Multiple-valued Logic Abstracts of the Talks

s of the Talks In the order in which the talks were given. Walter Carnielli { Non-deterministic Semantics 5 Matthias Baaz { Analytic Calculi for Many-valued Logics 5 Hiroakira Ono { Many-valued Logics as Logics without the Contraction Rule 6 Reiner H ahnle { Proof Theory of Many-valued Logic and Hardware Design 7 Gonzalo Escalada-Imaz { Determining the Truth Degree in Real Time Settings 7 Peter Vojt a s { Many-valued Logic Programming and Abduction 8 Rasim Egri { A New Fuzzy Approach to Unit Commitment in Power Systems 9 Hans J urgen Ohlbach { How to Augment a Logical System with a Boolean Algebra Component 9 Christian Ferm uller { Finite-valued Logics and Classical Proof Theory 10 Agata Ciabattoni { Cut-free Proof Systems for Logics of Weak Excluded Middle 10 Ewa Orlowska { Many-valued Substitutivity Principles 11 Felip Many a { Solving the SAT-problem in Regular CNF-formulas 11 Siegfried Weber { Conditional Objects Based on MV-algebras 12 Ulrich H ohle { Singletons and Fuzzy Partitions 12 Antonio Di Nola { One Chain Generated Varieties of MV-algebras 13 Zbigniew Stachniak { From Inferentially and Referentially Finitely-valued Systems to Resolution 14 Dirk Van Heule and Albert Hoogewijs { The Partial Predicate Calculus. A Three-valued Object Logic for the Theorem Prover Isabelle 15 Erik Rosenthal { A Linear Resolution Rule for Annotated Logics 16 3 Petr H ajek { Takeuti-Titani Logic Revisited 16 Marc Roubens { Some Basic Fuzzy Set Algebraic Operations Revisted 17 Bernard de Baets { Residuation in Fuzzy Set Theory 18 Radko Mesiar { Universal Operations in Fuzzy Logic 19 Jeffrey Paris { Semantics for Fuzzy Logic 19 Patrik Eklund { What is the Role of Logic in Biomedical Engineering? 20 Esko Turunen { BL-algebras of Basic Fuzzy Logic 21 Costas A. Drossos { Non-standard Methods in Many-valued Logics 22 Luisa Iturrioz { Non-functionally Complete n-Valued Systems Semantically Based on Posets 22 Teresa Alsinet Bernado { Fuzzy Uni cation 23 Didier Dubois { Fuzzy Logic = Many-valuedness + Partial Belief 24 Elena Tsiporkova { Possibility Theory in Modal Logic 25 Stephan Lehmke { A Comparison of Particular Logics of Graded Incomplete Truth and Graded Incomplete Knowledge 26 Brunella Gerla { The Ulam Game and MV-entropy 27 Stefano Aguzzoli {McNaughton Functions of One Variable for Automated Deduction in Lukasiewicz Logics 27 Manuel Ojeda Aciego { The TAS Reduction Method in MVL: A TAS Theorem Prover for Three-valued Logic 28 Neil Murray { Parameterized Prime Implicant/Implicate Computations for Regular Logics 28 4 Walter Carnielli, Campinas (Brazil) Non-deterministic Semantics This talk introduces a new form of combining semantics, with the double intention of, rst, to o er alternative semantic interpretations to certain less understood logic systems, and second, to combine simple logics so as to obtain other logics with richer structure. Let L be a propositional language. By a non-deterministic semantic framework for L wemean a pair ND = hT;Mi where T is a collection of transformations (called translations) from L to a family of languages L 2 governed by a set of axioms or conditions, and M = fM 2 g is a class of basic models, where L and M have the same type of similarity of L. It is possible then to de ne a non-deterministic forcing relation based on ND, generalizing from the conceptual framework of possible-worlds semantics (viewing the transformations in ND as abstractions of possible worlds, and the conditions on translations in ND as abstractions of accessibility relations). We discuss three example-cases of non-deterministic semantics, stressing the role of many-valued matrices as basic models: the rst one, showing how to associate multi-valued semantics to logics having non-truth-functional connectives (as in the paraconsistent logics), the second showing how to give interpretations to certain many-valued logics in terms of classes of logics having lower number of truth-values, and the third showing how to combine Kripke models to form new logics. Although the investigated examples involve just a nite set of basic models in M , it is very natural to extend these de nitions in terms of pre-sheaves of structures, with interesting consequences. In particular, a challenging problem is to de ne appropriate algebraic operations re ecting this construction, which would obtain new algebraic counterparts for the logics involved. Matthias Baaz, Vienna (Austria) Analytic Calculi for Many-valued Logics This lecture describes the impact of proof theoretic investigations on many-valued logics using three examples: 1. The rst-order variant of Avron's Hypersequent Calculus for in nitely valued Godel logic makes it possible to gain a clear understanding of the properties of the completeness proof of Takeuti-Titani. 2. The identi cation of fragments of Hajek's Basic Logic BL with Urquat's logic C and its extensions by residuation makes it easy to analyze derivations in BL and to demonstrate the independence of the \commutativity 5 of the minimum"-axiom from the other axioms of BL. A further prooftheoretic analysis of BL leads to an analytic formalization and consequently to decidability. 3. The possibility to deal with in nitary calculi/derivations in an e ective way is connected to quanti er elimination of the underlying metatheory. Hiroakira Ono, Ishikawa (Japan) Many-valued Logics as Logics without the Contraction Rule Logics lacking some or all of structural rules, when they are formulated in sequent calculi, are called substructural logics. The class of substructural logics includes Lambek calculus, logics without the contraction rule (BCK logics), linear logic, relevant logics and so on. The study of substructural logics will enable us to discuss these di erent kinds of logics within a uniform framework. Now, it is expected to be one of the central topics in non-classical logic. Here, we are trying to develop a general theory of logics without the contraction rule, i.e., extensions of the intuitionistic logic without the contraction rule, FLew (see e.g. [1,2] for the details). Our main tool is to use algebras which are related to the logic FLew, which are known as residuated lattices. First, we will develop a basic algebraic study of residuated lattices and will show that models of Lukasiewicz's many-valued logics and the model of product logic (from fuzzy set theory) come out naturally from their algebraic properties. Then we will discuss neighbours of the classical logic over FLew. By applying J onsson's Lemma, we can give a criterion for a logic to be a neighbour of the classical logic. As a consequence, we can show that there are in nitely many neighbours of the classical logic, each of which is di erent from any of Lukasiewicz m+ 1-valued logic with a prime number m. By discussing many-valued logics in such a broader context, we will be able to give a clearer view of them. Moreover, this approach will contribute to building a bridge between the study of substructural logics and that of many valued logics and fuzzy logic. For, though many of the results on the former logics are apparently related to the latter, as we have sketched in the above, they are sometimes unknown in the study of the latter logics. The whole contents of our study will appear in our forthcoming paper \Logics without the contraction rule and residuated lattices". References [1] H. Ono and Y. Komori: Logics without the contraction rule. Journal of Symbolic Logic, 50:169{201, 1985. [2] H. Ono: Semantics for substructural logics. In K. Do sen and P. Schroeder-Heister (eds.): Substructural Logic, pp. 259{291, Clarendon Press, Oxford, 1993. 