arXiv subjects

Daniel Penazzi

Publications and source records attributed to Daniel Penazzi.

3 recordsLinked to original sources

arXiv2019

The Complexity of Definability by Open First-Order Formulas

In this article we formally define and investigate the computational complexity of the Definability Problem for open first-order formulas (i.e., quantifier free first-order formulas) with equality. Given a logic $\mathbf{\mathcal{L}}$, the $\mathbf{\mathcal{L}}$-Definability Problem for finite structures takes as input a finite structure $\mathbf{A}$ and a target relation $T$ over the domain of $\mathbf{A}$, and determines whether there is a formula of $\mathbf{\mathcal{L}}$ whose interpretation in $\mathbf{A}$ coincides with $T$. We show that the complexity of this problem for open first-order formulas (open definability, for short) is coNP-complete. We also investigate the parametric complexity of the problem, and prove that if the size and the arity of the target relation $T$ are taken as parameters then open definability is $\mathrm{coW}[1]$-complete for every vocabulary $τ$ with at least one, at least binary, relation.

cs.CC↗

arXiv2017

The Lattice of Congruences of a Finite Line Frame

Let $\mathbf{F}=\left\langle F,R\right\rangle $ be a finite Kripke frame. A congruence of $\mathbf{F}$ is a bisimulation of $\mathbf{F}$ that is also an equivalence relation on F. The set of all congruences of $\mathbf{F}$ is a lattice under the inclusion ordering. In this article we investigate this lattice in the case that $\mathbf{F}$ is a finite line frame. We give concrete descriptions of the join and meet of two congruences with a nontrivial upper bound. Through these descriptions we show that for every nontrivial congruence $ρ$, the interval $[\mathrm{Id_{F},ρ]}$ embeds into the lattice of divisors of a suitable positive integer. We also prove that any two congruences with a nontrivial upper bound permute.

math.LO↗

arXiv2015

A new generalization of Hermite's reciprocity law

Given a partition $λ$ of $n$, the {\it Schur functor} $\mathbb{S}_λ$ associates to any complex vector space $V$, a subspace $\mathbb{S}_λ(V)$ of $V^{\otimes n}$. Hermite's reciprocity law, in terms of the Schur functor, states that $ \mathbb{S}_{(p)}\left(\mathbb{S}_{(q)}(\mathbb{C}^2)\right)\simeq \mathbb{S}_{(q)}\left(\mathbb{S}_{(p)}(\mathbb{C}^2)\right). $ We extend this identity to many other identities of the type $\mathbb{S}_λ\left(\mathbb{S}_δ(\mathbb{C}^2)\right)\simeq \mathbb{S}_μ\left(\mathbb{S}_ε(\mathbb{C}^2)\right)$.

math.CO↗