150 research outputs found
Learning algebraic structures from text
AbstractThe present work investigates the learnability of classes of substructures of some algebraic structures: submonoids and subgroups of given groups, ideals of given commutative rings, subfields of given vector spaces. The learner sees all positive data but no negative one and converges to a program enumerating or computing the set to be learned. Besides semantical (BC) and syntactical (Ex) convergence also the more restrictive ordinal bounds on the number of mind changes are considered. The following is shown: (a) Learnability depends much on the amount of semantic knowledge given at the synthesis of the learner where this knowledge is represented by programs for the algebraic operations, codes for prominent elements of the algebraic structure (like 0 and 1 fields) and certain parameters (like the dimension of finite-dimensional vector spaces). For several natural examples, good knowledge of the semantics may enable to keep ordinal mind change bounds while restricted knowledge may either allow only BC-convergence or even not permit learnability at all.(b) The class of all ideals of a recursive ring is BC-learnable iff the ring is Noetherian. Furthermore, one has either only a BC-learner outputting enumerable indices or one can already get an Ex-learner converging to decision procedures and respecting an ordinal bound on the number of mind changes. The ring is Artinian iff the ideals can be Ex-learned with a constant bound on the number of mind changes, this constant is the length of the ring. Ex-learnability depends not only on the ring but also on the representation of the ring. Polynomial rings over the field of rationals with n variables have exactly the ordinal mind change bound Ďn in the standard representation. Similar results can be established for unars. Noetherian unars with one function can be learned with an ordinal mind change bound aĎ for some a
Weakly complete axiomatization of exogenous quantum propositional logic
A weakly complete finitary axiomatization for EQPL (exogenous quantum
propositional logic) is presented. The proof is carried out using a non trivial
extension of the Fagin-Halpern-Megiddo technique together with three Henkin
style completions.Comment: 28 page
The Fundamental Theorems of Welfare Economics, DSGE and the Theory of Policy - Computable & Constructive Foundations
The genesis and the path towards what has come to be called the DSGE model is traced, from its origins in the Arrow-Debreu General Equilibrium model (ADGE), via Scarf's Computable General Equilibrium model (CGE) and its applied version as Applied Computable General Equilibrium model (ACGE), to its ostensible dynamization as a Recursive Competitive Equilibrium (RCE). It is shown that these transformations of the ADGE - including the fountainhead - are computably and constructively untenable. The policy implications of these (negative) results, via the Fundamental Theorems of Welfare Economics in particular, and against the backdrop of the mathematical theory of economic policy in general, are also discussed (again from computable and constructive points of view). Suggestions for going 'beyond DSGE' are, then, outlined on the basis of a framework that is underpinned - from the outset - by computability and constructivity considerationsComputable General Equilibrium, Dynamic Stochastic General Equilibrium, Computability, Constructivity, Fundamental Theorems of Welfare Economics, Theory of Policy, Coupled Nonlinear Dynamic
Computable structure theory on Banach spaces
In this dissertation we investigate computability notions on several different Banach spaces, namely the separable -spaces and .
It was demonstrated by McNicholl \cite{TM} that the halting problem is a necessary and sufficient condition for the existence of computable isometric isomorphisms between any two computable representations of the purely atomic -spaces (e.g. ) where the underlying measure space is generated by finitely many atoms. In the case where the underlying measure space is generated by finitely many atoms (such as in ), McNicholl also proved that it is always possible to find an algorithm that computes isometric isomorphisms between any two computable representations. Clanin, McNicholl, and Stull \cite{CMS} proved a similar result. Namely they proved that for any two computable representations of a non-atomic -space (e.g. ) there is always a computable isometric isomorphism between them. We both continue and complete the classification of the separable -spaces up to degree of categoricity by investigating the hybrid -spaces, whose underlying measure spaces consist of both atomic and non-atomic parts, and determine how much computational power is necessary and sufficient to compute isometric isomorphisms between any two copies of these spaces.
Secondly, we continue a line of inquiry initialized by Melnikov and Ng in 2014, who proved that for (i.e. the Banach space of all continuous functions on the closed unit interval) there is a pair of computable representations between which there is no computable isometric isomorphism. They achieved this by constructing one of the representations in such a manner that the constant unit function \textbf{1} is not computable, contrasting with the other representation in which \textbf{1} is computable. We show in Chapter 5 that given any computable representation of as a Banach space the halting set always computes \textbf{1}. We also determine how much extra computational power beyond that of the halting set is sufficient to compute the modulus operator within any computable representation. Lastly, we use these two results to determine how much power is sufficient to compute an isometric isomorphism between any two computable representations of a restricted class of representations of
A hierarchy of languages, logics, and mathematical theories
We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal.
We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic
This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence
Proceedings of JAC 2010. JournĂŠes Automates Cellulaires
The second Symposium on Cellular Automata âJourn´ees Automates Cellulairesâ (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route TurkuâMariehamnâTurku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku.
The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume.
The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible.
These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast
- âŚ