6 Reiner Hahnle, Karlsruhe (Germany) Proof Theory of Many-valued Logic and Hardware Design We show that tableau and sequent rules for many-valued logics are closely related to many-valued decision diagrams and generalized formula decompositions as used in logic design and hardware veri cation. Although some tools and methods used in linear optimization, automated theorem proving, logic design and hardware veri cation are common, distinct notation is used and there seems to be a negligible amount of communication among these communities. We endavour to show where parallels occur, but also where di ering points of view lead to a di erent emphasis, thereby establishing a basic concordance. Gonzalo Escalada-Imaz, Barcelona (Spain) Determining the Truth Degree in Real Time Settings Until the beginning of this decade, controlling real-world industrial processes was done by classical control theory. However, currently it is widely accepted that whenever the mathematical model of the process is complex, Classical control theory fails to design a suitable controller. Given this fact, and the capital relevance of controlling industrial processes, a fragment of the AI research community has turned its attention to this promising eld. The goal consists in designing KBS-based controllers to work within the strong conditions that have to be met in real time settings (RTS), imposed by the dynamic evolution process. Controllers working in RTS must remain continuously active because each time the process state changes, a control signal must be produced in a bounded time (reactivity property). Also, some kind of temporal and approximated reasoning is needed in order to follow the inherently dynamic nature of the process and to deal with inexact information. Our approach approximating reasoning relies on regular multiple-valued logic (RML) which enables to capture vague, imprecise and partial information. Thus, our KBS is based on RML rules enriched pertinently with a temporal structure to reason about time and so, to follow the dynamic changes of the process. Finally, the underlaying general model of the controller is an AND-connector tree where with each node a truth degree is associated. The truth degree of a father AND-connector is obtained by a function whose arguments are the truth degrees of the AND-connector's sibling nodes. A compiler running o -line obtains a table data structure tied to each ANDconnector. These tables have the crucial feature that, for each con guration of the truth values modeling the current state of the process, the truth value degree of each father AND-connector is computed in O(1) time from the child nodes. This 7 performance together with the O(1) complexity of the bottom-up propagation of truth values ensures the reactivity condition. As a nal remark, we add that the proposed controller has been inspired by a real-world application framework, a pediatric intensive care unit. Peter Vojt a s, Ko sice (Slovakia) Many-valued Logic Programming and Abduction In our talk we present declarative and procedural semantics of many-valued definite logic programming (MVDLP) and prove its completeness. In order to t real world data and multiple agent behaviour, our connectives are arbitrary and subject of learning/approximation. First decision is that our program consists of implications of form A _ ([&1(B1;&2(B2; B3))]; [&3(C1;&4(C2; C3))]):cf = a 2 [0; 1] So we skip the clausal notation in DNF (because in arbitrary many-valued logics the formula :B _A needs not to be equivalent to B ! A; for another approach see work of Mundici). Di erent &i;'s in the body correspond to di erent nature of datas, depending, e.g., on di erent user environments and/or stereotypes. _ in the body serves to aggregate single ndings to the global con dence. Note that rule is equipped with a con dence factor, hence our programs are fuzzy theories in the sense of Pavelka, Nov ak and H ajek. The deduction is based on backward usage of many-valued modus ponens (B;x); (B! A; y) (A; C!(x; y)) Namely a query A? using the above rule develops into the query C! ((a;_ ([&1(B1;&2(B2; B3))]; [&3(C1;&4(C2; C3))]))? The only conditions we have to assume are Pedrycz-Gottwald conditions 2(C!;!) and 3(C!;!). We prove completeness of our semantics according to all left continuous connections (both t-operators), their nite approximations (without associativity) and aggregation (compensatory) are the operators of Zimmermann. We also show that minimal solutions of many valued de nite abduction restricts to composition of MVDLP and a linear programming problem. Relevant TEX les are at http://leibniz.math.fu-berlin.de/~vojtas. 8 Rasim Egri, Ankara (Turkey) with Ismet Erkmen A New Fuzzy Approach to Unit Commitment in Power Systems In this study, a new approach to solve Unit Commitment, one of the basic problems in power systems, using fuzzy set theory is proposed. Unit Commitment procedure provides the most economic and feasible schedules of \on" and \o " time periods for the generating units in a power system for the near future. A generation schedule is feasible if the schedule successfully meets various operational constraints of the system. However, due to the uncertainties coming from the nature of the problem, like the unknown demand and the unit production cost, an exact analytic solution is di cult to obtain. Therefore these uncertainties are modeled using fuzzy logic in our approach. The performance and the sensitivity of the proposed technique are tested on a sample power system; the results are compared with the ones that are optained by the well established Dynamic Programming method. It is observed that the performance and the robustness of the proposed method is as good as the Dynamic Programming method. Considering the linguistic simplicity and the computational e ciency of the proposed technique, it can be said that it is a potential alternative for solving the Unit Commitment problem in power systems. The present extension to this study is on the tuning of the fuzzy logic controller and the application of the new technique to a real power system, namely the Turkish Power System. Hans J urgen Ohlbach, London (UK) with Jana Koehler, Freiburg (Germany) How to Augment a Logical System with a Boolean Algebra Component We investigate how to augment a given logical system, for example an arithmetical equation solver, with a Boolean component. The atomic decomposition technique proposed in this talk reduces reasoning about the Boolean component in the combined system to reasoning in the pure basic system only. A typical instance of this scheme is a linear programming system which is to be augmented with reasoning about cardinalities of sets, or other functions mapping sets to integers. The sets and their set-theoretic relationships are axiomatized with propositional logic. Atomic decomposition then reduces reasoning about numerical attributes of these sets to arithmetic equation solving. References [1] H. J. Ohlbach and J. Koehler: How to Augment a Logical System with a Boolean Algebra Component. To appear in W. Bibel and P. H. Schmitt (eds.): Automated 9 Deduction { A Basis for Applications, volume III, Kluwer, Dordrecht. Christian Ferm uller, Vienna (Austria) Finite-valued Logics and Classical