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Alternating TwoWay ACTree Automata
 IN PREPARATION
, 2002
"... We explore the notion of alternating twoway tree automata modulo the theory of finitely many associativecommutative (AC) symbols, some of them with a unit (AC1). This was prompted by questions arising in cryptographic protocol verification, where the emptiness question for intersections of such au ..."
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Cited by 11 (5 self)
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We explore the notion of alternating twoway tree automata modulo the theory of finitely many associativecommutative (AC) symbols, some of them with a unit (AC1). This was prompted by questions arising in cryptographic protocol verification, where the emptiness question for intersections of such automata is fundamental. We show that the use of conditional push clauses, or of alternation, leads to undecidability, already in the case of one AC or AC1 symbol, with only functions of arity zero. On the other hand, emptiness is decidable in the general case of many function symbols, including many AC or AC1 symbols, provided push clauses are unconditional and intersection clauses are final. To this end, extensive use of refinements of resolution is made.
Model Building with Ordered Resolution
 International Workshop on First Order Theorem Proving FTP'2000
, 2000
"... this paper, we propose to use clause sets to represent models. This idea is very simple and allows to combine in a well balanced way, expressive power with the \good" required properties of a reasonable model representation. Obviously, in order that the set of clauses specifying the model bring ..."
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Cited by 4 (1 self)
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this paper, we propose to use clause sets to represent models. This idea is very simple and allows to combine in a well balanced way, expressive power with the \good" required properties of a reasonable model representation. Obviously, in order that the set of clauses specifying the model brings new information w.r.t. the initial one, a basic requirement is that these clause sets must have exactly one Herbrand model (on a given signature). Such clause sets are straightforward representations of their Herbrand models
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...
Combining inference and disinference rules with enumeration for model building (Extended Abstract)
, 1997
"... ) Ricardo Caferra and Nicolas Peltier Laboratory LEIBNIZIMAG 46, Avenue F'elix Viallet 38031 Grenoble Cedex FRANCE Ricardo.Caferra@imag.fr, Nicolas.Peltier@imag.fr Phone: (33) (0)4 76 57 46 59 1 Introduction The possibility of systematic model building in firstorder logic exists at least si ..."
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Cited by 1 (1 self)
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) Ricardo Caferra and Nicolas Peltier Laboratory LEIBNIZIMAG 46, Avenue F'elix Viallet 38031 Grenoble Cedex FRANCE Ricardo.Caferra@imag.fr, Nicolas.Peltier@imag.fr Phone: (33) (0)4 76 57 46 59 1 Introduction The possibility of systematic model building in firstorder logic exists at least since the introduction of the tableaux method (Hintikka, Beth, Smullyan,: : : ), approximately 40 years ago. Some striking results in interactive model building have been obtained less than 20 years ago [21]. But it is only since less than 10 years that results on model building are regularly published [15, 7, 8, 5, 12, 13, 20, 22]. One important difference must be underlined between proof calculi and model building methods. Different calculi for firstorder logic (resolution, tableaux, connection method (matings), : : : ) are potentially able to prove the same class of theorems. In papers on model building methods what is emphasized in general is the class of models that the methods can (or cannot...
ContextFree Term Sets are Regular  and Some Applications to Logic
, 1997
"... . We prove that all contextfree sets of terms are regular tree languages, a natural conjecture formulated in [Mat97]. We motivate our interest in this languagetheoretic result with two issues. First in automated model building it is desirable to characterize the power of contextfree languages to ..."
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. We prove that all contextfree sets of terms are regular tree languages, a natural conjecture formulated in [Mat97]. We motivate our interest in this languagetheoretic result with two issues. First in automated model building it is desirable to characterize the power of contextfree languages to represent single Herbrand models, which we conclude to be exactly the finite models. Second in propositional logic our theorem serves to build finitely valued truth tables from an arbitrary contextfree specification of a logic's nvariable fragment, thus showing that a logic's nvariable fragment is finitely valued iff its tautologies with not more than n propositional variables form a contextfree language. Keywords. regular tree language, contextfree, grammar, propositional logic, manyvalued logic, automated model building Table of Contents 1 Introduction and Motivation 1 2 Regular Tree Automata and Context Free Languages 2 3 Consequences and Applications 8 4 Conclusion and Summary 10...
