework allowing us to prove them inductively proof trees are developed which simulate rewriting trees by narrowing and abstracting subterms our technique is constructive in the sense that proof trees can be used to infer an evaluation strategy for any given input the right computation branch is developed without using a costly breadth first strategy nor backtracking

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

orderings for term rewriting systems abstract methods of proving that a term rewriting system terminates are presented they are based on the intuitive notion of simplification orderings orderings in which any term that is syntactically simpler than another is smaller than the other as a consequence of kruskal s tree theorem any nonterminating system must be self embedding in the sense that it allows for the derivation of some term from a simpler one thus termination is guaranteed if every rule in the system is a reduction in some simplification ordering most of the orderings that have been used for proving termination are indeed simplication orderings using this notion often allows for much easier proofs a particularly useful class of simplification orderings the recursive path orderings is defined examples of the use of simplication orderings in termination proofs are given

fourth order two point boundary value problems estimates by two sided bounds on presente une methode simple pour transformer des problemes aux deux points limites d ordre 4 en problemes d ordre 2

ground reducibility is exptime complete we prove that ground reducibility is exptime complete in the general case exptime hardness is proved by encoding the emptiness problem for the intersection of recognisable tree languages it is more difficult to show that ground reducibility belongs to dexptime we associate first an automaton with disequality constraints ar t to a rewrite system r and a term t this automaton is deterministic and accepts at least one term iff t is not ground reducible by r the number of states of ar t is o 2 r t and the size of its constraints is polynomial in the size of r t then we prove some new pumping lemmas using a total ordering on the computations of the automaton thanks to these lemmas we can show that emptiness for an automaton with disequality constraints can be decided in a time which is polynomial in the number of states and exponential in the size of the constraints altogether we get a simply exponential time deterministic algorithm for ground reducibility decision

equational inference canonical proofs and proof orderings we describe the application of proof orderings a technique for reasoning about inference systems to various rewrite based theorem proving methods including refinements of the standard knuth bendix completion procedure based on critical pair criteria huet s procedure for rewriting modulo a congruence ordered completion a refutationally complete extension of standard completion and a proof by consistency procedure for proving inductive theorems

general 2 2 system of nonlinear integral equations and its approximate solution abstract in this note we consider a general 2 2 system of nonlinear volterra type integral equations the modified newton method modified nm is used to reduce the nonlinear problems into 2 2 linear system of algebraic integral equations of volterra type the latter equation is solved by discretization method nystrom method with gauss legendre quadrature is applied for the kernel integrals and newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained finally numerical examples are provided to show the validity and efficiency of the method presented numerical results reveal that the proposed methods is efficient and accurate comparisons with other methods for the same problem are also presented

proof by consistency abstract advances of the past decade in methods and computer programs for showing consistency of proof systems based on first order equations have made it feasible in some settings to use proof by consistency as an alternative to conventional rules of inference musser described the method applied to proof of properties of inductively defined objects refinements of this inductionless induction method were discussed by kapur goguen huet and hullot huet and oppen lankford dershowitz paul and more recently by jouannaud and kounalis as well as by kapur narendran and zhang this paper gives a very general account of proof by consistency and inductionless induction and shows how previous results can be derived simply from the general theory new results include a theorem giving characterizations of an unambiguity property that is key to applicability of proof by consistency and a theorem similar to the birkhoff s completeness theorem for equational proof systems but concerning inductive proof

existence uniqueness and construction of rewrite systems the construction of term rewriting systems specifically by the knuth bendix completion procedure is considered we look for conditions that might ensure the existence of a finite canonical rewriting system for a given equational theory and that might guarantee that the completion procedure will find it we define several notions of equivalence between rewriting systems in the ordinary and modulo case and examine uniqueness of systems and the need for backtracking in implementing completion

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

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

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

theorem proving using equational matings and rigid e unification in this paper it is shown that the method of matings due toandrews and bibel can be extended to first order languages withequality a decidable version of e unification called rigid e unification is introduced and it is shown that the method of equational matings remains completewhen used in conjunction with rigid e unification checking that a familyof mated sets is an equational mating is equivalent to the followingrestricted kind of e unification problem given e a e i m1 i n a family of n finite sets of equations and s u i v i m1 i n a set of n pairsof terms is there a substitution q such that treating each set q e i as a set of ground equations i e holding the variables in q e i rigid q u i and q v i are provably equal from q e i for i 1 n equivalently is there a substitution q such that q u i and q v i can be shown congruent from q e i by the congruence closure method for i 1 n a substitution q solving the above problem is called a rigid e a unifier of s anda pair e a s such that s hassome rigid e a unifier is called an equationalpremating it is show that deciding whether a pair e a s is an equational premating is an np completeproblem

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

tools for proving inductive equalities relative completeness and ω completeness abstract inductive theorems are properties valid in the initial algebra a now popular tool for proving them in equational theories or abstract data types is based on proof by consistency this method uses a completion procedure and requires two essential properties of the specification namely relative completeness and completeness this paper investigates ways of proving them for the first one the complement algorithm is presented it is based on unification and computation of coverings and complements for the second one a technique based on discrimination of pairs of normal forms is explained and illustrated through examples

abstract canonical presentations solving goals like proving properties deciding word problems or resolving constraints is much easier with some presentations of the underlying theory than with others typically what have been called completion processes in particular in the study of equational logic involve finding appropriate presentations of a given theory to more easily solve a given class of problems we provide a general proof theoretic setting that relies directly on the fundamental concept of good that is normal form proofs itself defined using well founded orderings on proof objects this foundational framework allows for abstract definitions of canonical presentations and very general characterizations of saturation and redundancy criteria

equational formulas with membership constraints we propose a set of transformation rules for first order formulae whose atoms are either equations between terms or membership constraints t can be interpreted as a regular tree language is called a sort in the algebraic specification community or as a tree language in any class of languages which satisfies some adequate closure and decidability properties this set of rules is proved to be correct terminating and complete this provides a quantifier elimination procedure for every regular tree language l the first order theory of some structure defining l is decidable this extends the results of several previously published results we also show how to apply our results to automatic inductive proofs in equational theories