Proof Theory We report on joint work of the VGML (Vienna Group in Many-valued Logics) consisting of M. Baaz, C.G. Ferm uller, G. Salzer, and R. Zach. Finitely-valued logics are used as a tool to illuminate classic concepts of proof theory. Claiming that there exists a systematic relation between two concepts like the classical sequent calculus LK and natural deduction for classical logic, is a void statement as long as one does not consider broad classes of concepts of which the ones to be compared are just particular instances. We use the family of all nitely-valued logics with arbitrary truth functional connectives and distribution quanti ers as basis to substantiate the claim that important concepts of Gentzen-style proof theory can be \derived" from each other. In particular we consider the relation between many-placed sequents and multi-conclusional natural deduction systems, truth tables and operators for reducing cuts with corresponding types of cut-formulas, and the relation between the well-known syntactical restriction that turns LK into LJ, on the one hand, and Kripke semantics for logics of intuitionistic type, on the other hand. Tools like signed resolution allow to automatize the various translations of concepts. The system Multlog demonstrates this strikingly: Given any nite set of truth tables as input Multlog outputs various types of calculi along with soundness and completeness proofs as a LATEX documents. Agata Ciabattoni, Bologna (Italy) with Dov Gabbay, London (UK), and Nicola Olivetti, Torino (Italy) Cut-free Proof Systems for Logics of Weak Excluded Middle In this talk we explore logics which arise from well known systems by weakening the excluded-middle principle. In particular, we introduce cut-free hypersequent calculi for the LQ logic, obtained by adding to Intuitionistic Logic the weak law of the excluded middle, that is :A _ ::A, and for systems, called Wn, obtained by adding to a ne Linear Logic (without exponential connectives) the n-weak law of the excluded middle, that is :A_(A A) (n 1 times). For n = 3, the system Wn coincides with 3-valued Lukasiewicz Logic; for n > 3, it is a proper subsystem of n-valued Lukasiewicz Logic. Then, our calculi can be seen as a 10 step forward in order to nd hypersequent calculi, in which the cut-elimination theorem holds, for nitely-valued Lukasiewicz Logics. Ewa Orlowska, Warsaw (Poland) with Mihir Chakraborty, Calcutta (India) Many-valued Substitutivity Principles We consider two families of many-valued logics: logics whose many-valuedness is numerical and logics with non-numerical many-valuedness. These classes of logics include, among others, many-valued information logics, Rosser-Turquette logic, fuzzy logic. For several logics from each of the two groups we propose a semantics for the identity predicate and we prove validity of the underlying substitutivity principles. We also discuss substitutivity principles that hold for p-compatible identities in algebraic theories and substitutivity principles in Euni cation theory. Felip Many a, Lleida (Spain) with Ram on B ejar, Lleida (Spain), and Gonzalo Escalada-Imaz, Barcelona (Spain) Solving the SAT-problem in Regular CNF-formulas First of all, we present a Davis-Putnam-style satis ability checking procedure for regular propositional formulas in conjunctive normal form (CNF-formulas). For the sake of e ciency, we have equipped the procedure with a suitable data structure for representing formulas. This data structure allows, for example, to detect the existence of unit clauses in constant time and to apply the regular unit clause rule with a linear-time worst-case complexity. Second, we de ne several regular branching rules: An optimized version of Hanhle's branching rule, a regular version of the positive clause rule and a branching rule based on the concept of maximal set of truth values. Third, we describe a generator of random k-SAT instances of the satis ability problem in regular CNF-formulas. Fourth, we show the experimental results obtained after executing an implementation of the algorithm on a distribution family of randomly generated regular 3-SAT instances: It turns out that for 3 truth values and 60 propositional variables, near the ratio C=V = 6:17 { where C and V are, respectively, the number of clauses and propositional variables { we nd the hardest instances. The number of nodes of the proof tree generated by the procedure increases exponentially near this ratio and quickly decreases beyond than. For 5 truth values and 60 propositional variables, we observe the same e ects near the ratio C=V = 8:17. 11 Finally, we show that the 2-satis ability problem in regular CNF-formulas canbe solved in polynomial time using a re nement of the proof procedure proposed.Siegfried Weber, Mainz (Germany)Conditional Objects Based on MV-algebrasAbstract not available.Ulrich Hohle, Wuppertal (Germany)Singletons and Fuzzy PartitionsLet M = (L; ; ) be a GL-monoid [1, Section 5] satisfying the additional dis-tributivity law(î2I i) = î2I(i) for all 2 L and all f i j i 2 Ig L :Typical examples are complete Heyting algebras or completeMV -algebras. TheM -valued interpretation of the formalized theory of identity and existence [2, Sub-section 3.2] leads to the following concept of MV -valued sets (X;E) where X isa non-empty set and E : X X 7 ! L is a map (so-called M-valued equalityon X) subjected to the following axioms(E1) E(x; y)E(x; x) ^ E(y; y)(Strictness)(E2) E(x; y) = E(y; x)(Symmetry)(E3) E(x; y) (E(y; y)! E(y; z))E(x; z)(Transitivity)A subset ffi j i 2 Ig of LX is called an L-fuzzy partition of the universe X iffi j i 2 Ig satis es the following disjointness condition:_x2X((_̀2I f`(x))! fi(x)) fj(x)[E(fi) (ŷ2X fi(y)! fj(y))] ^ [E(fj) (ŷ2X fj(y)! fi(y))]where E(g) = _fg(x) j x 2 Xg. Then the following theorem holds [2, Theo-rem 4.2.2]: A subset Z = ffi j2 Ig of LX is an L-fuzzy partition of X i thereexists an M -valued equality E on X such thatEvery map fi 2 Z is a singleton of (X;E),E(x; x) = Wi2I fi(x)for all x 2 X.12 References[1] U. Hohle: Commutative, residuated L-monoids. In U. Hohle and E. P. Klement(eds.): Non-classical Logics and Their Applications to Fuzzy Subsets, pp. 53{106,Kluwer, Boston, 1995.[2] U. Hohle: On the fundamentals of fuzzy set theory. J. Math. Anal. Appl.,201:786{826, 1996.Antonio Di Nola, Naples (Italy)with Ada LettieriOne Chain Generated Varieties of MV-algebrasMV-algebras constitute a generalization of Boolean Algebras and arise from themany-valued logic of Lukasiewicz in the same manner as boolean algebras arisefrom two-valued logics.An MV-algebra is an algebraic structure A = (A; ; ; 0) such that (A; ; 0)is an abelian monoid and the following identities hold: x = x; x 0 = 0 ;(x y) y = (y x) x.If we set x y = (x y ) , x ^ y = (x y ) y, and x _ y = (x y ) y,for every x; y 2 A, then (A;_;^; 0; 1) is a bounded distributive lattice, which iscalled the reduct of A and denoted by L(A). Boolean algebras coincide with MV-algebras satisfying the additional identity x x = x. Let A be an MV-algebra.The set B(A) = fx 2 A j x x = xg is a boolean algebra. Actually it is thegreatest boolean subalgebra of A.The variety MV of all MV-algebras coincides with the variety HSP[0,1] gen-erated by the MV-algebra de ned on the real unit [0; 1] as follows:x y = min(1; x+ y); x y = max(0; x+ y 1); x = 1 x :The main results presented here, are the following:1. A characterization of the sub-varieties satisfying the amalgamation