Using Term Schematizations in Automated Deduction
"... We propose a new method for using recurrent schematizations in Theorem Proving. We provide techniques for detecting cycles in proofs (via proof generalization), and we show how to take advantage of the expressive power of schematizations in order to avoid generating such cycles explicitly. This may ..."
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We propose a new method for using recurrent schematizations in Theorem Proving. We provide techniques for detecting cycles in proofs (via proof generalization), and we show how to take advantage of the expressive power of schematizations in order to avoid generating such cycles explicitly. This may shorten proofs and avoid divergence in some cases. These techniques are more general than existing ones, and unlike them, they can be used with any kind of proof procedure (using tableauxbased approaches as well as resolutionbased ones).
Algorithmic Aspects of Herbrand Models Represented by Ground Atoms with Ground Equations
, 2002
"... Automated model building has evolved as an important subdiscipline of automated deduction over the past decade. One crucial issue in automated model building is the selection of an appropriate (finite) representation of (in general infinite) models. Quite a few such formalisms have been proposed in ..."
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Automated model building has evolved as an important subdiscipline of automated deduction over the past decade. One crucial issue in automated model building is the selection of an appropriate (finite) representation of (in general infinite) models. Quite a few such formalisms have been proposed in the literature. In this paper, we concentrate on the representation of Herbrand models by ground atoms with ground equations (GAEmodels), introduced in [9]. For the actual work with any model representation, efficient algorithms for two decision problems are required, namely: The clause evaluation problem (i.e.: Given a clause C and a representation M of a model, does C evaluate to "true" in this model?) and the model equivalence problem (i.e.: Given two representations M1 and M2 , do they represent the same model?). Previously published algorithms for these two problems in case of GAEmodels require exponential time. We prove that the clause evaluation problem is indeed intractable (that is, coNPcomplete), whereas the model equivalence problem can be solved in polynomial time. Moreover, we show how our new algorithm for the model equivalence problem can be used to transform an arbitrary GAEmodel into an equivalent one with better computational properties.
Extending FirstOrder Unification by Tractable SecondOrder Features
, 2000
"... We present a new approach for solving certain infinite sets of first order unification problems represented by term schemes. Within the framework of secondorder equational logic solving such scheme unification problems amounts exactly to solving (variable)restricted unification problems. Finally ..."
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We present a new approach for solving certain infinite sets of first order unification problems represented by term schemes. Within the framework of secondorder equational logic solving such scheme unification problems amounts exactly to solving (variable)restricted unification problems. Finally, we show how this approach yields a generic solution technique for infinitely many ordinary firstorder unification problems.
A New Approach to Model Building: Searching for Clauses Sets with only one Herbrand Model
"... . A method is proposed to transform automatically some satis able clauses sets S into clauses sets S 0 having exactly one Herbrand model in a given signature, such that if M j= S 0 , then M j= S. These clauses sets with only one model can be used as representations of (in general innite) Her ..."
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. A method is proposed to transform automatically some satis able clauses sets S into clauses sets S 0 having exactly one Herbrand model in a given signature, such that if M j= S 0 , then M j= S. These clauses sets with only one model can be used as representations of (in general innite) Herbrand models of the initial set of clauses. Existing theorem provers may be used to evaluate literals and clauses in the models thus represented (using the standard proof by consistency mechanism) . We also prove that for some classes of clauses, nite models can be automatically extracted, thus showing that the evaluation of arbitrary formulae in the represented models is a decidable problem. Keyword: automated deduction, model building. 1 Introduction Automated Model building is one of the more fundamental tasks in Mathematics and is now widely recognized as a very important and dicult challenge in Automated Deduction (see for example [4, 9]). Models (resp. counterexamples) do no...