Luigi Santocanale

Dual characterizations for finite lattices via correspondence theory for monotone modal logic

We establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nations characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic. Indeed, we adapt the correspondence algorithm ALBA to the setting of monotone modal logic, and we use a certain duality-induced encoding of finite lattices as monotone neighbourhood frames to translate lattice terms into formulas in monotone modal logic.

Completions of μ-algebras

The characteristics of a weld made by electron beam welding are improved by introducing into the region of the pieces to be welded, which is subjected to remelting during welding, a material which is obtained by applying an arc welding process, or a process for remelting under slag, to a mild or slightly alloyed steel. The introduction of the material into the region of the weld increases the production of acicular ferrite which increases the tensile strength of the weld. The material may be introduced by providing a deposit of it over part or the whole of the surface of one or both of the pieces to be welded, by introducing a mass of it between the surfaces to be welded or as a filler wire.

Ambiguous classes in µ-calculi hierarchies

A classical result by Rabin states that if a set of trees and its complement are both Buchi definable in the monadic second order logic, then these sets are weakly definable. In the language of µ-calculi, this theorem asserts the equality between the complexity classes Σ2 ∩ Π2 and Comp(Σ1, Π1) of the fixed-point alternation-depth hierarchy of the µ-calculus of tree languages. It is natural to ask whether at higher levels of the hierarchy the ambiguous classes Σn+1 ∩ Πn+1 and the composition classes Comp(Σn, Πn) are equal, and for which µ-calculi.The first result of this paper is that the alternation-depth hierarchy of the games µ-calculus--whose canonical interpretation is the class of all complete lattices--enjoys this property. More explicitly, every parity game which is equivalent both to a game in Σn+1 and to a game in Πn+1 is also equivalent to a game obtained by composing games in Σn and Πn.The second result is that the alternation-depth hierarchy of the µ-calculus of tree languages does not enjoy the property. Taking into account that any Buchi definable set is recognized by a nondeterministic Buchi automaton, we generalize Rabins result in terms of the following separation theorem: if two disjoint languages are recognized by nondeterministic Πn+1 automata, then there exists a third language recognized by an alternating automaton in Comp(Σn, Πn) containing one and disjoint from the other.Finally, we lift the results obtained for the µ-calculus of tree languages to the propositional modal µ-calculus: ambiguous classes do not coincide with composition classes, but a separation theorem is established for disjunctive formulas.

Cuts for circular proofs: semantics and cut-elimination

One of the authors introduced in [Santocanale, FoSSaCS, 2002] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products, finite coproducts, initial algebras, final coalgebras. The calculus presented [Santocanale, FoSSaCS, 2002] is cut-free; even if sound and complete for provability, it lacked an important property for the semantics of proofs, namely fullness w.r.t. the class of intended categorical models (called mu-bicomplete categories in [Santocanale, ITA, 2002]). In this paper we fix this problem by adding the cut rule to the calculus and by modifying accordingly the syntactical constraint ensuring soundness of proofs. The enhanced proof system fully represents arrows of the canonical model (a free mu-bicomplete category). We also describe a cut-elimination procedure as a a model of computation arising from the above mentioned categorical operations. The procedure constructs a cut-free proof-tree with possibly infinite branches out of a finite circular proof with cuts.

Undirected graphs of entanglement 2

Entanglement is a complexity measure of directed graphs that origins in fixed point theory. This measure has shown its use in designing efficient algorithms to verify logical properties of transition systems. We are interested in the problem of deciding whether a graph has entanglement at most k. As this measure is defined by means of games, game theoretic ideas naturally lead to design polynomial algorithms that, for fixed k, decide the problem. Known characterizations of directed graphs of entanglement at most 1 lead, for k = 1, to design even faster algorithms. In this paper we give two distinct characterizations of undirected graphs of entanglement at most 2. With these characterizations at hand, we present a linear time algorithm to decide whether an undirected graph has this property.

Relational Lattices via Duality

The natural join and the inner union combine in different ways tables of a relational database. Tropashko [18] observed that these two operations are the meet and join in a class of lattices-called the relational lattices- and proposed lattice theory as an alternative algebraic approach to databases. Aiming at query optimization, Litak et al. [12] initiated the study of the equational theory of these lattices. We carry on with this project, making use of the duality theory developed in [16]. The contributions of this paper are as follows. Let A be a set of columns names and D be a set of cell values; we characterize the dual space of the relational lattice R(D, A) by means of a generalized ultrametric space, whose elements are the functions from A to D, with the P (A)-valued distance being the Hamming one but lifted to subsets of A. We use the dual space to present an equational axiomatization of these lattices that reflects the combinatorial properties of these generalized ultrametric spaces: symmetry and pairwise completeness. Finally, we argue that these equations correspond to combinatorial properties of the dual spaces of lattices, in a technical sense analogous of correspondence theory in modal logic. In particular, this leads to an exact characterization of the finite lattices satisfying these equations.

Lattices of regular closed subsets of closure spaces

For a closure space (P, φ) with φ(o) = o, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P, φ), extending the poset Clop(P, φ) of all clopen subsets. If (P, φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind–MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg(P, φ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces, • Reg(P, φ) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity. • If Reg(P, φ) is semidistributive, then it is a bounded homomorphic image of a free lattice. • Clop(P, φ) is a lattice if and only if every regular closed set is clopen. The extended permutohedron R(G) on a graph G and the extended permutohedron Reg S on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of all regular closed sets is, in the semilattice context, always the Dedekind–MacNeille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R(G) and Reg S are bounded homomorphic images of free lattices.

On the Join Dependency Relation in Multinomial Lattices

Aiming to understand equivalence relations that model concurrent computation, we investigate congruences of multinomial lattices

The Variable Hierarchy for the Lattice μ-Calculus

\mathcal{L}(v)

The equational theory of the weak order on finite symmetric groups

introduced by Bennett and Birkhoff (Algebra Univers. 32(1):115–144, 1994). Our investigation gives rise to an explicit description of the join dependency relation between two join irreducible elements and of its reflexive transitive closure. The explicit description emphasizes several properties and makes it possible to separate the equational theories of multinomial lattices by their dimensions. In their covering of non modular varieties Jipsen and Rose (Varieties of lattices, Lecture Notes in Mathematics, vol. 1533, Springer, Berlin, 1992) define a sequence of equations