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We present an implementation and verification in higher-order logic of Cooper's quantifier elimination for Presburger arithmetic. Reflection, i.e. the direct execution in ML, yields a speed-up of a factor of 200 over an LCF-style implementation and performs as well as a decision procedure hand-coded in ML. [ABSTRACT FROM AUTHOR]

Structural recursion over sets is meaningful only if the result is independent of the order in which the set's elements are enumerated. This paper outlines a theory of function definition for finite sets, based on the fold functionals often used with lists. The fold functional is introduced as a relation, which is then shown to denote a function under certain conditions. Applications include summation and maximum. The theory has been formalized using Isabelle/HOL . [ABSTRACT FROM AUTHOR]

Using the theorem prover Isabelle/HOL we have formalized and proved correct an executable bytecode verifier in the style of Kildall's algorithm for a significant subset of the Java Virtual Machine. First an abstract framework for proving correctness of data flow based type inference algorithms for assembly languages is formalized. It is shown that under certain conditions Kildall's algorithm yields a correct bytecode verifier. Then the framework is instantiated with a model of the JVM. [ABSTRACT FROM AUTHOR]

This paper presents proof terms for simply typed, intuitionistic higher order logic, a popular logical framework. Unification-based algorithms for the compression and reconstruction of proof terms are described and have been implemented in the theorem prover Isabelle. Experimental results confirm the effectiveness of the compression scheme. [ABSTRACT FROM AUTHOR]

A formal verification of the soundness and completeness of Milner's type inference algorithm W for simply typed lambda-terms is presented. Particular attention is paid to the notorious issue of "new" variables. The proofs are carried out in Isabelle/HOL, the HOL instantiation of the generic theorem prover Isabelle. [ABSTRACT FROM AUTHOR]

This paper presents the development and verification of a (very simple) lexical analyzer generator that takes a regular expression and yields a functional lexical analyzer. The emphasis is on simplicity and executability. The work was carried out with the help of the theorem prover Isabelle/HOL. [ABSTRACT FROM AUTHOR]

We examine different possibilities of coupling saturationbased theorem provers by exchanging positive/negative information. Positive information is given by facts that should be employed for proving a proof goal, negative information is represented by facts that do not appear to be useful. We introduce a basic model for cooperative theorem proving employing both kinds of information. We present theoretical results regarding the exchange of positive/negative information as well as practical methods that allow for a gain of efficiency in comparison with sequential provers. Finally, we report on experimental studies conducted in the areas unfailing completion and superposition. [ABSTRACT FROM AUTHOR]

We demonstrate the usefulness of higher-order rewriting techniques for specializing programs, i.e., for partial evaluation. More precisely, we demonstrate how casting program specializers as combinatory reduction systems (CRSs) makes it possible to formalize the corresponding program transformations as meta-reductions, i.e., reductions in the internal "substitution calculus." For partial-evaluation problems, this means that instead of having to prove on a case-by-case basis that one's "two-level functions" operate properly, one can concisely formalize them as a combinatory reduction system and obtain as a corollary that static reduction does not go wrong and yields a well-formed residual program. We have found that the CRS substitution calculus provides an adequate expressive power to formalize partial evaluation: it provides sufficient termination strength while avoiding the need for additional restrictions such as types that would complicate the description unnecessarily (for our purpose). In addition, partial evaluation provides a number of examples of higher-order rewriting where being higher order is a central (rather than an occasional or merely exotic) property. We illustrate this by demonstrating how standard but non-trivial partial-evaluation examples are handled with higher-order rewriting. [ABSTRACT FROM AUTHOR]