prop-erty; actually, they are those that have exactly one generator.2. There exists a categorical equivalence between the sub-variety generated bya nite chain MV-algebra with n elements and the category whose objectsare pairs (B;Rn) where B = (B;+; ; ; 0; 1) is a boolean algebra and RnBn such that:i) If (b0; b1; : : : ; bn 1) 2 Rn, then b0 b1bn 1;ii) If (b0; b1; : : : ; bn 1) 2 Rn, then (bn 1; bn 2; : : : ; b0) 2 Rn;13 iii) If (b0; b1; : : : ; bn 1); (c0; c1; : : : ; cn 1) 2 Rn, then(b0 + c0;...bk + ck +Pi+j=k 1 bi cj;...bn 1 + cn 1 +Pi+j=n 2 bi cj)is an element of Rn;iv) (b; b; : : : ; b) 2 Rn for all b 2 B.3. Every relation Rn can be represented by a boolean algebra B and a vectorof ideals of B.4. A characterization of the automorphisms of B(A) which can be extendedto automorphisms of A.5. Following I. R. Goodman, H. T. Nguyen, and E. A. Walker, we introducethe notion of Abstract Conditional MV-Space, as an MV-algebra having thereduct lattice to be an Abstract Conditional Space. These seem to be thesuitable algebraic structures coping with the notion of conditional events inthe framework of Lukasiewicz logic. Indeed we provide a characterizationof such a class of MV-algebras.6. Finally we characterize the group of automorphisms of an n-valued algebra.Zbigniew Stachniak, Toronto (Canada)From Inferentially and Referentially Finitely-valued Systems to Res-olutionAre there many-valued logics or just logics with many-valued semantics? Is thesemantic apparatus of many truth-values a distinctive feature of many-valuedlogic or is it the intended interpretation of an applied language, a philosophicalcommitment, that necessitates the choice of multiple-valued interpretation forthe language? These issues seem to be still unresolved in spite of almost 80 yearsof continuous research in the area of many-valued logics.In the maze of opinions concerning the de ning features for the class of many-valued logics, there is still one path that can be explored. The bivalent nature ofclassical logic is clearly represented in the theses of this calculus. Other logicalsystems (such as Lesniewski's Protothetic or Ontology) are formalized in such away as to make the intended interpretation of an applied language evident fromthe start, at the proof theoretic level. 14 In this talk we are concerned with the search for a proof-theoretical evidenceof nite-valuedness within the class of cumulative inference systems. We de neand investigate the notion of an inferentially many-valued inference system. Inthis de nition we try to capture the idea of the proof-theoretical representationof truth-functional many-valued semantics. We contrast the notion of inferentialmany-valuedness with semantic (or referential) many-valuedness.Although the notions of inferential and referential many-valuedness do notcoincide, standard (i.e., structural and compact) inferentially or referentiallynitely-valued logics are resolution logic in the sense of [1]. Hence, every suchsystem P has a resolution counterpart, i.e., there exists a deductive proof systembased on the (non-clausal) resolution principle which is refutationally equivalentto P.References[1] Z. Stachniak: Resolution Proof Systems: An Algebraic Theory. Kluwer AcademicPublishers, 1996.Dirk Van Heule and Albert Hoogewijs, Gent (Belgium)The Partial Predicate Calculus. A Three-valued Object Logic for theTheorem Prover IsabelleIn this talk we de ne the 3-valued Partial Predicate Calculus (PPC) as an objectlogic for the generic theorem prover Isabelle. We will focus on the propositionallogic. It will be necessary to add some new de nitions and rules to the semanticsof PPC in order to use the proving mechanisms of Isabelle in natural deductionstyle.The truth-tables for the logical operators negation (:) and conjunction (^)correspond to Kleene's notation. The logical operators are extended with a non-monotone operator to express the de nedness (T ) or unde nedness(F ) of a formula.:T F TF T TU U F^ T F UT T F UF F F FU U F UThe de nition of validity and consequence di ers from most other 3-valued logics.In PPC, a valuation is a mapping V : FormPPC 7! fT;F;Ug with FormPPC beingthe well-formed formulas of PPC. M is a model for a set of formulas i forall 2 : V( ) T . A formula is valid for a model M (M j= ) iV( ) 2 fT;Ug. A formula is valid i it is valid for all models of PPC. A formula15 is a consequence of a set of formulas ( j= ) i for all models M of ,M j= .By choosing Isabelle as theorem proving engine for PPC, we had to transformPPC into a natural deduction calculus, PPC nat. Because the \assumption cal-culus" of PPC is not really a sequent calculus nor a natural deduction calculus,we had to reorganize, rede ne and modify most of the rules. Also we had to addnew rules.Due to the de nition of validity and consequence in PPC, we had to expressthe di erence between a formula A at the left-hand side and a formula B at theright-hand side of `, but also the di erence between A ` B and ` A` B . Thiswas done by introducing a new connective \!", where !A (A^ A) means \A isa true formula (trueF)".In this talk we discuss the translation of the inference rules of PPC into thoseof PPC nat. We show some results including the use of the cut-rule and themodi cation of the modus ponens.Erik Rosenthal, New Haven (USA)A Linear Resolution Rule for Annotated LogicsSigned logics provide a classical logic framework for reasoning about multiple-valued logics. In particular, signed resolution restricted to regular sets { upsetsor complements of upsets { uni es the resolution and reduction rules of annotatedlogics. However, with these inference rules, the linear restriction, which is desir-able in a variety of settings, including logic programming, precludes restriction toregular signs. A hyperresolution-like extension that preserves regularity is avail-able, but linearity is arti cial since, in e ect, the reduction steps are \hidden."The inference rule f-resolution, currently under development, may be re-garded as a replacement for annotated resolution/reduction. If the domain oftruth values is linear, reduction is never necessary, and f-resolution is identicalto annotated resolution. Otherwise, f-resolution admits linear proofs, and pre-liminary analysis indicates that it is more e cient than either signed resolutionor annotated resolution/reduction with regard to both proof space and searchspace.Petr Hajek, Prague (Czech Republic)Takeuti-Titani Logic RevisitedIn my forthcoming book [1], I pay much attention to the study of three fuzzy logics(both propositional and predicate) given by the well known most important con-tinuous t-norms ( Lukasiewicz, Godel, product). The t-norm de nes the semanticsof conjunction &, its residuum is the truth function of implication !; various16 other connectives are de nable. Analyzing the approach of Takeuti and Titani [2]I develop (in Chap. IX, Sect. 1 of [1]) a predicate logic having three di erent con-junctions (as above), three corresponding implications and two correspondingnegations as well as rational truth constants (as in Pavelka logic). This logic hasobvious semantics and is not recursively axiomatizable (since Lukasiewicz predi-cate logic over [0; 1] is not). I present an axiomatization whose axioms are thoseof the three logics for the respective connectives, nitely many axiom schemesfor truth constants and for connectives from di erent logics. Deduction rules aremodus ponens, (double) generalization and an in nitary rule whose simplest casereads:'! _ r for all r > 0'!Here _ is strong Lukasiewicz disjunction. Completeness of the logic is proved.References[1] P. Hajek: Metamathematics of Fuzzy Logic. Kluwer, 1998.