Habilitation Report (abstract)

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1 Habilitation Report (abstract) N. Peltier February Personal Information Born on March 13, Nationality: French. Professional Address: Laboratoire d Informatique de Grenoble (LIG). 46, Avenue Félix Viallet 38031, Grenoble Cedex France Phone: Nicolas.Peltier@imag.fr Current Position Since October 99 : Research Fellow at CNRS, Leibniz/LIG Laboratory. (Grenoble). CAPP team (Calculi, Algorithms, Programs, Proofs). Education October 1997 : PhD Thesis, INPG (Institut National Polytechnique de Grenoble). Advisor: Ricardo Caferra. New Techniques for Finite and Infinite Model Building in Automated Deduction. June 94 : Diploma in Computer Science Engineering from ENSIMAG (Ecole Nationale Supérieure d Informatique et de Mathématiques Appliquées de Grenoble) with highest honours. 1

2 2 Scientific Publications Most papers are available on: peltier. Books Ricardo Caferra, Alexander Leitsch and Nicolas Peltier. Automated Model Building. Applied Logic Series, vol 31, Kluwer Academic Publisher, International Journals Nicolas Peltier. Increasing Model Building Capabilities by Constraint Solving with Terms with Integer Exponents. Journal of Symbolic Computation, vol. 24, pages , Nicolas Peltier. Tree Automata and Automated Model Building. Fundamenta Informaticae, vol. 30 (1), pages Ricardo Caferra and Nicolas Peltier. A New Technique for Verifying and Correcting Logic Programs. Journal of Automated Reasoning, vol. 19 (3), pages , Christophe Bourely, Gilles Défourneaux and Nicolas Peltier. Semantic Generalizations for Proving and Disproving Conjectures by Analogy. Journal of Automated Reasoning. Special issue of JELIA 96 best papers, vol. 20 (1 & 2), pages 27 45, Nicolas Peltier. Pruning the Search Space and Extracting More Models in Tableaux. Logic Journal of the IGPL, vol 7 (2), pages , Nicolas Peltier. A New Method for Automated Finite Model Building Exploiting Failures and Symmetries. Journal of Logic and Computation, vol. 8 (4), pages , Ricardo Caferra and Nicolas Peltier. Combining Enumeration and Deductive Techniques in order to Increase the Class of Constructible Infinite Models. Journal of Symbolic Computation (special issue on advances in First-Order Theorem Proving), vol. 29, pages ,

3 Nicolas Peltier. On the Decidability of the PVD Class with Equality. Logic Journal of the IGPL, vol 9 (4), pages , Nicolas Peltier. A Calculus Combining Resolution and Enumeration for Building Finite Models. Journal of Symbolic Computation, vol. 36 (1 & 2), pages 49 77, Nicolas Peltier. Model Building with Ordered Resolution: Extracting Models from Saturated Clause Sets. Journal of Symbolic Computation, vol. 36, (1 & 2), pages 5 48, Nicolas Peltier. Constructing Decision Procedures in Equational Clausal Logic. Fundamenta Informaticae, vol 54 (1), pages 17 65, Nicolas Peltier. Building Infinite Models for Equational Clause Sets: Constructing Non-Ambiguous Formulae. Logic journal of the IGPL, vol 11 (1), pages , Nicolas Peltier. Extracting Models from Clause Sets Saturated under Semantic Refinements of the Resolution Rule. Information and Computation, vol 181, pages , Nicolas Peltier. The First Order Theory of Primal Grammars is Decidable. Theoretical Computer Science, 323, pages , Nicolas Peltier. A Proof Procedure for Functional First Order Logic Programs with non-deterministic Lazy Functions and Built-in Predicates. Journal of Functional and Logic Programming, 3, Nicolas Peltier. A Resolution Calculus for Shortening Proofs. Logic Journal of the Interest Group in Pure and Applied Logics, vol. 13, pages , Nicolas Peltier. Representing and Building Models for Decidable Subclasses of Equational Clausal Logic. Journal of Automated Reasoning, vol. 33 (2), pages , Nicolas Peltier. Some Techniques for Proving Termination of the Hyperresolution Calculus. Journal of Automated Reasoning, vol. 35 (4),