We introduce an intersection typing system for combinatory logic. We prove the soundness and completeness for the class of partial combinatory algebras. We derive that a term of combinatory logic is typeable iff it is SN. Let F be the class of non-empty filters which consist of types. Then F is an extensional non-total partial combinatory algebra. Furthermore, it is a fully abstract model with respect to the set of sn c terms of combinatory logic. By F, we can solve Bethke-Klop's question; "find a suitable representation of the finally collapsed partial combinatory algebra of P". Here, P is a partial combinatory algebra, and is the set of closed sn terms of combinatory logic modulo the inherent equality. Our solution is the following: the finally collapsed partial combinatory algebra of P is representable in F. To be more precise, it is isomorphically embeddable into F. [ABSTRACT FROM AUTHOR]

An extension of a modular termination result for term rewriting systems (TRSs, for short) by A. Middeldorp (1989) is presented. We intended to obtain this by adapting the dummy elimination transformation by M. C. F. Ferreira and H. Zantema (1995) under the presence of a non-collapsing non-duplicating terminating TRS whose function symbols are all to be eliminated. We propose a tree lifting ordering induced from a reduction order and a set G of function symbols, and use this ordering to transform a TRS R into R'; termination of R' implies that of RUS for any non-collapsing non-duplicating terminating TRS S whose function symbols are contained in G, provided that for any l → r in R (1) the root symbol of r is in G whenever that of l is in G; and (2) no variable appears directly below a symbol from G in l when G contains a constant. Because of conditions (1) and (2), our technique covers only a part of the dummy elimination technique; however, even when S is empty, there are cases that our technique has an advantage over the dummy elimination technique. [ABSTRACT FROM AUTHOR]

We present a transformation technique called freezing to facilitate automatic termination proofs for left-linear term rewriting systems. The significant merits of this technique lie in its simplicity, its amenability to automation and its effectiveness, especially, when combined with other well-known methods such as recursive path orderings and polynomial interpretations. We prove that applying the freezing technique to a left-linear term rewriting system always terminates. We also show that many interesting TRSs in the literature can be handled with the help of freezing while they elude a lot of other approaches aiming for generating termination proofs automatically for term rewriting systems. We have mechanically verified all the left-linear examples presented in this paper. [ABSTRACT FROM AUTHOR]

A new criterion for termination of rewriting has been described by Arts and Giesl in 1997. We show how this criterion can be generalized to rewriting modulo associativity and commutativity. We also show how one can build weak AC-compatible reduction orderings which may be used in this criterion. [ABSTRACT FROM AUTHOR]

The framework of dependency pairs allows automated termination and innermost termination proofs for many TRSs where such proofs were not possible before. In this paper we present a refinement of this framework in order to prove termination in a modular way. Our modularity results significantly increase the class of term rewriting systems where termination resp. innermost termination can be proved automatically. Moreover, the modular approach to dependency pairs yields new modularity criteria which extend previous results in this area. In particular, existing results for modularity of innermost termination can easily be obtained as direct consequences of our new criteria. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the class of co-definite set constraints. This is a natural subclass of set constraints which, when satisfiable, have a greatest solution. It is practically motivated by the set-based analysis of logic programs with the greatest-model semantics. We present an algorithm solving co-definite set constraints and show that their satisfiability problem is DEXPTIME-complete. [ABSTRACT FROM AUTHOR]

The system FT≤of ordering constraints over feature trees has been introduced as an extension of the system FT of equality constraints over feature trees. We investigate decidability and complexity questions for fragments of the first-order theory of FT≤. It is well-known that the first-order theory of FT is decidable and that several of its fragments can be decided in quasi-linear time, including the satisfiability problem of FT and its entailment problem with existential quantification ϕ ⊨ ∃x1 .∃xnϕR', Much less is known on the first-order theory of FT≤. The satisfiability problem of FT≤can be decided in cubic time, as well as its entailment problem without existential quantification. Our main result is that the entailment problem of FT≤with existential quantifiers is decidable but PSPACE-hard. Our decidability proof is based on a new technique where feature constraints are expressed in second-order monadic logic with countably many successors SΩS. We thereby reduce the entailment problem of FT≤with existential quantification to Rabin's famous theorem on tree automata. [ABSTRACT FROM AUTHOR]