[2] G. Takeuti, S. Titani: Fuzzy logic and fuzzy set theory. Archive for Math. Logic32 (1992) 1-32.Marc Roubens, Liege (Belgium)Some Basic Fuzzy Set Algebraic Operations RevistedThe most elementary operations for usual sets, as the union and the intersectionof any two sets and the complement of any set, can be generalized for the sameoperations on fuzzy sets with the use of a de Morgan triple (T; S; n) where T isa t-norm and S the associated t-conorm de ned with the strict negation n.It is well known that these extensions give impossibility results if one wantsthe idempotence property and the contradiction law to hold simultaneously.Pexiderized extensions of distributivity properties and the classical booleanrelation (A \ B) [ (A \BC) = A are considered.In terms of functional analysis, one has to solve the following equation ifdistributivity is considered:S1[x; T4(y; z)] = T5[S2(x; y); S3(x; z)]which is equivalent toS[x;min(y; z)] = min[S(x; y); S(x; z)] :Classically extending the relation (A \B) [ (A \ BC) = A givesS[T (x; y); T (x;ny)] = x(1)17 which is known (Alsina, 1985) to have no de Morgan solution.A pexiderized extension of (1) givesS[p(x; ny); i(x; y)] = x :(2)Solutions of (2) are explored for the case that S is a continuous t-norm asso-ciated to a strict negation n, p and i being symmetric functions on [0; 1]2.Bernard de Baets, Gent (Belgium)Residuation in Fuzzy Set TheoryAn extensively studied topic in fuzzy set theory is the solution of fuzzy relationalequations. We put these equations in a broader perspective and study them froma purely lattice-theoretic point of view. We discuss the more general sup-O andinf-O equations, where O a logical operator. These equations can be consideredas polynomial lattice equations. The main issues that have to be addressed arethe following:the logical operators that can be considered,the types of complete lattices in which these equations can be solved,additional conditions that have to be imposed on these logical operators,a representation of the solution sets of these equations.We deal with four primitive logical operators: conjunctors, disjunctors, impli-cators and co-implicators. The complete lattices in which these equations canbe solved include the distributive, complete lattices of which all elements areeither join-irreducible or join-decomposable, and/or meet-irreducible or meet-decomposable. Important representatives of the latter class of lattices are theproduct lattices ([0; 1]n; ). The additional conditions imposed on the logical op-erators are mostly quite natural, and concern some morphism behaviour or sometype of surjectivity of their partial mappings.The representation of the solution sets of these equations is the key issuein solving fuzzy relational equations. If one succeeds in discovering an order-theoretic structure that can be used to fully describe the solution set of individualsup-O or inf-O equations, and that moreover behaves nicely under intersectionand Cartesian product, then one is able to tackle more complex problems suchas systems of equations, families of independent equations, : : : We will showthat crowns and root systems are the concepts one has been looking for. Thisrepresentation is based on our knowledge of the extremal solutions. A compactdescription of these speci c solutions requires a generalization of the theory ofresiduation from triangular norms to more general logical operators.18 Radko Mesiar, Bratislava (Slovakia)Universal Operations in Fuzzy LogicThe associativity as well as the commutativity of basic operations in fuzzy logicbecomes a non-trivial problem when we have to deal with in nitely many inputs.The problem of an in nite extension of a conjunction operator (modelled by at-norm) or of a disjunction operator (modelled by a t-conorm) may meet di cul-ties similar to those ones which occur by the extension of the common additionon the real line to the series sum. An axiomatic approach to a universal con-junction (disjunction) operator acting on arbitrary system of inputs from [0; 1] isproposed. Any universal conjunction operator when acting on nite systems ofinputs is shown to coincide with a t-norm, i.e. each universal conjunction operatoris an extension of some t-norm. By duality, universal disjunction operators areextensions of t-conorms. Two kinds of possible extensions of a given t-norm Tare proposed, namely the weakest T and the strongest T . Each T is shown tobe a universal conjunction operator. However, min 6= inf. On the other hand,min = inf, but T needs not to be a universal conjunction operator. Conditionsunder which T = T are given. Complete characterization of t-norms for whichT is a universal conjunction operator is given, as well as a necessary condition,and a su cient condition (namely, the right-continuity of T ). The operator Twith respect to a continuous T is described.Je rey Paris, Manchester (UK)Semantics for Fuzzy LogicThe assumption that an agent's belief value, w( ) (or any of the other termsused to denote the agent's degree of con dence, or certainty) of a conjunctionis a xed function F^ of the belief values of the two conjuncts (and similarlyfor disjunctions and negations) is rather commonly applied in the constructionof expert systems. Similarly one nds the same assumption being made in socalled logics of vagueness where the values assigned to sentences are intended torepresent their \degrees of truth". This raises the question of explaining whatthese values w( ) actually mean, and why the functions F employed to combinethem are appropriate. Conventionally the choice of, say F^, has been guidedby desirable properties (e.g. associativity, monotonicity) and such considerationslead us naturally to focus on the continuous t-norms for F^.In my talk I consider possible semantics. or meanings, for w( ) and thecorresponding functions F^, etc. both for \belief values" and for \degrees oftruth". In the former case w( ) is interpreted as an approximation to the expectedprobability, Prob( ) of and F^ etc. chosen to minimise the error resulting fromusing w( ) in place of Prob( ). In the second case, of w( ) as \degree of truth"of , a meaning is given to w( ) by identifying it with a notion of the acceptability19 of , more precisely the proportion of independent arguments which an agentcould muster for , as opposed to against . Versions of \independent" areconsidered which gave min and product as the appropriate F^ etc.Patrik Eklund, Umea (Sweden)What is the Role of Logic in Biomedical Engineering?Human reasoning within clinical decision making, or in general, within decisionmaking and observations in the health care domain, involves not only intelligence,but also ethics and law that constrain decision making in various ways. Tradi-tionally, statistics and probabilistic approaches, as means for supporting decisionmaking processes, are well established in the health care domain, even in de ningboundaries related to legality of actions. Compared to statistical methods, alsonumerical analytic means, as represented, e.g., by pattern recognition, clusteringand also neural networks, provide tools for approximation and optimisation thatare widely accepted by health care professionals. However, logical approaches,as building upon notions within resolution, algebra and