4 Nicolas Peltier. A Resolution Calculus with Shared Literals. Fundamenta Informaticae, vol. 76 (4) Nicolas Peltier. Extended Resolution Simulates Binary Decision Diagrams. Discrete Applied Mathematics, accepted, to appear. Ricardo Caferra and Nicolas Peltier. Accepting/rejecting Proposition from Accepted/Rejected Propositions: a Unifying Overview. International Journal of Intelligent Systems, accepted, to appear. Book Chapters Ricardo Caferra, Nicolas Peltier. The Connection Method, Constraints and Model Building. Intellectics and Computational Logic, pages Ricardo Caferra and Nicolas Peltier. Disinference rules, Model Building and Abduction. in Logic at Work. Essays dedicated to the memory of Helena Rasiowa (partie 5 : Logic in Computer Science, Chapitre 20), pages , Ewa Orłowska (ed). Physica-Verlag, International Conferences Christophe Bourely, Ricardo Caferra and Nicolas Peltier. A Method for Building Models Automatically: Experiments with an Extension of OTTER. CADE 94 (12th Conference on Automated Deduction). Springer Verlag, LNAI 814, pages 72-86, Nancy, France, July Ricardo Caferra and Nicolas Peltier. Extending Semantic Resolution via Automated Model Building: applications. IJ- CAI 95 (International Joint Conference of Artificial Intelligence), Morgan Kaufmann, pages Montreal, Canada, August Ricardo Caferra and Nicolas Peltier. Model Building and Interactive Theory Discovery. TABLEAUX 95 (4th Workshop on Theorem Proving with Analytic Tableaux and Related Methods). Springer Verlag, LNAI 918, pages , Schloss Rheinfels, St. Goar, Germany, May

5 Ricardo Caferra and Nicolas Peltier. Decision Procedures using Model Building Techniques. CSL 95 (Computer Science Logic). Springer Verlag, LNCS 1092, pages , Paderborn, Germany, September Ricardo Caferra and Nicolas Peltier. A Significant Extension of Logic Programming by Adapting Model Building Rules. ELP 96 (5th International Workshop on Extensions of Logic Programming). Springer Verlag, LNAI 1050, pages 51-65, Leipzig, Germany, March Christophe Bourely and Nicolas Peltier. DISC-ATINF: a general framework for implementing calculi and strategies. DISCO 96 (Design and Implementation of Symbolic Computational Systems). Springer Verlag, 1128, pages 34-45, Karlsruhe, Germany, September Christophe Bourely, Gilles Défourneaux and Nicolas Peltier. Building Proofs or Counter-Examples by Analogy in a Resolution Framework. JELIA 96 (5th European Workshop on Logics in AI). Springer Verlag, LNAI 1126, pages 34-49, Evora, Portugal, September Nicolas Peltier. Simplifying and Generalizing formulae in tableaux. Pruning the search space and building models. TABLEAUX 97 (Automated Reasoning with Analytic Tableaux and Related Methods). Springer, LNAI 1227, pages , Pont-à-Mousson, France, May Ricardo Caferra and Nicolas Peltier. Combining inference and disinference rules with enumeration for model building. Workshop on Model-based Reasoning, Technical Report, Nagoya, Japan, August Gilles Défourneaux and Nicolas Peltier. Analogy and Abduction in Automated Reasoning. IJCAI 97 (International Joint Conference on Artificial Intelligence). Morgan Kaufmann, pages , Nagoya, Japan, August

