Logic Prizes <Emphasis Type="Italic">et Cætera</Emphasis>

I discuss the origin and development of logic prizes around the world. In a first section I describe how I started this project by creating the Newton da Costa Logic Prize in Brazil in 2014. In a second section I explain how this idea was extended into the world through the manifesto A Logic Prize in Every Country! and how was organized the Logic Prizes Contest at the 6th UNILOG (World Congress and School on Universal Logic) in Vichy in June 2018 with the participation of 9 logic prizes winners from 9 countries. In a third section I discuss how this project will develop in the future with the creation of more logic prizes, an Encyclopædia of Logic, the book series Logic PhDs, as well as the creation of a World Logic Day, January 14, day of birth of Alfred Tarski and of death of Kurt Gödel.     

Canonical formulas for <Emphasis Type="Italic">k</Emphasis>-potent commutative,integral, residuated lattices

Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Indeed, they provide a uniform and semantic way of axiomatising all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective, canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper, we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for k-potent, commutative, integral, residuated lattices (k-CIRL). We show that any subvariety of k-CIRL is axiomatised by canonical formulas. The paper ends with some applications and examples.     

n-Contractive BL-logics

In the field of many-valued logics, Hájek??s Basic Logic BL was introduced in Hájek (Metamathematics of fuzzy logic, trends in logic. Kluwer Academic Publishers, Berlin, 1998). In this paper we will study four families of n-contractive (i.e. that satisfy the axiom ${\phi^n\rightarrow\phi^{n+1}}$ , for some ${n\in\mathbb{N}^+}$ ) axiomatic extensions of BL and their corresponding varieties: BL ⁿ , SBL ⁿ , BL _n and SBL _n . Concerning BL ⁿ we have that every BL ⁿ -chain is isomorphic to an ordinal sum of MV-chains of at most n?+?1 elements, whilst every BL _n -chain is isomorphic to an ordinal sum of MV _n -chains (for SBL ⁿ and SBL _n a similar property holds, with the difference that the first component must be the two elements boolean algebra); all these varieties are locally finite. Moving to the content of the paper, after a preliminary section, we will study generic and k-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties. Finally, we will analyze the first-order versions of these logics, from the point of view of completeness and arithmetical complexity.     

Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted in larger scale by philosophers.     

Some Properties of Approximately Dual Frames in Hilbert Spaces

In this paper, a new characterization is obtained for approximately dual frames of a given frame. Among other things, it is proved that if the sequence \({\Psi=(\psi_n)_n}\) is sufficiently close to the frame \({\Phi=(\varphi_n)_n}\), then \({\Psi}\) is a frame for \({\mathcal{H}}\) and approximately dual frames \({\Phi^{ad}=(\varphi^{ad}_n)_n}\) and \({\Psi^{ad}=(\psi^{ad}_n)_n}\) can be found which are close to each other and \({T_\Phi U_{\Phi^{ad}}=T_\Psi U_{\Psi^{ad}}}\), where T_X and U_X denote the synthesis and analysis operators of the frame X, respectively. Finally, the results are applied to Gabor systems to obtain some practical examples.     

On the Modal Logic of Jeffrey Conditionalization

We continue the investigations initiated in the recent papers (Brown et al. in The modal logic of Bayesian belief revision, 2017; Gyenis in Standard Bayes logic is not finitely axiomatizable, 2018) where Bayes logics have been introduced to study the general laws of Bayesian belief revision. In Bayesian belief revision a Bayesian agent revises (updates) his prior belief by conditionalizing the prior on some evidence using the Bayes rule. In this paper we take the more general Jeffrey formula as a conditioning device and study the corresponding modal logics that we call Jeffrey logics, focusing mainly on the countable case. The containment relations among these modal logics are determined and it is shown that the logic of Bayes and Jeffrey updating are very close. It is shown that the modal logic of belief revision determined by probabilities on a finite or countably infinite set of elementary propositions is not finitely axiomatizable. The significance of this result is that it clearly indicates that axiomatic approaches to belief revision might be severely limited.     

Regularity of powers of bipartite graphs

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all \(s \ge 1\), we obtain upper bounds for \({\text {reg}}(I(G)^s)\) for bipartite graphs. We then compare the properties of G and \(G'\), where \(G'\) is the graph associated with the polarization of the ideal \((I(G)^{s+1} : e_1\cdots e_s)\), where \(e_1,\cdots , e_s\) are edges of G. Using these results, we explicitly compute \({\text {reg}}(I(G)^s)\) for several subclasses of bipartite graphs.     

