70 research outputs found
First Steps in Synthetic Computability Theory
AbstractComputability theory, which investigates computable functions and computable sets, lies at the foundation of computer science. Its classical presentations usually involve a fair amount of Gödel encodings which sometime obscure ingenious arguments. Consequently, there have been a number of presentations of computability theory that aimed to present the subject in an abstract and conceptually pleasing way. We build on two such approaches, Hyland's effective topos and Richman's formulation in Bishop-style constructive mathematics, and develop basic computability theory, starting from a few simple axioms. Because we want a theory that resembles ordinary mathematics as much as possible, we never speak of Turing machines and Gödel encodings, but rather use familiar concepts from set theory and topology
A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis
The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable. We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finitenes
On Generalized Computable Universal Priors and their Convergence
Solomonoff unified Occam's razor and Epicurus' principle of multiple
explanations to one elegant, formal, universal theory of inductive inference,
which initiated the field of algorithmic information theory. His central result
is that the posterior of the universal semimeasure M converges rapidly to the
true sequence generating posterior mu, if the latter is computable. Hence, M is
eligible as a universal predictor in case of unknown mu. The first part of the
paper investigates the existence and convergence of computable universal
(semi)measures for a hierarchy of computability classes: recursive, estimable,
enumerable, and approximable. For instance, M is known to be enumerable, but
not estimable, and to dominate all enumerable semimeasures. We present proofs
for discrete and continuous semimeasures. The second part investigates more
closely the types of convergence, possibly implied by universality: in
difference and in ratio, with probability 1, in mean sum, and for Martin-Loef
random sequences. We introduce a generalized concept of randomness for
individual sequences and use it to exhibit difficulties regarding these issues.
In particular, we show that convergence fails (holds) on generalized-random
sequences in gappy (dense) Bernoulli classes.Comment: 22 page
Kolmogorov Complexity and Solovay Functions
Solovay proved that there exists a computable upper bound f of the
prefix-free Kolmogorov complexity function K such that f (x) = K(x) for
infinitely many x. In this paper, we consider the class of computable functions
f such that K(x) <= f (x)+O(1) for all x and f (x) <= K(x) + O(1) for
infinitely many x, which we call Solovay functions. We show that Solovay
functions present interesting connections with randomness notions such as
Martin-L\"of randomness and K-triviality
Trakhtenbrot's Theorem in Coq, A Constructive Approach to Finite Model Theory
We study finite first-order satisfiability (FSAT) in the constructive setting
of dependent type theory. Employing synthetic accounts of enumerability and
decidability, we give a full classification of FSAT depending on the
first-order signature of non-logical symbols. On the one hand, our development
focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as
the signature contains an at least binary relation symbol. Our proof proceeds
by a many-one reduction chain starting from the Post correspondence problem. On
the other hand, we establish the decidability of FSAT for monadic first-order
logic, i.e. where the signature only contains at most unary function and
relation symbols, as well as the enumerability of FSAT for arbitrary enumerable
signatures. All our results are mechanised in the framework of a growing Coq
library of synthetic undecidability proofs
Categoricity
After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those involving the distinction between characterizing a system and axiomatizing the truths of a syste
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