equalities, do not seemto have experienced a breakthrough and acceptance so as to be recognised asproviding supporting techniques in development of decision support systems.Considering decision making as supported, respectively, by statistics, numer-ics and logic, we certainly agree that the conglomerates of methods provided ineach domain overlap to some extent, and also that non-overlapping parts com-plement each other. However, very little seems to be known about similarities,formal relationships and interplay between various techniques. Some bridges,however, do exist. Propositional logic inference and neural network feed-forwardcan be seen as identical in a general framework, thus providing logical understand-ings of numerical computations, and similarly enabling parameter optimisationswithin expert systems. Conditionality in probabilistics is obviously related toimplication, even if it is still not clear how formal transformation rules, rewritingprobabilistic networks in to rule bases and vice versa, should be de ned.The di culties within the many-valued logic communities to demonstrate ap-plicabilitity, partly stems from hubris within the AI community some decades ago,when (heuristic) logic based expert systems, as strongly connected to knowledgeacquisition techniques, were expected to have huge potentials for future decisionsupport not only in health care but in general for almost any kind of decision sce-narios. However, the logic community is also still to blaim in that the objectivesin development of logical systems relate exclusively to soundness/completeness,satis ability, NP-completeness, and notions alike. Sometimes it even seems inlogic that we restrict to reaching understandings of what happens if we change\this" to \that" or \these" to \those" in the syntactic or semantic apparatus ofsome logical system, thus with very little relation to the objectives as understoodand being enforced in human reasoning for a particular decision making situation.20 Esko Turunen, Lappeenranta (Finland)BL-algebras of Basic Fuzzy LogicBasic logic algebras (BL-algebras) have been invented recently by Hajek [3] inorder to provide an algebraic proof of the completeness theorem of a class of[0; 1]-valued logics familiar in fuzzy logic framework. BL-algebras arise as Lin-denbaum algebras from certain logical axioms in a similar manner as MV-algebras(cf. [1, 2, 4]) do from the axioms of Lukasiewicz logic. In fact, MV-algebras areBL-algebras. The converse, however, is not true. It follows from a result of Hohlethat BL-algebras with involutory complement are MV-algebras. In this studywe start a similar study of BL-algebras as Belluce [1], Gluschankof [2], Hoo [4]and others have done in the theory of MV-algebras; there the basic tool is idealtheory while in BL-algebras, because of lack of a suitable algebraic addition, wehave to deal with deductive systems. In MV-algebra theory, deductive systemsand ideals are dual notions; there deductive systems are also called lters but, inorder to avoid confusion, we prefer to talk about deductive systems. We intro-duce locally nite BL-algebras and prove that such algebras are MV-algebras. Asone may expect, there is a one-to-one correspondence between deductive systemsand congruence relations of a BL-algebra. We prove that a deductive systemis maximal if, and only if, the corresponding quotient algebra is a locally niteMV-algebra. This fact implies one of the main result of our study: semi-simpleMV-algebras are, in the sense of Chang and Belluce, the only BL-algebras thatare representable by a system of fuzzy subsets of a set. However, as proved byHajek [3], all BL-algebras are representable by linear BL-algebras. We have somepreliminary results in order to characterize all linear BL-algebras. We introduceco-annihilators and prove some of their elementary properties; all these resultswill be an introduction for a future, more detailed analysis on BL-algebras. Anextended article containing all proofs will be published in Mathware and SoftComputer.References[1] L. P. Belluce, Semi-simple and complete MV-algebras, Algebra Universalis 29(1992), 1{9.[2] D. Gluschankof, Prime deductive systems and injective objects in the algebrasof Lukasiewicz in nite-valued calculi, Algebra Universalis 29 (1992), 354{377.[3] P. Hajek, Metamathematics of Fuzzy Logic. Inst. of Comp. Science, Academy ofScience of the Czech Republic, Technical report 682 (1996).[4] C. S. Hoo, MV-algebras, ideals and semisimplicity, Math. Japonica 34 (1989),563{583.21 Costas A. Drossos, Patras (Greece)Non-standard Methods in Many-valued LogicsIn this talk, we examine the relationships of MV-algebras and non-standard math-ematics (in nitesimal and Boolean analysis).MV-algebras include, in an essential way, in nitesimals, e.g., Rad(A) consistsof pure in nitesimals and the standard in nitesimal which is 0. Also Di Nola'srepresentation theorem involves the nonstandard unit interval. After all these,it was felt that a systematic study of relationships of non-standard methods andMV-algebras was needed. We try to lay down some basic frameworks to studyMV-algebras from a non-standard point of view. We introduce hyper nite MV-algebras, and it is expected that these MV-algebras are the image through theMundici Gamma functor, of a nite S-dense subset of R. One can see thatthere are a lot of -concepts. We have -completeness, -simple, -Archimedian,-semi-simplicity, -good sequence, etc. It is shown that the -Belluce represen-tation theorem is not equal to the Di Nola representation theorem. It is shownthat there is a hyper nite MV-algebra which includes all rational points in [0; 1]but no irrational appears in it. This is very important in de ning hyper niteMcNaughton functions. It is shown also that Chang's MV-algebra is external.A di erent proof is given that the Boolean power of [0; 1] is a semi-simple MV-algebra. Finally, we examine in which way stochastic MV-algebras can be de ned.Luisa Iturrioz, Lyon (France)Non-functionally Complete n-Valued Systems Semantically Basedon PosetsThere are|at least|two traditional ways to study n-valued formal systems:functionally complete or not; namely, Post logics of order n or Lukasiewicz log-ics of order n (or even Lukasiewicz-Moisil logics of order n). In the best knownn-valued systems, the basic algebra A given functionally completeness is an n-element chain A = fe0; e1; : : : ; en 1g.In the past, many computer applications needed n-valued functionally com-pleteness and this is one of the reasons why the theories of Post algebras andgeneralized Post algebras have been developed. Applications were closely relatedto the design of switching and electronic circuits, and in circuits usually the con-stant operations are at our disposal (at no extra cost). Also, let p be a primenumber; in the Post algebra of order p the arithmetic operations of addition andmultiplication can be de ned. In this way, n-valued logics have been used inarithmetic units.The contents of the book [1] is a typical example of our proposal. Even \thenomenclature for the so-called `constants' fe0; e1; : : : ; en 1g was chosen with nvoltage levels in mind" [1, page 35]. 