The permutation of integers with small least common multiple of two subsequent terms

Erd?s, Freud and Hegyvári [1] constructed a permutation a ₁,a ₂,… of positive integers with \([a_{i}, a_{i+1}]< i\exp \left\{c\sqrt{\log i}\log\log i\,\right\}\) for an absolute constant c>0 and all i≧3. In this note, we construct a permutation of all positive integers such that for any ε>0 there exists an i ₀ with \([a_{i}, a_{i+1}]\allowbreak < i\exp \left\{\left(2\sqrt{2}+\varepsilon\right) \sqrt{\log i\log\log i}\,\right\}\) for all i≧i ₀.     

On the equality and comparison problem of a class of mean values

Let \({I\subset \mathbb {R}}\) be a nonvoid open interval. A function \({K:I^2\to I}\) is called an M-conjugate mean if there exists \({(p,q)\in [0,1]^2}\) and a continuous strictly monotone real valued function \({\varphi}\) on I such that

$K(x,y)=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(M(x,y)))=:M_ \varphi^{(p,q)}(x,y)$

holds for all \({x,y\in I}\). In this paper, we investigate the equality and comparison problem in the class of M-conjugate means, in the case when

$M(x,y):=\min\{x,y\}\quad (x,y\in I)$

.

     

On the least distance eigenvalues of the second power of a graph

The k-th power of a graph G, denoted by \(G^k\), is the graph obtained from G by adding an edge between each pair of vertices with distance at most k. This paper investigates the least distance eigenvalues of the second power of a connected graph, and determine the trees and unicyclic graphs with least distance eigenvalues of the second power in \([-\,3,-\,2]\) and \((-\,\frac{3+\sqrt{5}}{2}, -\,1]\), respectively.     

The cohomology of orbit spaces of certain free circle group actions

Suppose that \(G =\mathbb{S}^1\) acts freely on a finitistic space X whose (mod p) cohomology ring is isomorphic to that of a lens space \(L^{2m-1}(p;q_1,\ldots,q_m)\) or \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). The mod p index of the action is defined to be the largest integer n such that α ⁿ ?≠?0, where \(\alpha \,\epsilon\, H^2(X/G;\mathbb{Z}_p)\) is the nonzero characteristic class of the \(\mathbb{S}^1\)-bundle \(\mathbb{S}^1\hookrightarrow X\rightarrow X/G\). We show that the mod p index of a free action of G on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\) is p???1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.     

Classification of Klein four symmetric pairs of holomorphic type for $$\mathrm {E}_{6(-14)}$$

The author classifies Klein four symmetric pairs of holomorphic type for non-compact Lie group \(\mathrm {E}_{6(-14)}\), which gives a class of pairs \((G,G')\) of real reductive Lie group G and its reductive subgroup \(G'\) such that there exist irreducible unitary representations \(\pi \) of G, which are admissible upon restriction to \(G'\).     

Logical laws for existential monadic second-order sentences with infinite first-order parts

We consider existential monadic second-order sentences ?X φ(X) about undirected graphs, where ?X is a finite sequence of monadic quantifiers and φ(X) ∈ +_∞ω^ω is an infinite first-order formula. We prove that there exists a sentence (in the considered logic) with two monadic variables and two first-order variables such that the probability that it is true on G(n, p) does not converge. Moreover, such an example is also obtained for one monadic variable and three first-order variables.     

Complete Diagnostic Length 2 Tests for Logic Networks under Inverse Faults of Logic Gates

It is proved that any Boolean function can be implemented by a logic network in the basis {x&y &z, x ⊕ y, 1} in such a way that this logic network admits a complete diagnostic test of length at most 2 with respect to inverse faults at the outputs of logic gates.     

Pseudo f-Manifolds with Complemented Frames

In this paper we study pseudo \({\varphi}\)-structure on a differential manifold in which is a tensor field of type (1, 1) satisfying \({\varphi^{3}\,-\,\varphi=0}\). We obtain basic curvature tensor fields for pseudo f-manifolds with complemented frame.     

Approximate Solutions of Variational Inequalities on Sets of Common Fixed Points of a One-Parameter Semigroup of Nonexpansive Mappings

Let \(\mathcal{T}\) be a one-parameter semigroup of nonexpansive mappings on a nonempty closed convex subset C of a strictly convex and reflexive Banach space X. Suppose additionally that X has a uniformly Gâteaux differentiable norm, C has normal structure, and \(\mathcal{T}\) has a common fixed point. Then it is proved that, under appropriate conditions on nonexpansive semigroups and iterative parameters, the approximate solutions obtained by the implicit and explicit viscosity iterative processes converge strongly to the same common fixed point of \(\mathcal{T}\), which is a solution of a certain variational inequality.     

