Mathematics

The blog focusses on algorithmic randomness and its connections to quantum information theory, group theory and its connections to logic, and computability analogs of cardinal characteristics.

This year's blog has focused on the connections of group theory with logic and algorithms. The first post is on automata presentable groups. Then there are several posts related to topological groups, for instance Ivanov and Majcher showing that extreme amenability of closed subgroups of S ??is a Borel property. One post due to Harrison-Trainor and Nies reviews notes by Segal on pseudofinite groups, and attempts an effective version. About 25 percent is on computability and randomness, in particular equivalence of reducibilities weaker than Turing on the K-trivials by Greenberg, Nies and Turetsky, and the effective SMB theorem in the quantum setting by Nies and Tomamichel.

Quantum field theory has successfully generated a number of general conclusions. It seems meaningful to disclose the logical forms of these conclusions. The present paper reports two results. The first result shows the logic of local gauge symmetry and indefinability of mass. The second result shows the logic of Higgs mechanism and definability of mass. The results are obtained by integrating four components, namely, gauge symmetry and Higgs mechanism in quantum field theory, and incompleteness theorem and indefinability theorem in mathematical logic. Godel numbering is the key for arithmetic modeling applied in this paper.

Generalized orthomodular posets were introduced recently by D. Fazio, A. Ledda and the first author of the present paper in order to establish a useful tool for studying the logic of quantum mechanics. They investigated structural properties of these posets. In the present paper we study logical and algebraic properties of these posets. In particular, we investigate conditions under which they can be converted into operator residuated structures. Further, we study their representation by means of algebras (directoids) with everywhere defined operations. We prove congruence properties for the class of algebras assigned to generalized orthomodular posets and, in particular, for a subvariety of this class determined by a simple identity. Finally, in contrast to the fact that the Dedekind-MacNeille completion of an orthomodular poset need not be an orthomodular lattice we show that the Dedekind-MacNeille completion of a stronger version of a generalized orthomodular poset is nearly an orthomodular lattice.

It is customary to expect from a logical system that it can be algebraizable, in the sense that an algebraic companion of the deductive machinery can always be found. Since the inception of da Costa's paraconsistent calculi C n , algebraic equivalents for such systems have been sought. It is known, however, that these systems are not self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, several systems belonging to this class of logics are only characterizable by semantics of a non-deterministic nature. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by extending with rules several LFIs weaker than C 1 , thus obtaining the replacement property (that is, such LFIs turn out to be self-extensional). Moreover, these logics become algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with additional operations. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied. In addition, a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. the proposed algebraic semantics.

We investigate the logical structure of intuitionistic Kripke-Platek set theory IKP, and show that the first-order logic of IKP is intuitionistic first-order logic IQC.

An involutive Stone algebra (IS-algebra) is a structure that is simultaneously a De Morgan algebra and a Stone algebra (i.e. a pseudo-complemented distributive lattice satisfying the well-known Stone identity ~xv~~x=1). IS-algebras have been studied algebraically and topologically since the 1980's, but a corresponding logic (here denoted IS ??) has been introduced only very recently. The logic IS ??is the departing point for the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that IS ??is a conservative expansion of the Belnap-Dunn four-valued logic (i.e. the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic so as to obtain an extension of IS ??. We show that every logic thus defined can be axiomatized by adding a fixed finite set of rule schemata to the corresponding super-Belnap base logic. We also consider a few sample extensions of IS ??that cannot be obtained in the above-described way, but can nevertheless be axiomatized finitely by other methods. Most of our axiomatization results are obtained in two steps: through a multiple-conclusion calculus first, which we then reduce to a traditional one. The multiple-conclusion axiomatizations introduced in this process, being analytic, are of independent interest from a proof-theoretic standpoint. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of IS ??. Indeed, as in the super-Belnap case, we establish that the finitary extensions of IS ??are already uncountably many.

For an ordinal α , an α -ITRM is a machine model of transfinite computability that operates on finitely many registers, each of which can contain an ordinal ?<α ; they were introduced by Koepke in \cite{KM}. In \cite{alpha itrms}, it was shown that the α -ITRM-computable subsets of α are exactly those in a level L β(α) of the constructible hierarchy. It was conjectured in \cite{alpha itrms} that β(α) is the first limit of admissible ordinals above α . Here, we show that this is false; in particular, even the computational strength of ? ? -ITRMs goes far beyond ? CK ? .

We show that a computable function f:R→R has Luzin's property (N) if and only if it reflects Π 1 1 -randomnes, if and only if it reflects Δ 1 1 (O) -randomness, and if and only if it reflects O -Kurtz randomness, but reflecting Martin-Löf randomness or weak-2-randomness does not suffice. Here a function f is said to reflect a randomness notion R if whenever f(x) is R -random, then x is R -random as well. If additionally f is known to have bounded variation, then we show f has Luzin's (N) if and only if it reflects weak-2-randomness, and if and only if it reflects ∅ ′ -Kurtz randomness. This links classical real analysis with algorithmic randomness.

We prove that in the Miller model, every M -separable space of the form C p (X) , where X is metrizable and separable, is productively M -separable, i.e., C p (X)×Y is M -separable for every countable M -separable Y .