6 Gilles Défourneaux and Nicolas Peltier. Partial Matching for Analogy Discovery in Proofs and Counterexamples. CADE 97 (Conference on Automated Deduction). Springer, LNAI 1249, pages , Townsville, Australia, July, Ricardo Caferra and Nicolas Peltier. Model building in the crossroads of consequence and non-consequence relations. FTP 97 (International Workshop First-Order Theorem Proving). Technical Report RISC-Linz Report Series No , pages 40-44, Schloss Hagenberg, Austria, October Nicolas Peltier. An Equational Constraints Solver. CADE 98 ( Conference on Automated Deduction), Springer LNAI 1421, pages , Lindau, Germany, July Nicolas Peltier. Proof Generalization and Function Introduction. FTP 98 (International Workshop First-Order Theorem Proving). Technical Report, Schloss Wilhelminenberg, Vienna, Austria, November Ricardo Caferra, Nicolas Peltier, and François Puitg. Emphazing Human Techniques in Geometry Automated Theorem Proving: a Practical Realization. Workshop on Automated Deduction in Geometry., Springer, LNAI 2061, pages , Zurich, Switzlerland, September Nicolas Peltier. Combining Resolution and Enumeration for Finite Model Building. FTP 00 (Third International Workshop First-Order Theorem Proving). Technical Report, Universität Koblenz- Landau, pages , St-Andrews, Scotland, July Nicolas Peltier. Model Building with Ordered Resolution. FTP 00 (Third International Workshop First-Order Theorem Proving). Technical Report, Universität Koblenz-Landau, pages , St-Andrews, Scotland, July Nicolas Peltier. A General Method for Using Terms Schematizations in Automated Deduction. Proceedings of the International Joint Conference on Automated Reasoning (IJCAR 01). Springer, LNCS 2083, pages , Sienna, Italy,

7 Nicolas Peltier. A Resolution-based Model Building Algorithm for a Fragment of OCC1N =. FTP 2003 (4th International Workshop on First-Order Theorem Proving). Electronic Notes in Theoretical Computer Science, Elsevier, vol 86, n 1, Valencia, Spain, June 03. Nicolas Peltier. A More Efficient Tableau Procedure for Simultaneous Search for Refutations and Finite Models. TABLEAU 03 (International Conference on Automated Reasoning with Analytic Tableaux and Related Methods). Springer LNAI 2796, pages , Rome, Italy, September Nicolas Peltier. Some Techniques for Branch-Saturation in Free- Variable Tableaux. JELIA 04 (Logic in Artificial Intelligence, 9th European Conference), Springer LNCS 3229, pages , Lisbon, Portugal, September Ricardo Caferra, Rachid Echahed and Nicolas Peltier. A Graph Clausal Logic. FTP 2005 (international workshop on First-Order Theorem Proving), Technical Report, Universität Koblenz-Landau, Germany, September Ricardo Caferra, Rachid Echahed and Nicolas Peltier. Rewriting Term-Graphs with Priority. PPDP 06 (Eighth ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming), ACM Press, page , Venice, Italy, July Rachid Echahed and Nicolas Peltier. Narrowing Data-Structures with Pointers. ICGT 06 (International Conference of Graph Transformation), Springer LNCS 4718, pages , Natal, Brazil, September

8 3 Research Activities Our research is mainly devoted to the field of automated deduction, which is closely related to theoretical computer science, mathematical logic and artificial intelligence. Most research efforts in this domain have been devoted to the definition of powerful proof procedures. The goal is to improve the efficiency of theorem provers, by defining efficient calculi, by reducing their branching factor (i.e. the number of inferences available at a given point) and by eliminating redundant computations (i.e. inferences that are useless, either because they cannot lead to a proof or because they are subsumed by another inference). To this aim, inference rules are defined and strategies are designed to guide or to restrict the application of these rules. The properties of the obtained procedures are formally proven (mainly soundness and refutational completeness). Efficient implementation techniques are also developed (for instance term indexing or compilation of clauses) and their practical performances are investigated using experiments and case studies. For specific subclasses, decision procedures can be designed and their complexity can be investigated. These approaches are obviously essential, but we believe that it is also important for the future of automated deduction to enlarge the scope of the domain by adding to automated theorem provers some reasoning capabilities that are strictly speaking beyond proof search (or proof verification). Thus, our long-term goal is to integrate essential aspects of human reasoning that are generally absent or very informally treated in existing tools: model or counter-example building, use of analogy, discovery of regularities in the search space, proof planning 1, proof presentation,... The relative inefficiency of computers when proving difficult theorems (beside the cases in which the combinatorial aspect plays the crucial role, i.e. in which the proof merely consists in exploring a huge number of easy cases) is clearly related to a lack of understanding. Automated theorem provers do not use proof planning, do not exploit the information extracted during proof search in order to adapt their strategy, have little or no learning capabilities,... It is important to mention that some of these problems were 1 By proof planning, we mean the ability to reason at a more abstract level than the level of logical inference. The aim is to construct a (necessarily partial) proof schema, that has to be instantiated or completed (if possible) afterwards in order to yield a proof. One may refer to [14] for a pioneer work in this domain. 8

