On Frankl conjecture. Coherence in predicate logic. Algebraic theory of fuzzy languages and automata

Transcription

1 On Frankl conjecture Vladimir Božin University of Warwick, Coventry, United Kingdom Frankl conjecture states that for every finite family of sets closed under intersections there is an element which belongs to no more than half of sets in the family. The conjecture is known to be true for many special cases. We discuss some new cases in which the conjecture holds, a strenghtened version of the conjecture and a counterexample construction. Coherence in predicate logic Kosta Došen kosta@turing.mi.sanu.ac.yu We give a syntactical description of the categories corresponding to the fragment of linear predicate logic covered by proof nets. For these fragments we can prove coherence in the sense of Mac Lane and Kelly. Algebraic theory of fuzzy languages and automata Jelena Ignjatović, Miroslav Ćirić University of Niš jejaign@yahoo.com, ciricm@bankerinter.net One of the most significant branches of the algebraic theory of languages and automata is Myhill-Nerode s theory, in which recognizability of languages by deterministic finite automata is studied through right congruences and congruences of finite index on a free monoid. In this paper we develop a Myhill-Nerode s type theory for fuzzy languages recognizable by deterministic finite automata. We show that this theory does not hold in general for fuzzy languages recognizable by fuzzy finite automata, except if the underlying structure of truth values has a locally finite semiring reduct. This 1

2 has shown oneself to be closely related to the problem of determinization of fuzzy automata, which will be also considered here. We will present a new method for determinization of fuzzy automata, which gives better results than the previous methods developed by Bělohlávek (2002) and Li and Pedrycz (2005). We will show that for any fuzzy language recognizable by a deterministic finite automaton there exists a minimal deterministic finite automaton recognizing it, which is unique up to an isomorphism, and we will give a method for its construction, as well as an efficient algorithm for minimization of deterministic fuzzy recognizers. Unlike deterministic automata, the state minimization of non-deterministic and fuzzy automata is computationally hard. For that reason, we have developed certain efficient size reduction methods for fuzzy automata which do not necessarily give minimal automata, but which give small enough fuzzy automata. We will also show that reduction of fuzzy automata by means of fuzzy equivalences is closely related to the problem of solving of a particular system of fuzzy relation equations. Finally, we will show that regular operations on fuzzy languages, used by Li and Pedrycz (2005) in the proof of the Kleene s type theorem for fuzzy languages, can be defined through regular operations on crisp languages. This will be done using the concept of a formal power series over a quantale, with coefficients in the algebra of crisp languages, that we introduce here. We will also show that a fuzzy language is recognizable by a fuzzy finite automaton if and only if it can be represented by a rational power series, and that it is recognizable by a deterministic finite automaton if and only if it can be represented by a polynomial whose coefficients are regular crisp languages. 2

3 Formal calculi corresponding to intuitionistic sequent calculus Jelena Ivetić Curry-Howard correspondence exhibits the well-known connection between the intuitionistic natural deduction and λ-calculus. Although it was clear that this connection can be extended to intuitionistic sequent calculus, only recently the corresponding formal calculus was constructed and this topic is still quite live. In this talk we will present the most significant sequent formal calculi, with particular focus on λ-gentzen calculus, for which we will present the type assignment system with intersection types. Formal models of concurrent processes Svetlana Jakšić Concurrence is a property of the systems in which several processes are being executed at the same time, and possibly interact with each other. Therefore, the transition to the next state of the system is not deterministic. Various transitions are possible, making the analysis of this type of behavior very complex. The systems of this kind today are occurring more and more often and are becoming more complex all the time: the Internet, WWW, operational systems, network systems, and also in economy, biology... Properties characterizing concurrent processes differ from the ones associated with the sequential processes. Depending on the properties they are describing, various mathematical models have been developed, such as Calculus of Comunicationg systems, Communicating Sequential Processes, Algebra of Communicating Processes, Petri nets, π-calculus, Mobile ambients,... Mobility of processes, i.e. possibility of changing the connectedness between processes within a system, is what differentiates π-calculus from other models. Mobile processes are those processes which are able to change links and so dynamically change the configuration of the system. Security is another 3

