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Higherorder Unification via Explicit Substitutions (Extended Abstract)
 Proceedings of LICS'95
, 1995
"... Higherorder unification is equational unification for βηconversion. But it is not firstorder equational unification, as substitution has to avoid capture. In this paper higherorder unification is reduced to firstorder equational unification in a suitable theory: the λσcal ..."
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Cited by 103 (13 self)
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Higherorder unification is equational unification for βηconversion. But it is not firstorder equational unification, as substitution has to avoid capture. In this paper higherorder unification is reduced to firstorder equational unification in a suitable theory: the λσcalculus of explicit substitutions.
On Unification of Terms with Integer Exponents
 Mathematical Systems Theory
, 1995
"... this paper, we got informed of the work of G. Salzer, which has been done independently from ours [6]. He uses a quite different formalism, but he shows essentially the same results as ours, except that his formalism is slightly more powerful since he can express for example the set of all complete ..."
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Cited by 24 (2 self)
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this paper, we got informed of the work of G. Salzer, which has been done independently from ours [6]. He uses a quite different formalism, but he shows essentially the same results as ours, except that his formalism is slightly more powerful since he can express for example the set of all complete finite binary trees with internal nodes labeled with f . Indeed, his syntax allows for multilple holes in the terms, in which case, the semantics is different from what we considered in section 7. In order to express it shortly, you can assume that the holes
Tree Automata and Automated Model Building
, 1997
"... . The use of regular tree grammars to represent and build models of formulae of firstorder logic without equality is investigated. The combination of regular tree grammars with equational constraints provides a powerful and general way of representing Herbrand models. We show that the evaluation pr ..."
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Cited by 9 (4 self)
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. The use of regular tree grammars to represent and build models of formulae of firstorder logic without equality is investigated. The combination of regular tree grammars with equational constraints provides a powerful and general way of representing Herbrand models. We show that the evaluation problem (i.e. the problem of finding the truth value of a formula in a given model) is decidable when models are represented in the way we propose. We also define a method to build such representations of models for firstorder formulae. These results are a powerful extension of our former method for simultaneous search for refutations and models. Keywords: Automated Deduction, Model Building, Tree Automata, Regular Tree Grammars. 1. Introduction The problem of building models or counterexamples of firstorder formulae is a very important one, particularly in the field of automated deduction. Besides their intrinsic interest for disproving conjectures, counterexamples (models) have numerous...
Computational Representations of Herbrand Models Using Grammars
 Computer Science Logic, 10th International Workshop, CSL'96, volume 1258 of LNCS
, 1997
"... . Finding computationally valuable representations of models of predicate logic formulas is an important subtask in many fields related to automated theorem proving, e.g. automated model building or semantic resolution. In this article we investigate the use of contextfree languages for representin ..."
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Cited by 9 (4 self)
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. Finding computationally valuable representations of models of predicate logic formulas is an important subtask in many fields related to automated theorem proving, e.g. automated model building or semantic resolution. In this article we investigate the use of contextfree languages for representing single Herbrand models, which appear to be a natural extension of "linear atomic representations" already known from the literature. We focus on their expressive power (which we find out to be exactly the finite models) and on algorithmic issues like clause evaluation and equivalence test (which we solve by using a resolution theorem prover), thus proving our approach to be an interesting base for investigating connections between formal language theory and automated theorem proving and model building. 1 Introduction Representing single models of predicate logic formulas plays an important role in various subfields of automated theorem proving. We just mention two of them: 1. An interesti...
Increasing Model Building Capabilities by Constraint Solving on Terms with Integer Exponents
 Journal of Symbolic Computation
, 1997
"... this paper the decidability of first order theory of the language of Iterms. ..."
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Cited by 6 (1 self)
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this paper the decidability of first order theory of the language of Iterms.
A Note on Simple Termination of Infinite Term Rewriting Systems
, 1992
"... this paper we investigate the relationship of some properties of TRSs which are equivalent in the finite case and are thus (in the finite case) correctly used synonymously. However, these properties differ in the infinite case. This is a potential source of errors  inaccurate citations may lead to ..."
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Cited by 4 (2 self)
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this paper we investigate the relationship of some properties of TRSs which are equivalent in the finite case and are thus (in the finite case) correctly used synonymously. However, these properties differ in the infinite case. This is a potential source of errors  inaccurate citations may lead to inaccurate or false "results". One of the main objects of this paper is the careful analysis of the differences that occur in the infinite case which may help to avoid mistakes. Note that not all of the results and examples are claimed to be new but are included for completeness and clarity. The notion "simple termination" was first introduced by Kurihara and Ohuchi in [KO90a]. They call
Comparing Computational Representations of Herbrand Models
 Computational Logic and Proof Theory, 5th Kurt Godel Colloquium, KGC'97, volume 1289 of LNCS
, 1997
"... . Finding computationally valuable representations of models of predicate logic formulas is an important issue in the field of automated theorem proving, e.g. for automated model building or semantic resolution. In this article we treat the problem of representing single models independently of buil ..."