We show that simultaneous rigid E-unification, or SREU for short, is decidable and in fact EXPTIME-complete in the case of one variable. This result implies that the ∀*∃∀* fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the ∃∃-fragment, we obtain a complete classification of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix. It is also proved that SREU with one variable and a constant bound on the number of rigid equations is P-complete. [ABSTRACT FROM AUTHOR]

In this paper we initiate a study of polynomial-time reductions for some basic decision problems of rewrite systems. We then give a polynomial-time algorithm for Unique-normal-form property of ground systems for the first time. Next we prove undecidability of these problems for a fixed string rewriting system using our reductions. Finally, we prove partial decidability results for Confluence of commutative semi-thue systems. The Confluence and Unique-normal-form property are shown Expspace-hard for commutative semi-thue systems. We also show that there is a family of string rewrite systems for which the word problem is trivially decidable but confluence undecidable, and we show a linear equational theory with decidable word problem but undecidable linear equational matching. [ABSTRACT FROM AUTHOR]

We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations of sets of descendants and sets of normal forms provide tools for analysing a few properties of programs: we show how to compute a superset of results, to prove the sufficient completeness property, or to find a criterion for proving termination under a specific strategy, the sequential reduction strategy. The main technical contribution of the paper is the construction of an approximation automaton which recognises a superset of the set of normal forms of terms in a set E, w.r.t. a Term Rewriting System R. [ABSTRACT FROM AUTHOR]

The combinator S has the reduction rule Sx y z → x z (y z). We investigate properties of ground terms built from S alone. We show that it is decidable whether such an S-term has a normal form. The decision procedure makes use of rational tree languages. We also exemplify and summarize other properties of S-terms and hint at open questions. [ABSTRACT FROM AUTHOR]

This paper gives a procedure for solving disequations modulo equational theories, and to decide existence of solutions. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. The procedure represents the possibly infinite set of solutions thanks to a grammar, and decides existence of solutions thanks to an emptiness test. As a consequence, checking whether a linear equality is an inductive theorem is decidable, if assuming moreover sufficient completeness. [ABSTRACT FROM AUTHOR]

This paper is concerned with the unification problem in the path logics associated by the optimised functional translation method with the propositional modal logics K, KD, KT, KD4, S4 and S5. It presents improved unification algorithms for certain forms of the right identity and associativity laws. The algorithms employ mutation rules, which have the advantage that terms are worked off from the outside inward, making paramodulating into terms superfluous. [ABSTRACT FROM AUTHOR]

We consider the compatibility checking problem for a simple fragment of CCS, called BCCSP[12], using equational unification techniques. Two high-level specifications given as two process algebraic terms with free variables are said to be compatible modulo some equivalence relation if a substitution on the free variables can make the resulting terms equivalent modulo that relation. We formulate this compatibility (modulo an equivalence relation) checking problems as unification problems in the equational theory of the the corresponding equivalence relation. We use van Glabbeek's equational axiomatizations [12] for some interesting process algebraic relations. Specifically, we consider equational axiomatizations for bisimulation equivalence and trace equivalence and establish complexity lower bounds and upper bounds for the corresponding unification and matching problems. We also show some special cases for which efficient algorithmic solutions exist. [ABSTRACT FROM AUTHOR]

We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equational theories is introduced. This class is a natural extension of tree automata with equality constraints between brother subterms as well as shallow sort theories. We show that saturation under sorted superposition is effective on sorted shallow equational theories. So called semi-linear equational theories can be effectively transformed into equivalent sorted shallow equational theories and generalize the classes of shallow and standard equational theories. [ABSTRACT FROM AUTHOR]

Context unification is a generalisation of string unification where words are generalized to terms with one hole. Though decidability of string unification was proved by Makanin, the decidability of context unification is currently an open question. This paper provides a step in understanding the complexity of context unification and the structure of unifiers. It is shown, that if a context unification problem of size d is unifiable, then there is also a unifier with an exponent of periodicity smaller than O(21.07d). We also prove NP-hardness for restricted cases of the context unification problem and compare the complexity of general context unification with that of general string unification. [ABSTRACT FROM AUTHOR]