9 already identified as very important at the beginning of the field: for instance, the usefulness of model building is recognized since the 60 s [33], but it is only since the 90 s that automated model building procedures are regularly published [3]. In particular, one of the main limitations of theorem provers is that they are not able to introduce their own definitions and lemmata. The language used by automated provers is in some sense closed and almost entirely determined by the initial formulation of the problem. The inference systems handle all formulae in a almost uniform way, without reusing and structuring logical knowledge. This is a crucial difference with human beings, who introduce their own lemmata, definitions, languages, proof strategies, etc. in accordance to the considered problem (and reusing intensively prior knowledge). It is obvious that the choice of the appropriate definitions and lemmata lies at the very heart of mathematical activity, and constitutes what can be considered as its most creative part. Clearly, some problems are easier to solve if they are appropriately reformulated and/or connected to other (sometimes more general) problems. These aspects are currently neglected by running systems, due to the fact that non analytic proof procedures (for instance sequent calculus with cut) are usually not suitable for automation. We also believe that significant progresses could proceed from the conception of systems able not only to perform logical inferences but also to reason on these inferences (using a kind of meta-reasoning ). One could for instance analyse the search space in order to detect loops or to dynamically adapt the strategy. This qualitative approach to automated deduction has been developed in the Atinf projet, led by R. Caferra and now integrated in the Capp team from the Leibniz/LIG Laboratory, to which we contribute since Research Themes We work mainly on resolution-based calculi, although some results have also been obtained for tableaux-based proof procedures, or for techniques combining both approaches. We try to improve these calculi on a qualitative basis, i.e. by extending the scope and capabilities of reasoning procedures. The following lines of research have been considered. 9

10 Automated Model Building Automated model building is now recognized as a very important subfield of automated deduction. Models have many applications: they allow one for instance to verify that a theory is consistent, to prove the independence of axioms, to detect bugs in formal specifications, or to guide the search for a proof [33]. We defined several procedures for building automatically models of firstorder formulae. In contrast to proof search (for which complete procedures exist), there is no hope to find any general algorithm, since first-order logic is not decidable. Even worst, there is no single formalism able to denote all (Herbrand) models on a non-trivial signature (since there are nondenumerably many Herbrand interpretations). We distinguish two kinds of approaches. The first one we call direct, in which the models are constructed since the beginning of the search (possibly in parallel with proof search) and the indirect one in which models are obtained as a side-effect of the search for a proof, in case the theorem prover terminates without detecting unsatisfiability. In the context of tableaux-based approaches, the possibility of extracting models is related to the problem of branch-saturation, which is a very difficult issue for free-variables tableaux. Most existing tableaux-based procedures backtrack on the different instantiations [1], hence the possibility of branch-saturation and model building is lost (the procedures are only weakly complete). We propose a solution to this problem in [27]. The idea is to restrict the application of the γ-rule (i.e. the rule corresponding to universal quantification) in such a way that blind instantiation is avoided, but that repeated applications of this rule with the same term is avoided. We also defined hybrid methods combining different aspects of resolution calculi, of tableaux procedures and of enumeration-based finite model constructors [25, 16, 21]. We also defined algorithms to extract models from clause sets saturated under some restriction of the resolution calculus (i.e. from clause sets from which resolution cannot derive new, non redundant, information). Again, no general approach is possible and we studied this problem by considering successively several restriction strategies [17, 24, 20, 23]. The reader can also refer to the book that we have written on this subject with A. Leitsch and R. Caferra [3] for a more detailed description. 10