On semistable principal bundles over a complex projective manifold,II

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and \({E_G\,\longrightarrow\,X}\) a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over \({\mathbb{C}}\). Let Z(G) denote the center of G. We prove that the following three statements are equivalent:

1. (1)
   There is a parabolic subgroup \({P\,\subset\,G}\) and a holomorphic reduction of structure group \({E_P\,\subset\,E_G}\) to P, such that the corresponding L(P)/Z(G)–bundle

   $E_{L(P)/Z(G)}\,:=\,E_P(L(P)/Z(G))\,\longrightarrow\,X$

   admits a unitary flat connection, where L(P) is the Levi quotient of P.
    
2. (2)
   The adjoint vector bundle ad(E _G ) is numerically flat.
    
3. (3)
   The principal G–bundle E _G is pseudostable, and

   $\int\limits_X c_2({\rm ad}(E_G))\omega^{d-2}\,=\,0.$
    

If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that E _G is semistable with c ₂(ad(E _G )) = 0.

     

A Characterisation of Some $$\mathbf {}$$-Like Logics

In Béziau (Log Log Philos 15:99–111, 2006) a logic \(\mathbf {Z}\) was defined with the help of the modal logic \(\mathbf {S5}\). In it, the negation operator is understood as meaning ‘it is not necessary that’. The strong soundness–completeness result for \(\mathbf {Z}\) with respect to a version of Kripke semantics was also given there. Following the formulation of \(\mathbf {Z}\) we can talk about \(\mathbf {Z}\)-like logics or Beziau-style logics if we consider other modal logics instead of \(\mathbf {S5}\)—such a possibility has been mentioned in [1]. The correspondence result between modal logics and respective Beziau-style logics has been generalised for the case of normal logics naturally leading to soundness–completeness results [see Marcos (Log Anal 48(189–192):279–300, 2005) and Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 34(4):229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 37(3–4):185–196, 2008), (Bull Sect Log 38(3–4):189–203, 2009) some partial results for non-normal cases are given. In the present paper we try to give similar but more general correspondence results for the non-normal-worlds case. To achieve this aim we have to enrich original Beziau’s language with an additional negation operator understood as ‘it is necessary that not’.     

Intrinsic Hölder Continuity of Harmonic Functions

We consider the stochastic differential equation (SDE) of the form

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{rcl} dX^ x(t) &=& \sigma(X(t-)) dL(t) \\ X^ x(0)&=&x,\quad x\in{\mathbb{R}}^ d, \end{array}\right. \end{array} $$

where \(\sigma :{\mathbb {R}}^ d\to {\mathbb {R}}^ d\) is globally Lipschitz continuous and L={L(t):t≥0} is a Lévy process. Under this condition on σ it is well known that the above problem has a unique solution X. Let \((\mathcal {P}_{t})_{t\ge 0}\) be the Markovian semigroup associated to X defined by \(\left ({\mathcal {P}}_{t} f\right ) (x) := \mathbb {E} \left [ f(X^ x(t))\right ]\), t≥0, \(x\in {\mathbb {R}}^{d}\), \(f\in \mathcal {B}_{b}({\mathbb {R}}^{d})\). Let B be a pseudo–differential operator characterized by its symbol q. Fix \(\rho \in \mathbb {R}\). In this article we investigate under which conditions on σ, L and q there exist two constants γ>0 and C>0 such that

$$\left| B {\mathcal{P}}_{t} u \right|_{H^{\rho}_{2}} \le C \, t^{-\gamma} \,\left| u \right|_{H^{\rho}_{2}}, \quad \forall u \in {H^{\rho}_{2}}(\mathbb{R}^{d} ),\, t>0. $$

     

Regularity for free interface variational problems in a general class of gradients

We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form

$$\begin{aligned} (u,A) \quad \mapsto \quad \int _\Omega 2fu \,\mathrm {d}x \; - \int _{\Omega \cap A} \sigma _1\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; - \int _{\Omega {\setminus } A} \sigma _2\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; + \; \text {Per }(A;\overline{\Omega }), \end{aligned}$$

where \(\Omega \) is a bounded Lipschitz domain, \(A\subset \mathbb {R}^N\) is a Borel set, \(u:\Omega \subset \mathbb {R}^N \rightarrow \mathbb {R}^d\), \(\mathscr {A}\) is an operator of gradient form, and \(\sigma _1, \sigma _2\) are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on f, we show for a solution (w, A), that the topological boundary of \(A \cap \Omega \) is locally a \(\mathrm {C}^1\)-hypersurface up to a closed set of zero \(\mathscr {H}^{N-1}\)-measure.