arXiv subjects

Luigi Santocanale

Publications and source records attributed to Luigi Santocanale.

33 records · Page 2Linked to original sources

arXiv2014

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 Nation's 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.

math.LO↗

arXiv2013

Lattices of regular closed subsets of closure spaces

For a closure space (P,f) with f(\emptyset)=\emptyset, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P,f), extending the poset Clop(P,f) of all clopen subsets. If (P,f) is a finite convex geometry, then Reg(P,f) 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,f) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces, (1) Reg(P,f) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity. (2) If Reg(P,f) is semidistributive, then it is a bounded homomorphic image of a free lattice. (3) Clop(P,f) is a lattice iff 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 regular closed sets is, in the semilattice context, always the Dedekind Mac-Neille 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.

math.CO↗

arXiv2013

Sublattices of associahedra and permutohedra

Grätzer asked in 1971 for a characterization of sublattices of Tamari lattices (associahedra). A natural candidate was coined by McKenzie in 1972 with the notion of a bounded homomorphic image of a free lattice---in short, bounded lattice. Urquhart proved in 1978 that every associahedron is bounded (thus so are its sublattices). Geyer conjectured in 1994 that every finite bounded lattice embeds into some associahedron. We disprove Geyer's conjecture, by introducing an infinite collection of lattice-theoretical identities that hold in every associahedron, but not in every finite bounded lattice. Among those finite counterexamples, there are the permutohedron on four letters P(4), and in fact two of its subdirectly irreducible retracts, which are Cambrian lattices of type A. For natural numbers m and n, we denote by B(m,n) the (bounded) lattice obtained by doubling a join of m atoms in an (m+n)-atom Boolean lattice. We prove that B(m,n) embeds into an associahedron iff min(m,n) is less than or equal to 1, and that B(m,n) embeds into a permutohedron iff min(m,n) is less than or equal to 2. In particular, B(3,3) cannot be embedded into any permutohedron. Nevertheless we prove that B(3,3) is a homomorphic image of a sublattice of the permutohedron on 12 letters.

math.CO↗

arXiv2013

The extended permutohedron on a transitive binary relation

For a given transitive binary relation e on a set E, the transitive closures of open (i.e., co-transitive in e) sets, called the regular closed subsets, form an ortholattice Reg(e), the extended permutohedron on e. This construction, which contains the poset Clop(e) of all clopen sets, is a common generalization of known notions such as the generalized permutohedron on a partially ordered set on the one hand, and the bipartition lattice on a set on the other hand. We obtain a precise description of the completely join-irreducible (resp., completely meet-irreducible) elements of Reg(e) and the arrow relations between them. In particular, we prove that (1) Reg(e) is the Dedekind-MacNeille completion of the poset Clop(e); (2) Every open subset of e is a set-theoretic union of completely join-irreducible clopen subsets of e; (3) Clop(e) is a lattice iiff every regular closed subset of e is clopen, iff e contains no "square" configuration, iff Reg(e)=Clop(e); (4) If e is finite, then Reg(e) is pseudocomplemented iff it is semidistributive, iff it is a bounded homomorphic image of a free lattice, iff e is a disjoint sum of antisymmetric transitive relations and two-element full relations. We illustrate the strength of our results by proving that, for n greater than or equal to 3, the congruence lattice of the lattice Bip(n) of all bipartitions of an n-element set is obtained by adding a new top element to a Boolean lattice with n2^{n-1} atoms. We also determine the factors of the minimal subdirect decomposition of Bip(n).

math.CO↗

arXiv2011

Varieties of lattices with geometric descriptions

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann, Pickering, and Roddy proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite, join-semidistributive variety.

math.RA↗

arXiv2009

On the word problem for SP-categories, and the properties of two-way communication

The word problem for categories with free products and coproducts (sums), SP-categories, is directly related to the problem of determining the equivalence of certain processes. Indeed, the maps in these categories may be directly interpreted as processes which communicate by two-way channels. The maps of an SP-category may also be viewed as a proof theory for a simple logic with a game theoretic intepretation. The cut-elimination procedure for this logic determines equality only up to certain permuting conversions. As the equality classes under these permuting conversions are finite, it is easy to see that equality between cut-free terms (even in the presence of the additive units) is decidable. Unfortunately, this does not yield a tractable decision algorithm as these equivalence classes can contain exponentially many terms. However, the rather special properties of these free categories -- and, thus, of two-way communication -- allow one to devise a tractable algorithm for equality. We show that, restricted to cut-free terms s,t : X --> A, the decision procedure runs in time polynomial on |X||A|, the product of the sizes of the domain and codomain type.

cs.LO↗

arXiv2009

A Nice Labelling for Tree-Like Event Structures of Degree 3 (Extended Version)

We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the labelling number of an event structure of degree 3 is bounded by a linear function of the height. The main theorem we present in this paper states that event structures of degree 3 whose causality order is a tree have a nice labelling with 3 colors. Finally, we exemplify how to use this theorem to construct upper bounds for the labelling number of other event structures of degree 3.