22 In the n-valued case, constants are added to obtain functionally completeness.Another useful property given by the constants fe0; e1; : : : ; en 1g is the followingadditional symmetry (George Epstein, 1960): A Post algebra of order n is duallyisomorphic with itself under the mapping: (x) = Wn 1i=1 (di(x) ^ ei) where thedi(x) are generalized complementation operators.At the present, computer science applications are more diversi ed than in thepast, and more exible (discrete) tools are needed. Following the non-functionallycomplete way, we intend to generalize the interesting and powerful works of He-lena Rasiowa concerning Perception Logics: rst, in Lukasiewicz style, that is,losing constants but keeping the symmetry; second, losing constants and symme-try. In both cases, only for nite posets. These rst order systems are semanti-cally based on posets which are interpreted as posets of co-operating intelligentagents.References[1] David C. Rine (ed.): Computer Science and Multiple-valued Logic, Theory andApplications. North-Holland, 1977.Teresa Alsinet Bernado, Lleida (Spain)Fuzzy Uni cationThe uni cation process is usually associated with logic programming languages,which lets us compare the arguments of two predicates and infer new informationthrough some deduction rule. In classical logic the uni cation problem is well-de ned, and the set of techniques that implements it di ers on the complexityassociated with the process. A new problem arises when we add vague, incompleteor imprecise information to the system. It is necessary to de ne some measurein order to establish the degree of similarity between two fuzzy sets.Our aim is to include the data type fuzzy set in a logic programming systembased on a family of in nitely-valued logics. Within this context, we addressthe problem of uni cation involving fuzzy constants and linguistic variables insystems where a separation between general and speci c knowledge can be made.We describe an algorithm for the construction of the most general fuzzy uni erof a pair of atomic formulae based on the semantics of terms. For this purpose,taking account of the distinction between general and speci c knowledge, wede ne the semantic structure associated with variables and linguistic variables of alogic program and we specify the data structures used by the uni cation algorithmfor its representation. With the aim of quantifying the inclusion degree betweentwo fuzzy constants, we establish a similarity measure based on the interpretationof the membership function of a fuzzy set as the degree of similarity between theelements of the universal set and the set of prototypes. Moreover, we formalize23 the de nitions of fuzzy substitution, fuzzy uni er and most general fuzzy uni er.Finally, taking the origin of the variable instances into account, we develop ameasure of the uni cation degree of two atomic formulae through their mostgeneral fuzzy uni er.With reference to future lines of work, we must consider the inclusion offunctions in the fuzzy uni cation algorithm and the integration of the algorithminto the declarative programming environment for in nitely-valued logics. Therst problem lies in de ning the semantics of functions and the relationshipsbetween them and the fuzzy constants and the linguistic variables. To deal withthe second problem, we must de ne the semantics of multiple-valued facts andrules of a logic program and develop a proof procedure. For this purpose, we mustformalize the notion of interpretation, model, logical consequence and logicalinference of a rst-order multiple-valued language when the terms express vague,incomplete or imprecise information.Finally, we would like to point out that the system may be particularly use-ful within the framework of fuzzy deductive data bases and diagnosis systems inwhich symptoms are not always described precisely. It may be also valuable forthe implementation of dialog agents in multi-agent systems when vague, incom-plete or imprecise information about the real world and approximately speci edrules are needed.Didier Dubois, Toulouse (France)with Stefan Lehmke, Dortmund (Germany)Fuzzy Logic = Many-valuedness + Partial BeliefTwenty years ago Zadeh [3] claimed that fuzzy logic: : : is a logic of approximate reasoning [: : : ] whose distinguishing featuresare (i) fuzzy truth-values expressed by linguistic terms, (ii) imprecise truth-tables.This statement has been misregarded by some logicians who assume that fuzzylogic equals many-valued logics. Of course it is clear that the algebraic settingof fuzzy set theory is precisely the one of many-valued logics. However, it shouldnot lead to a confusion between fuzziness in the wide sense and many-valuednessstricto sensu. When Zadeh points at fuzzy truth-values, he really refers to fuzzysets of truth-values expressing imprecision, and not only to graded truth-values.Fuzzy truth-values are possibility distributions that combine many-valued truthand many-valued plausibility levels.To support this claim, it can be shown that each logic possesses an embeddedrepresentation of belief, and not only a calculus of truth-values. In propositionallogic, belief is two-valued (certainty versus lack of certainty) and is not self dual(to say that a proposition p is not believed is not equivalent to say that :p24 is believed, contrary to probabilistic belief). Belief values are closely relatedto consequencehood and are not compositional [1], even in classical logic. Wepropose to consider fuzzy logic as relying on two ordered sets: a truth-set T thatdescribes the range of propositional variables, and a plausibility set D that is therange of uncertainty levels. Fuzzy logic handles labelled formulas [p; ] where pis a formula in a given language and is a non-decreasing function from T to D,such that full truth is fully plausible. To write [p; ] is a means of declaring thatp holds in some sense. Semantic evaluation involves both D and T . Namely, aninterpretation of I is a model of [p; ] to a degree that represents the plausibilityof the truth-value of p in I, as computed with . The degrees of modelhood arein the plausibility set and (I(p)) represents the degree of plausibility that I isthe correct state of the world when the agent's knowledge is described by [p; ].It must be stressed that the set of fuzzy truth-values is not a truth-set. Labelsare not compositional truth-values in a special many-valued logic. Fuzzy logic isboth a many-valued logic and a logic of partial belief.The proposed framework with labels attached to formulas encompasses thestatement of propositions in classical logic (where D = Tf0; 1g), of some signedformulas in many-valued logics (where truth is many-valued but plausibility isbinary), and of weighted formulas in possibilistic logic (where truth is binary andplausibility is many-valued). It also o ers a natural setting to extend possibilisticlogic and many-valued logics conjointly [2] into a genuine fuzzy logic, that logi-cians may acknowledge as being a logic, and that remains faithful to the originalmotivation of Zadeh.References[1] D. Dubois and H. Prade: Can we enforce full compositionality in uncertaintycalculi? In: Proceedings of the 12th National Conference on Arti cial Intelligence(AAAI '94), Seattle, USA, pages 149{154, 1994.[2] S. Lehmke: On general uncertainty logics. Research Report 635, Department ofInformatics, University of Dortmund, Germany, 1997.