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Unification of Infinite Sets of Terms Schematized by Primal Grammars
 Theoretical Computer Science
, 1996
"... Infinite sets of terms appear frequently at different places in computer science. On the other hand, several practically oriented parts of logic and computer science require the manipulated objects to be finite or finitely representable. Schematizations present a suitable formalism to manipulate fin ..."
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Cited by 23 (3 self)
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Infinite sets of terms appear frequently at different places in computer science. On the other hand, several practically oriented parts of logic and computer science require the manipulated objects to be finite or finitely representable. Schematizations present a suitable formalism to manipulate finitely infinite sets of terms. Since schematizations provide a different approach to solve the same kind of problems as constraints do, they can be viewed as a new type of constraints. The paper presents a new recurrent schematization called primal grammars. The main idea behind the primal grammars is to use primitive recursion as the generating engine of infinite sets. The evaluation of primal grammars is based on substitution and rewriting, hence no particular semantics for them is necessary. This fact allows also a natural integration of primal grammars into Prolog, into functional languages or into other rewritebased applications. Primal grammars have a decidable unification problem and ...
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...
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.
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,
Relative Equational Specification and Semantics
, 1997
"... Abstract: Standard concepts of initial and final algebra semantics are generalised in a modular hierarchical manner. The resulting relative formalism allows a unified view on the relationship between initial and final algebra semantics and gives a dualised notion of consistency. Using this, a modula ..."
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Abstract: Standard concepts of initial and final algebra semantics are generalised in a modular hierarchical manner. The resulting relative formalism allows a unified view on the relationship between initial and final algebra semantics and gives a dualised notion of consistency. Using this, a modular hierarchical approach to proof by consistency is taken by which only toplevel equations need be considered at any level. The formalism also allows nonhomogeneous specification schemes and different proof methods at each level.
More Flexible Term Schematisations via Extended Primal Grammars
"... We propose an extension of primal grammars (Hermann & Galbavý 1997). Primal grammars are term grammars with a high expressive power and good computational properties. The extended grammars have exactly the same properties but are more modular, more concise, and easier to use, as shown by some exampl ..."
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We propose an extension of primal grammars (Hermann & Galbavý 1997). Primal grammars are term grammars with a high expressive power and good computational properties. The extended grammars have exactly the same properties but are more modular, more concise, and easier to use, as shown by some examples. An algorithm transforming any extended primal grammar into an equivalent primal grammar is provided. 1
Computing Overlappings by Unification in the Deterministic Lambda Calculus LR with letrec, case, constructors, seq and variable chains
, 2011
"... Abstract. We investigate the possibilities to automatize correctness proofs of program transformations in an extended lambda calculus LR. The calculus is equipped with an operational semantics, a standardized form of evaluation and based on that a notion of contextual equivalence which is used to de ..."
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Abstract. We investigate the possibilities to automatize correctness proofs of program transformations in an extended lambda calculus LR. The calculus is equipped with an operational semantics, a standardized form of evaluation and based on that a notion of contextual equivalence which is used to define when a program transformations is considered as correct. A successful approach to proving correctness of program transformations is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We describe an effective unification algorithm to determine all overlaps of transformations with reduction rules for the lambda calculus LR which comprises a recursive letexpressions, constructor applications, case expressions and a seq construct for strict evaluation. The unification algorithm uses manysorted terms, the equational theory of leftcommutativity to model multisets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive letexpressions. The algorithm computes a finite set of overlappings for the reduction rules of the calculus LR that serve as a starting point for the automation of the analysis of program transformations. This author is supported by the DFG under grant SCHM 986/91.2 C. Rau and M. SchmidtSchauß