cs.DC↗

arXiv2008

Completeness for Flat Modal Fixpoint Logics

This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γof modal formulas of the form γ(x, p1, . . ., pn), where x occurs only positively in γ, the language L\sharp (Γ) is obtained by adding to the language of polymodal logic a connective \sharp\_γfor each γε. The term \sharp\_γ(\varphi1, . . ., \varphin) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ(x, φ1, . . ., φn). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L\sharp (Γ) on Kripke frames. We prove two results that solve this problem. First, let K\sharp (Γ) be the logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective \sharp\_γ. Provided that each indexing formula γsatisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K+ (Γ) of K\_\sharp (Γ). This extension is obtained via an effective procedure that, given an indexing formula γas input, returns a finite set of axioms and derivation rules for \sharp\_γ, of size bounded by the length of γ. Thus the axiom system K+ (Γ) is finite whenever Γis finite.

cs.LO↗

arXiv2008

Derived Semidistributive Lattices

For L a finite lattice, let C(L) denote the set of pairs g = (g_0,g_1) such that g_0 is a lower cover of g_1 and order it as follows: g <= d iff g_0 <= d_0, g_1 <= d_1, but not g_1 <= d_0. Let C(L,g) denote the connected component of g in this poset. Our main result states that C(L,g) is a semidistributive lattice if L is semidistributive, and that C(L,g) is a bounded lattice if L is bounded. Let S_n be the permutohedron on n letters and T_n be the associahedron on n+1 letters. Explicit computations show that C(S_n,a) = S_{n-1} and C(T_n,a) = T_{n-1}, up to isomorphism, whenever a is an atom. These results are consequences of new characterizations of finite join semidistributive and finite lower bounded lattices: (i) a finite lattice is join semidistributive if and only if the projection sending g in C(L) to g_0 in L creates pullbacks, (ii) a finite join semidistributive lattice is lower bounded if and only if it has a strict facet labelling. Strict facet labellings, as defined here, are generalization of the tools used by Barbut et al. to prove that lattices of Coxeter groups are bounded.

math.LO↗

arXiv2008

The Variable Hierarchy for the Games mu-Calculus

Parity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games mu-calculus into the class of all complete lattices. We answer this question negatively by providing, for each n >= 1, a parity game Gn with these properties: it unravels to a mu-term built up with n fixed-point variables, it is semantically equivalent to no game with strictly less than n-2 fixed-point variables.

cs.LO↗

arXiv2007

Undirected Graphs of Entanglement Two

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 present an explicit characterization of undirected graphs of entanglement at most 2. With such a characterization at hand, we devise a linear time algorithm to decide whether an undirected graph has this property.

cs.LO↗

arXiv2007

A Nice Labelling for Tree-Like Event Structures of Degree 3

We address the problem of ﬁnding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the labelling number of an event structure of degree 3 is bounded by a linear function of the height. The main theorem we present in this paper states that event structures of degree 3 whose causality order is a tree have a nice labelling with 3 colors. Finally, we exemplify how to use this theorem to construct upper bounds for the labelling number of other event structures of degree 3.

cs.DC↗

arXiv2005

Congruences of Multinomial Lattices

We study the congruence lattices of the multinomial lattices L(v) introduced by Bennett and Birkhoff. Our main motivation is to investigate Parikh equivalence relations that model concurrent computation. We accomplish this goal by providing 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 define a sequence of equations SD_{n}(\land), for n \geq 0. Our main result sounds as follows: if v = (v_{1},...,v_{n}) \in N^{n} and v_{i} > 0 for i = 1,..., n, then the multinomial lattice L(v) satisfies SD_{n-1}(\land) and fails SD_{n-2}(\land).

math.CO↗

arXiv2005

Completions of mu-algebras

A $μ$-algebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms $(f,μ_{x}.f)$ where $μ_{x}.f$ is axiomatized as the least prefixed point of $f$, whose axioms are equations or equational implications. Standard $μ$-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of $μ$-algebras contains a $μ$-algebra that has no embedding into a complete $μ$-algebra. We focus then on modal $μ$-algebras, i.e. algebraic models of the propositional modal $μ$-calculus. We prove that free modal $μ$-algebras satisfy a condition -- reminiscent of Whitman's condition for free lattices -- which allows us to prove that (i) modal operators are adjoints on free modal $μ$-algebras, (ii) least prefixed points of $Σ_{1}$-operations satisfy the constructive relation $μ_{x}.f = \bigvee_{n \geq 0} f^{n}(\bot)$. These properties imply the following statement: {\em the MacNeille-Dedekind completion of a free modal $μ$-algebra is a complete modal $μ$-algebra and moreover the canonical embedding preserves all the operations in the class $Comp(Σ_{1},Π_{1})$ of the fixed point alternation hierarchy.}

math.RA↗

arXiv2004

Logical Construction of Final Coalgebras

We prove that every finitary polynomial endofunctor of a category $C$ has a final coalgebra if $C$ is locally Cartesian closed, has finite disjoint coproducts and a natural number object. More generally, we prove that the category of coalgebras for such an endofunctor has all finite limits.

math.CT↗