[3] L. A. Zadeh: Fuzzy logic and approximate reasoning. Synthese, 30:407{428,1975.Elena Tsiporkova, Plovdiv (Bulgaria)with Bernard De Baets and Veselka Boeva, Gent (Belgium)Possibility Theory in Modal LogicIn this talk, the modal logic interpretation of plausibility and belief measuresde ned on an arbitrary universe of discourse, as proposed by Harmanec, Klir,Resconi, St. Clair, and Wang, is further developed by applying notions from set-valued analysis. The attractiveness of this multi-valued interpretation lies in25 its natural analogy to the original approach of Dempster. In fact, as far as wemanage to construct a multi-valued mapping that establishes a correspondencebetween the set of possible worlds and the set of atomic propositions, we auto-matically obtain without imposing any restrictions on the accessibility relation,that this mapping induces a plausibility measure and a belief measure. Thismulti-valued interpretation is explained in detail in the case of possibility andnecessity measures. Moreover, it is shown that by restricting a model of modallogic in a non-trivial way, the possibility and necessity measures induced by thisrestricted model coincide with the conditional possibility and necessity measuresof the measures induced by the original model, corresponding to the set used forrestricting.Stephan Lehmke, Dortmund (Germany)with Didier Dubois and Henri Prade, Toulouse (France)A Comparison of Particular Logics of Graded Incomplete Truth andGraded Incomplete KnowledgeWe study the relationships between logics of graded incomplete knowledge (aprominent representative being possibilistic logic) and logics of graded incompletetruth (for which logical systems in the style of J. Pavelka are examples).The two di erent types of logics are intended for the representation of twodi erent aspects of uncertainty (in the linguistic sense) in natural language, whichcan be given by degrees.In a logic which is intended purely for the representation of graded incompleteknowledge, it is assumed that a logical proposition is always either completelytrue or completely false, but for the knowledge to be represented, we are able toassign degrees of uncertainty about (respectively trust in) whether a statementfrom a knowledge base is indeed true.In a logic which is intended purely for the representation of graded incompletetruth, we assume that not only can every proposition be true to a degree lyingbetween complete truth and complete falsehood, but for the knowledge to berepresented, we are able to assign degrees of truth which have to be assumed forthe statements from a knowledge base to be considered valid.In the present talk, we compare and di erentiate the speci c and unique prop-erties of logics of graded incomplete truth and logics of graded incomplete knowl-edge, by demonstrating these for two speci c examples: possibilistic logic withnecessity-weighted formulae and Lee's fuzzy logic with truth values as weights.26 Brunella Gerla, Salerno (Italy)The Ulam Game and MV-entropyIn the Ulam game playerAmust guess an element of a nite set S, asking playerBquestions, who can only answer \yes" or \no" but can lie up to k times. Thetask is to nd the secret element with as few questions as as possible. The Ulamgame can be represented in Lukasiewicz logic and in particular it can be seen asa combination of classical information (questions are subsets of the set S) andmulti-valued information (answers are elements of an appropriate MV-algebra).This game is strongly connected with the problem of nding an error-correct-ing code, and in particular one can refer to a set on which a probability distribu-tion is de ned, in order to simulate better the random nature of the source. Inthis case it is possible to extend a result holding in the classical case. In fact, inthe classical case, where the game is also called Twenty Questions Game, the beststrategy is the balanced one, i.e., the strategy in which the questions "divide"the probability distribution; this can be shown with some observations on theentropy of the system.In accordance with the de nition recently developed in the literature, it ispossible to introduce a generalization of the notions of probability and conditionalprobability, and so it is possible to give a de nition of entropy that extends theclassical one.The result is that in the non classical case, too, best strategies (in the averagecase) are those that balance the probability.Stefano Aguzzoli, Siena (Italy)McNaughton Functions of One Variable for Automated Deductionin Lukasiewicz LogicsMcNaughton functions play the same role in Lukasiewicz logics as boolean func-tions do in classical logic. Formulas in one variable are an important ingredientof automated deduction in many-valued logics. Here we present how it is possibleto develop resolution calculi for the in nite-valued logic of Lukasiewicz based onsuitable classes of formulas in one variable: these classes of formulas correspondto classes of McNaughton functions characterized by nice geometric properties.We introduce normal forms built upon these classes of formulas.27 Manuel Ojeda Aciego, Malaga (Spain)with Gabriel Aguilera Venegas and Inmaculada P. de Guzman,Malaga (Spain)The TAS Reduction Method in MVL: A TAS Theorem Prover forThree-valued LogicWe present a new prover for propositional 3-valued logics, named TAS-M3, whichis an extension of the TAS-D prover for classical propositional logic. As areduction-based method, the power of TAS-M3 is based on processes which re-duce the size of the formula. These processes lter the information contained inthe syntactic structure of the formula to avoid as much distributions as possible.Roughly speaking, the idea is to get the information given by unitary satisfyingassignments; this idea has proven to be extremely useful. TAS applies a sequenceof transformations to the formula being considered. It is worth to note that thetransformations are not just applied one after the other; through the e cient de-termination and manipulation of sets of unitary models of a formula, the methodinvestigates exhaustively the formula, to detect if it is possible to decrease thesize of the formula being analysed. Sets of unitary models are associated to eachnode in the syntactic tree of the formula, they can be considered the key tool ofTAS methodology, since they are used to conclude whether the structure of thesyntactic tree has or has not direct information about the validity of the formula.This way, either the method ends giving this information or, otherwise, it de-creases the size of the problem before applying the next transformation. So, it ispossible to decrease the number of distributions or, even, to avoid them all. Thepower of the method is based on the fact that every reduction process reduces thesize of the formula and the branching part of the method follows a lazy strategy.Neil Murray, Albany (USA)Parameterized Prime Implicant/Implicate Computations for Regu-lar LogicsPrime implicant/implicate generating algorithms for multiple-valued logics areintroduced. Techniques from classical logic not requiring large normal formsor truth tables are adapted to certain \regular" multiple-valued logics. This isaccomplished by means of signed formulas, a meta-logic for multiple valued logics;the formulas are normalized in a way analogous to negation normal form. Thelogic of signed formulas is classical in nature.The presented method is based on path dissolution, a strongly complete infer-ence rule. The generalization of dissolution that accommodates signed formulasis described. The method is rst characterized as a procedure iterated over thetruth value domain = f0; 1; : : : ; n 1g of the multiple-valued logic. The com-28 putational requirements are then reduced via parameterization with respect tothe elements and the cardinality of .29