Ottavio M. D'Antona

On the equivalence and range of applicability of graph-based representations of logic programs

Logic programs under Answer Sets semantics can be studied, and actual computation can be carried out, by means of representing them by directed graphs. Several reductions of logic programs to directed graphs are now available. We compare our proposed representation, called Extended Dependency Graph, to the Block Graph representation recently defined by Linke [Proc. IJCAI-2001, 2001, pp. 641-648]. On the relevant fragment of well-founded irreducible programs, extended dependency and block graph turns out to be isomorphic. So, we argue that graph representation of general logic programs should be abandoned in favor of graph representation of well-founded irreducible programs, which are more concise, more uniform in structure while being equally expressive.

A simple combinatorial interpretation of certain generalized Bell and Stirling numbers

In a series of papers, P. Blasiak et al. developed a wide-ranging generalization of Bell numbers (and of Stirling numbers of the second kind) that is relevant to the so-called boson normal ordering problem. They provided a recurrence and, more recently, also offered a (fairly complex) combinatorial interpretation of these numbers. We show that by restricting the numbers somewhat (but still widely generalizing Bell and Stirling numbers), one can supply a much more natural combinatorial interpretation. In fact, we offer two different such interpretations, one in terms of graph colourings and another one in terms of certain labelled Eulerian digraphs.

Whitney numbers of some geometric lattices

It is a well known fact that the Whitney numbers of a sequence of ranked posets P0, P1, P2,-.of rank 0, l, 2 .... such that every r-ranked filter of P, is isomorphic to Pr are the connection constants between the sequence of powers and the sequence of the characteristic polynomials of the posets [-Dow]. We note here that, if the posets are indeed supersolvable geometric lattices, then the sequence of characteristic polynomials is persistent in the sense of [Dam2]. Thus, in Sections 3 and 4, we will be able to transfer the two term recursion and the explicit formula for connection constants [Daml] as well as several log-concavity properties of symmetric functions [Sag2] to the Whitney numbers of those lattices. Moreover, since the roots of the characteristic polynomial of a supersolvable lattice have been given a nice combinatorial meaning [Sta2], we can also give a simple semantics for those formulas. As a consequence of our results, we obtain (Section 5) unifying proofs of several properties enjoyed by Whitney numbers of Boolean algebras, subspace lattices, partition lattices, and Dowling lattices. It turns out that these lattices form the only infinite sequences of modularly complemented geometric lattices satisfying the conditions mentioned at the beginning.

Independent subsets of powers of paths, and Fibonacci cubes

Abstract We provide a formula for the number of edges of the Hasse diagram of the independent subsets of the hth power of a path ordered by inclusion. For h = 1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself.

Open Partitions and Probability Assignments in Gödel Logic

In the elementary case of finitely many events, we generalise to Godel (propositional infinite-valued) logic -- one of the fundamental fuzzy logics in the sense of Hajek -- the classical correspondence between partitions, quotient measure spaces, and push-forward measures. To achieve this end, appropriate Godelian analogues of the Boolean notions of probability assignment and partition are needed. Concerning the former, we use a notion of probability assignment introduced in the literature by the third-named author et al. Concerning the latter, we introduce and use open partitions , whose definition is justified by independent considerations on the relational semantics of Godel logic (or, more generally, of the finite slice of intuitionistic logic). Our main result yields a construction of finite quotient measure spaces in the Godelian setting that closely parallels its classical counterpart.

Propositional Gödel Logic and Delannoy Paths

Godel propositional logic is the logic of the minimum triangular norm, and can be axiomatized as propositional intuitionistic logic augmented by the prelinearity axiom (alpha rarr beta) V (beta rarr alpha). Its algebraic counterpart is the subvariety of Heyting algebras satisfying prelinearity, known as Godel algebras. A Delannoy path is a lattice path in Z2 that only uses northward, eastward, and northeastward steps. We establish a representation theorem for free n-generated Godel algebras in terms of the Boolean n-cube {0,1}n, enriched by suitably generalized Delannoy paths.

A combinatorial interpretation of the connection constants for persistent sequences of polynomials

We give a combinatorial interpretation of the connection constants for persistent sequences of polynomials in terms of weighted binary paths. In this way we give bijective proofs for many formulas which generalize several classical identities and recurrences, such as the upper index sum, the Lagrange and the Vandermonde sum and Eulers theorem on the coefficients of Gaussian coefficients.

Decompositions of B n and P n using symmetric chains

We review the Green/Kleitman/Leeb interpretation of de Bruijns symmetric chain decomposition of Bn, and explain how it can be used to find a maximal collection of disjoint symmetric chains in the nonsymmetric lattice of partitions of a set.

The Euler Characteristic of a Formula in Godel Logic

Using the lattice-theoretic version of the Euler characteristic introduced by V. Klee and G.-C. Rota, we define the Euler characteristic of a formula in Gödel logic (over finitely or infinitely many truth-values). We then prove that the information encoded by the Euler characteristic is classical, i.e., coincides with the analogous notion defined over Boolean logic. Building on this, we define k-valued versions of the Euler characteristic of a formula φ, for each integer k ≥ 2, and prove that they indeed provide information about the logical status of φ in Gödel k-valued logic. Specifically, our main result shows that the k-valued Euler characteristic is an invariant that separates k-valued tautologies from non-tautologies.

Best Approximation of Ruspini Partitions in Gödel Logic

A Ruspini partition is a finite family of fuzzy sets {f 1 , ..., f n }, f i : [0, 1] i¾?[0, 1], such that