4 property which is increasingly important with expansion of concurrent systems. By introducing type systems, it is possible to control access rights. Robinson diagrams in algebra Miodrag Kapetanović A general method based on the model theoretic notion of diagram is used to prove versions of some well known algebraic results. As a first example a well known theorem of Sikorski, concerning the existence of a homomorphism between boolean algebras, is derived by merely checking the conditions from the Robinson s Diagram Lemma. The next example is another result of Sikorski about the existence of a homomorphism extension which is proved along the same lines and in the more general form. Further examples may be offered from group theory etc. Lattice of varieties defined by quadratic quasigroup identities with four variables Aleksandar Krapež sasa@mi.sanu.ac.yu We consider the class of identities with one binary operation which is assumed to be a quasigroup. Identities are quadratic with four variables whose occurrences all have equal height in the term tree. So, the identities are of the form x 1 x 2 x 3 x 4 = x 5 x 6 x 7 x 8, where x i {x, y, u, v}(1 i 8) and each variable occurs exactly two times in the identity. There are exactly 105 such identties. They can be sorted into 19 equivalent classes, defining in this way 19 varieties of quasigroups. 4

5 Deciding the formulas with (multi)sets and linear arithmetic Viktor Kunčak Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland We consider the problem if formulas containing operations over finite sets and multisets of arbitrary size are valid. Such operations can be naturally described by formulas of linear arithmetic over values of characteristic functions of multisets (e.g. set union corresponds to maximum and multiset union corresponds to sum). We also allow the cardinality operation, defined as sum of all values of the characteristic function of a multiset, to occur in our formulas. Deciding the resulting class of formulas has applications in automatic program verification, in interactive theorem provers, and also in languages for formal representation of knowledge on Internet. We will show that the problem of deciding if such formulas are valid without presence of quantifiers is co-np complete. We will also observe that in the presence of quantifiers the problem with multisets becomes undecidable. (The problem will remain decidable if the cardinality operator is applied only on sets.) additional information: kuncak/papers/piskackuncak08decisionproceduresmultisetscardinality.html Constraint satisfaction problem, graph theory, primitive positive formulas and weak near-unanimity terms Petar Marković pera@im.ns.ac.yu This lecture will present at an elementary level a well-known problem in complexity theory which also attracts logicians, graph theorists and universal algebraists. The problem, known as the Feder-Vardi Dichotomy Conjecture says that the fixed-template Constraint Satisfaction Problem can have exactly two complexities, NP-complete and tractable, depending on the template. This lecture will focus on the ideas which have been applied so far and 5

6 their interaction attempting to give a birds-eye view of the efforts towards resolving the conjecture. On a new type of homogeneity Dragan Mašulović masul@im.ns.ac.yu A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this talk we discuss homomorphismhomogeneous partially ordered sets and homomorphism-homogeneous finite tournaments where vertices are allowed to have loops. On descriptive logic CIF Milenko Mosurović University of Podgorica milenko@cg.ac.yu Descriptive logics (DL) are developed as the formal language for representing a knowledge base and concluding based on this knowledge base. One DL is CIF, considered by De Giacomo in his PhD thesis, which is obtained from logic CI by addition of constructor of restriction of functions. In his proofs, De Giacomo encodes CIF formulas in CI and thus obtains that the problem of satisfiability of a CIF formula is EXPTIME-complete. The present paper gives easy examples which prove that certain steps in De Giacomo s proofs are incorrect and how to correct these errors to prove complexity of the logic CIF with modal operators. 6

7 Equality of proofs in predicate logic Zoran Petrić After successful solution of the problem of equality of proofs in central fragment of linear predicate logic, we consider the questions of possible extensions of these results to intuitionistic and classical predicate logic. On positive quasi-antiorder in semigroup with apartness Daniel A. Romano (joint work with S. Crvenković) University of Banja Luka In the present paper we give a definition and some basic properties of positive quasi0antiorder in semigroup with apartness. Besides, we give definitions and basic properties of following positive quasi-antiorders: consistent, linear and lower-potent positive quasi-antiorders. Connection between lattice Q(S) of all quasi-antiorders, lattice Q p (S) of all positive quasi-antiorders and lattice Q lp (S) of lower-potent positive quasi-antiorders on semigroup S with apartness are presented. Probability on a universe of discourse Djordje Vukomanović University of Belgrade vdjordje@grf.bg.ac.yu In this approach to probability, following Boole, probabilities are attributed to statements rather than events. Technically speaking, probability is defined on Lindenbaum algebra of a propositional calculus which is a free Boolean algebra in which free generators represent stochastically independent statements (events). To these free generators we arbitrarily assign probabilities which are then defined recursively for all Boolean combinations of statements. For thus defined probability, among other things, hold the (weak) laws of large numbers of Bernoulli and Poisson and the (strong) laws of large numbers of Borel and Kolmogorov. 7