11 Resolution Calculi and Decision Procedures The indirect resolution-based model building methods mentioned above can only be used if the resolution calculus terminates. Thus, it is very natural to study the termination behavior of resolution calculi on some decidable subclasses of first-order logic [12, 8]. The termination of a complete resolution strategy on a given class obviously implies that this class is decidable. We try to identify syntactic (decidable) criteria ensuring termination. We have considered in particular equational extensions of known decidable classes PVD [19] and OCC1N [28], and more generally we have extended the termination criteria defined in [13] to a large class of equational formulae [22]. We also considered some classes decidable by hyperresolution [30], including some known classes such as BU [9] (a class obtained by translation of some modal or description logics). Avoiding Divergence of Proof Procedures: Term Schematisation In the cases in which resolution (or any other proof procedure) does not terminate, some techniques can be developed in order to avoid divergence and to construct a structural representation of the search space i.e. in the case of the resolution calculus, of the set of derivable clauses. The aim is to define extensions of the language of first-order logic ensuring termination. This is the purpose of term schematisation languages, that have been introduced in [5] in order to describe sequences of structurally similar terms such as a, f(a), f(f(a)),..., f n (a). Our work was initially motivated by the search for representation formalisms for denoting infinite Herbrand models, more expressive than the equational formulae used in [7, 4] but with similar decidability properties. One can view this approach as a first step toward an automated analysis of the search space of proof procedures. Our aim is twofold: First, we study the decidability of some problems on the term schematisation languages. Unification is in general decidable, but as in [7] we are interested in disunification problems, i.e. in more expressive theories involving quantifiers and negations [15, 26]. We solved this problem for I-terms [6] and primal grammars [11]. 11

12 Second, we define techniques for using these schematisation languages in order to improve the capabilities of proof procedures [18]. There are very few works in this field. Using term schematisation, one can detect cycles during proof search and avoid divergence by introducing adequate definitions. Reducing the Length of Resolution Proofs It is well-known that the size of the shortest resolution proof may be much longer (possibly non elementary longer) than the proofs in a more expressive calculus such as the sequent calculus or natural deduction [34]. Thus, resolution performs many redundant computations. For instance, the literals occurring in the parent clauses are systematically duplicated during the application of the resolution rule. This is very striking for particular problems such as the pigeonhole [10] for which no polynomial resolution proof exists, due to the fact that resolution alone is not able to exploit the natural symmetries between the objects occurring in the problem (all the permutations between the pigeons and holes). In contrast, any approach able to introduce lemmata such as the extended resolution of Tseitin [35] or the sequent calculus allows one to derive a proof with a polynomial number of steps. However, these approaches are not suitable for automation because the blind introduction of arbitrary definitions is not realistic. We try to overcome this problem by proposing some techniques allowing one to shorten resolution proofs, without increasing too much the branching factor of the calculus. In [29] we presented an approach allowing to share non-factorisable literals in the same clause, yielding a double-exponential reduction of proof length in some cases. In [32], we introduced another approach, able to share identical literals belonging to distinct clauses. The obtained calculus reduces the size of the proof by an exponential factor. We have also shown [31] that binary decision diagrams [2] can be simulated by extended resolution rule of [35]. This result is significant because it is known that standard decision diagrams and resolution are not comparable from a theoretical point of view. 12

13 References [1] B. Beckert and J. Posegga. Lean-TAP: Lean tableau-based deduction. Journal of Automated Reasoning, 15(3): , [2] R. Bryant. Symbolic boolean manipulation with ordered binary decision diagrams. ACM Computing Surveys, 23(3), [3] R. Caferra, A. Leitsch, and N. Peltier. Automated Model Building, volume 31 of Applied Logic Series. Kluwer Academic Publishers, [4] R. Caferra and N. Zabel. A method for simultaneous search for refutations and models by equational constraint solving. Journal of Symbolic Computation, 13: , [5] H. Chen, J. Hsiang, and H. Kong. On finite representations of infinite sequences of terms. In Conditional and Typed Rewriting Systems, 2nd International Workshop, pages Springer, LNCS 516, [6] H. Comon. On unification of terms with integer exponents. Technical report, LRI, Orsay, France, [7] H. Comon and P. Lescanne. Equational problems and disunification. Journal of Symbolic Computation, 7: , [8] C. Fermüller, A. Leitsch, T. Tammet, and N. Zamov. Resolution Methods for the Decision Problem. LNAI 679. Springer, [9] L. Georgieva, U. Hustadt, and R. Schmidt. A new clausal class decidable by hyperresolution. In A. Voronkov, editor, Automated Deduction CADE-18, volume 2392 of LNCS, pages Springer-Verlag, July [10] A. Haken. The Intractability of Resolution. Theoretical Computer Science, 39: , [11] M. Hermann and R. Galbavý. Unification of Infinite Sets of Terms schematized by Primal Grammars. Theoretical Computer Science, 176(1 2): , [12] W. Joyner. Resolution strategies as decision procedures. Journal of the ACM, 23: ,

