Giulia Simi

Product logic and probabilistic Ulam games

There is a well-known game semantics for Lukasiewicz logic, introduced by Daniele Mundici, namely the Renyi-Ulam game. Records in a Reny-Ulam game are coded by functions, which constitute an MV-algebra, and it is possible to prove a completeness theorem with respect to this semantics. In this paper we investigate some probabilistic variants of the Renyi-Ulam game, and we prove that some of them constitute a complete game semantics for product logic, whilst some other constitute a game semantics for a logic between @PMTL and product logic.

Logic and probabilistic systems

Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set ℐ of the sentences that get limit probability 1 is a Π3—theory, in fact ℐ can be a Π3—complete set. We prove incompleteness results for this setting, by showing for instance that for every k > 0 there are true Π3—sentences that get limit probability less than 1/2k. No set ℐ as above can contain the set of all true Π3—sentences, although we exhibit some ℐ containing all the true Σ2—sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent.

On learning to coordinate: random bits help, insightful normal forms, and competency isomorphisms

A mere bounded number of random bits judiciously employed by a probabilistically correct algorithmic coordinator is shown to increase the power of learning to coordinate compared to deterministic algorithmic coordinators. Furthermore, these probabilistic algorithmic coordinators are provably not characterized in power by teams of deterministic ones. An insightful, enumeration technique based, normal form characterization of the classes that are learnable by total computable coordinators is given. These normal forms are for insight only since it is shown that the complexity of the normal form of a total computable coordinator can be infeasible compared to the original coordinator. Montagna and Osherson showed that the competence class of a total coordinator cannot be strictly improved by another total coordinator. It is shown in the present paper that the competencies of any two total coordinators are the same modulo isomorphism. Furthermore, a completely effective, index set version of this competency isomorphism result is given, where all the coordinators are total computable. We also investigate the competence classes of total coordinators from the points of view of topology and descriptive set theory.

A note on algebras of languages

We study the Boolean algebras R,CS,D of regular languages, context-sensitive languages and decidable languages, respectively, over any alphabet. It is well known that R?CS?D, with proper inclusions. After observing that these Boolean algebras are all isomorphic, we study some immunity properties: for instance we prove that for every coinfinite decidable language L there exists a decidable language L? such that L?L?, L??L is infinite, and there is no context-sensitive language L?, with L??L? unless L??L is finite; similarly, for every coinfinite regular language L there exists a context-sensitive language L? such that L?L?, L??L is infinite, and there is no regular language L? such that L??L?, unless L??L is finite.

TRIAL AND ERROR MATHEMATICS I: DIALECTICAL AND QUASIDIALECTICAL SYSTEMS

We define and study quasidialectical systems, which are an extension of Magari’s dialectical systems, designed to make Magari’s formalization of trial and error mathematics more adherent to the real mathematical practice of revision: our proposed extension follows, and in several regards makes more precise, varieties of empiricist positions a la Lakatos. We prove several properties of quasidialectical systems and of the sets that they represent, called quasidialectical sets. In particular, we prove that the quasidialectical sets are

The reconstruction of a subclass of domino tilings from two projections

{\rm{\Delta }}_2^0

Paradigms in measure theoretic learning and in informant learning

sets in the arithmetical hierarchy. We distinguish between “loopless” quasidialectal systems, and quasidialectical systems “with loops”. The latter ones represent exactly those coinfinite c.e. sets, that are not simple. In a subsequent paper we will show that whereas the dialectical sets are ω -c.e., the quasidialectical sets spread out throughout all classes of the Ershov hierarchy of the

PAC learning of probability distributions over a discrete domain

{\rm{\Delta }}_2^0

On Learning to Coordinate

sets.

The NP-completeness of a tomographical problem on bicolored domino tilings

We present a new way of studying the classical and still unsolved problem of the reconstruction of a domino tiling from its row and column projections. After giving a simple greedy strategy for solving the problem from one projection, we introduce the concept of degree of a domino tiling. We generalize an algorithm for the reconstruction of domino tilings of degree two from two projections, to domino tilings of degree three and four.