There is a close relationship between word unification and second-order unification. This similarity has been exploited for instance for proving decidability of monadic second-order unification. Word unification. can be easily decided by transformation rules (similar to the ones applied in higher-order unification procedures) when variables are restricted to occur at most twice. Hence a well-known open question was the decidability of second-order unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid E-unification to second-order unification. This reduction, together with an inverse reduction found by Degtyarev and Voronkov, states an equivalence relationship between both unification problems. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid E-unification is decidable. This happens, for example, for one-variable problems where the variable occurs at most twice (because rigid E-unification is decidable for just one equation). We also prove decidability when no variable occurs more than once, hence significantly narrowing the gap between decidable and undecidable second-order unification problems with variable occurrence restrictions. [ABSTRACT FROM AUTHOR]

Due to their many nice properties groups with automatic structure (automatic groups) have received a lot of attention in the literature. The multiplication of an automatic group can be realized through finite automata based on a regular set of (not necessarily unique) representatives for the group, and hence, each automatic group has a tractable word problem and low derivational complexity. Consequently it has been asked whether corresponding results also hold for monoids with automatic structure. Here we show that there exist finitely presented monoids with automatic structure that cannot be presented through finite and convergent string-rewriting systems, thus answering a question in the negative that is still open for the class of automatic groups. Secondly, we present an automatic monoid that has an exponential derivational complexity, which establishes another difference to the class of automatic groups. In fact, both our example monoids are bi-automatic. In addition, it follows from the first of our examples that a monoid which is given through a finite, noetherian, and weakly confluent string-rewriting system need not have finite derivation type. [ABSTRACT FROM AUTHOR]

A very powerful method for proving the Church-Rosser property for abstract rewriting systems has been developed by van Oostrom. In this paper, his technique is extended in two ways to abstract rewriting modulo an equivalence relation. It is shown that known Church-Rosser theorems can be viewed as special cases of the new criteria. Moreover, applications of the new criteria yield several new results. [ABSTRACT FROM AUTHOR]

We introduce simultaneous critical pairs, which account for simultaneous overlapping of several rewrite rules. Based on this, we introduce a new CR-criterion widely applicable to arbitrary left-linear term rewriting systems. Our result extends the well-known criterion given by Huet (1980), Toyama (1988), and Oostrom (1997) and incomparable with other well-known criteria for left-linear systems. [ABSTRACT FROM AUTHOR]

The notion of descendants or residuals with its inverse notion of ancestors is classical in the theory of rewriting, both in first order term rewriting and in higher-order rewriting, such as lambda calculus. Recently this classical notion has been given much attention. On the one hand, the notion has been studied in an abstract, axiomatic way, in order to isolate the essential properties of the descendant concept. On the other hand descendants were studied in a very concrete way, inspired by practical considerations such as error recovery in program executions and program slicing. In the latter endeavour the emphasis is on tracing back the symbols constituting an expression to their 'causes' in an earlier expression in the rewrite sequence. This method is also known as origin tracking. The corresponding descendant notion is a refinement of the classical one. In our talk we present some basic properties of origin tracking. It will be apparent that various labeled versions of the rewrite systems concerned are important to establish such properties. We treat three (theoretical) applications of the origin tracking technique: first, a simplified proof of the classical theorem of Huet and Le'vy about needed reduction; second, another proof of the Genericity Lemma in pure lambda calculus; third, a simple proof of Berry's sequentially theorem for lambda calculus. The third application actually takes place in the setting of infinitary lambda calculus, where Boehm trees are normal forms; and Berry's the-orem will easily generalize to lazy lambda trees (or Levy-Longo trees), and to Berarducci trees. [ABSTRACT FROM AUTHOR]