14 [13] A. Leitsch. Deciding clause classes by semantic clash resolution. Fundamenta Informaticae, 18: , [14] R. Milner. The use of machines to assist in rigorous proof. In C. Hoare and E. J.C. Shepherdson, editors, Mathematical Logic and Programming Languages, pages Prentice-Hall, [15] N. Peltier. Increasing the capabilities of model building by constraint solving with terms with integer exponents. Journal of Symbolic Computation, 24:59 101, [16] N. Peltier. Combining resolution and enumeration for finite model building. In P. Baumgartner and H. Zhang, editors, FTP 00 (Third International Workshop First-Order Theorem Proving), pages Technical Report, Universität Koblenz-Landau., July St-Andrews, Scotland. [17] N. Peltier. Model building with ordered resolution. In P. Baumgartner and H. Zhang, editors, FTP 00 (Third International Workshop First- Order Theorem Proving)., pages Technical Report, Universität Koblenz-Landau., July St-Andrews, Scotland. [18] N. Peltier. A General Method for Using Terms Schematizations in Automated Deduction. In Proceedings of the International Joint Conference on Automated Reasoning (IJCAR 01), pages Springer LNCS 2083, [19] N. Peltier. On the decidability of the PVD class with equality. Logic Journal of the IGPL, 9(4): , [20] N. Peltier. Building infinite models for equational clause sets: Constructing non-ambiguous formulae. Logic Journal of the IGPL, 11(1): , [21] N. Peltier. A calculus combining resolution and enumeration for building finite models. Journal of Symbolic Computation, 36(1-2):49 77, [22] N. Peltier. Constructing decision procedures in equational clausal logic. Fundamenta Informaticae, 54(1):17 65,

15 [23] N. Peltier. Extracting models from clause sets saturated under semantic refinements of the resolution rule. Information and Computation, 181:99 130, [24] N. Peltier. Model building with ordered resolution: extracting models from saturated clause sets. Journal of Symbolic Computation, 36(1-2):5 48, [25] N. Peltier. A more efficient tableau procedure for simultaneous search for refutations and finite models. In TABLEAUX 03 (International Conference on Automated Reasoning with Analytic Tableaux and Related Methods). Springer LNAI 2796, September Roma, Italy. [26] N. Peltier. The First Order Theory of Primal Grammars is Decidable. Theoretical Computer Science, 323: , [27] N. Peltier. Some techniques for branch-saturation in free-variable tableaux. In J. Alferes and J. Leite, editors, JELIA 04 (Logics in Artificial Intelligence, Ninth European Conference). Springer LNCS, September [28] N. Peltier. Representing and building models for decidable subclasses of equational clausal logic. Journal of Automated Reasoning, 33(2): , [29] N. Peltier. A Resolution Calculus for Shortening Proofs. Logic Journal of the Interest Group in Pure and Applied Logics, 13: , [30] N. Peltier. Some Techniques for Proving Termination of the Hyperresolution Calculus. Journal of Automated Reasoning, 35: , [31] N. Peltier. Extended resolution simulates binary decision diagrams. Discrete Applied Mathematics, To appear. [32] N. Peltier. A resolution calculus with shared literals. Fundamenta Informaticae, To appear. [33] J. R. Slagle. Automatic theorem proving with renamable and semantic resolution. Journal of the ACM, 14(4): , October [34] R. Statman. Lower Bounds on Herbrand s Theorem. In Proc. AMS 75, pages ,

16 [35] G. S. Tseitin. On the Complexity of Derivation in Propositional Calculus. In A. Slisenko, editor, Studies in Constructive Mathematics and Mathematical Logics