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Cited by 3 (2 self)
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. Finding computationally valuable representations of models of predicate logic formulas is an important issue in the field of automated theorem proving, e.g. for automated model building or semantic resolution. In this article we treat the problem of representing single models independently of building them and discuss the power of different mechanisms for this purpose. We start with investigating contextfree languages for representing single Herbrand models. We show their computational feasibility and prove their expressive power to be exactly the finite models. We show an equivalence with "ground atoms and ground equations" concluding equal expressive power. Finally we indicate how various other well known techniques could be used for representing essentially infinite models (i.e. models of not finitely controllable formulas), thus motivating our interest in relating model properties with syntactical properties of corresponding Herbrand models and in investigating connections betwe...
Making a Productive Use of Failure to Generate Witnesses for Coinduction from Divergent Proof Attempts
 RR0004 in the Informatics Report Series
, 2000
"... this paper. Corresponding Author. 2 Witnesses for Coinduction witness relation is a fundamental step in the process of proof by coinduction. These techniques are based on middle{out reasoning (delaying the choice of witness for as long as possible by using meta{variables and higher order unicati ..."
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Cited by 2 (1 self)
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this paper. Corresponding Author. 2 Witnesses for Coinduction witness relation is a fundamental step in the process of proof by coinduction. These techniques are based on middle{out reasoning (delaying the choice of witness for as long as possible by using meta{variables and higher order unication) and proof critics (exploiting information from failed proof attempts to modify witnesses). Coinduction is the dual of induction and is used to deal naturally with innite processes. It was rst investigated seriously in the eld of concurrency [25] where looping communication networks are commonplace. It is also used in so{ called \lazy" functional languages where the evaluation procedure only evaluates functions when they are required and may not fully evaluate 1 them. In this way a potentially innite process may be present in a program without forcing the entire program to be non{terminating. The semantics of lazy languages are generally expressed in an operational style. This work concentrates on the use of coinduction with the operational semantics of a lazy functional language. Coinduction has also been proposed for use with object{oriented languages [20], cryptographic protocols [1] and the calculus of mobile ambients [21]. Tools have been provided for coinduction in several theorem proving environments. One of these, the Edinburgh Concurrency Workbench [12], is fully automated. This deals with problems described in Process Algebras. In other areas, such as functional languages, automation has not been attempted. The choice of the bisimulation needed by a proof is equivalent to the choice of induction scheme in inductive proofs [15]. Like the choice of induction scheme, the choice of bisimulation is a hard step in coinductive proof. This work presents an auto...
Overview of Existing Recurrent Schematizations
 Proc. of the CADE13 Workshop on Term Schematization and their Applications
, 1996
"... le, but the constraint formalism lacks schematization power, (2) the constraint formalism is powerful enough to represent the infinite families but the unification problem for these constraints is undecidable, and (3) the constraint formalism is powerful enough and the unification problem is decidab ..."
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Cited by 1 (0 self)
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le, but the constraint formalism lacks schematization power, (2) the constraint formalism is powerful enough to represent the infinite families but the unification problem for these constraints is undecidable, and (3) the constraint formalism is powerful enough and the unification problem is decidable, but the corresponding constraint solving unification algorithm produces an infinite family of constraints. In practice, the manipulated sets must be finite, unless there is a possibility to use constraints and implicit representations. The previous considerations clearly indicate the need for a formalism that allows to express explicitly infinite families by finite means, that has a decidable unification problem, and that has a terminating unification algorithm yielding a finite representation of the results. Moreover, such formalism should have a semantics compatible with their domain of application so that it can be easily incorporated into different theoretical developments,
SecondOrder Principles in Specification Languages for ObjectOriented Programs
"... Abstract. Within the setting of objectoriented program specification and verification, pointers and object references can be considered as relations between the elements of a data structure. When we specify properties of these data structures, we often describe properties of relations. Hence it is ..."
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Abstract. Within the setting of objectoriented program specification and verification, pointers and object references can be considered as relations between the elements of a data structure. When we specify properties of these data structures, we often describe properties of relations. Hence it is important to be able to talk about relations and their properties when specifying objectoriented programs or programs with pointers. Many interesting properties of relations such as transitive closure, finiteness, and generatedness are not expressible in firstorder logic (FOL); hence neither are they expressible in firstorder fragments of specification languages. In this paper we give an overview of the different ways such properties can be expressed in various logics, with a particular emphasis on extensions of FOL, i.e. transitive closure logic, fixedpoint logic, and firstorder dynamic logic. Within the paper we also discuss which of these extensions already are – or in fact should be – implemented within specification languages. We feel that such a discussion is necessary since it is often the case that when an extension of FOL is implemented within a specification language it is done so in an ad hoc manner or the underpinning logical concepts are not well documented.. 1