Godel's Incompleteness Phenomenon - Computationally

We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of such theories are undecidable

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

An empirical resource for discovering cognitive principles of discourse organisation: the ANNODIS corpus

International audienceThis paper describes the ANNODIS resource, a discourse-level annotated corpus for French. The corpus combines two perspectives on discourse: a bottom-up approach and a top-down approach. The bottom-up view incrementally builds a structure from elementary discourse units, while the top-down view focuses on the selective annotation of multi-level discourse structures. The corpus is composed of texts that are diversified with respect to genre, length and type of discursive organisation. The methodology followed here involves an iterative design of annotation guidelines in order to reach satisfactory inter-annotator agreement levels. This allows us to raise a few issues relevant for the comparison of such complex objects as discourse structures. The corpus also serves as a source of empirical evidence for discourse theories. We present here two first analyses taking advantage of this new annotated corpus --one that tested hypotheses on constraints governing discourse structure, and another that studied the variations in composition and signalling of multi-level discourse structures

Epistemic virtues, metavirtues, and computational complexity

I argue that considerations about computational complexity show that all finite agents need characteristics like those that have been called epistemic virtues. The necessity of these virtues follows in part from the nonexistence of shortcuts, or efficient ways of finding shortcuts, to cognitively expensive routines. It follows that agents must possess the capacities – metavirtues –of developing in advance the cognitive virtues they will need when time and memory are at a premium

Strong Turing Degrees for Additive BSS RAM's

For the additive real BSS machines using only constants 0 and 1 and order tests we consider the corresponding Turing reducibility and characterize some semi-decidable decision problems over the reals. In order to refine, step-by-step, a linear hierarchy of Turing degrees with respect to this model, we define several halting problems for classes of additive machines with different abilities and construct further suitable decision problems. In the construction we use methods of the classical recursion theory as well as techniques for proving bounds resulting from algebraic properties. In this way we extend a known hierarchy of problems below the halting problem for the additive machines using only equality tests and we present a further subhierarchy of semi-decidable problems between the halting problems for the additive machines using only equality tests and using order tests, respectively

Characterizing the strongly jump-traceable sets via randomness

We show that if a set $A$ is computable from every superlow 1-random set, then $A$ is strongly jump-traceable. This theorem shows that the computably enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets computable from every superlow 1-random set. We also prove the analogous result for superhighness: a c.e.\ set is strongly jump-traceable if and only if it is computable from every superhigh 1-random set. Finally, we show that for each cost function $c$ with the limit condition there is a 1-random $\Delta^0_2$ set $Y$ such that every c.e.\ set $A \le_T Y$ obeys $c$. To do so, we connect cost function strength and the strength of randomness notions. This result gives a full correspondence between obedience of cost functions and being computable from $\Delta^0_2$ 1-random sets.Comment: 41 page