Dick de Jongh

Formal Learning Theory

Disputes about human nature inevitably evolve into disputes about the acquisition of knowledge, since the core issue in each controversy is the interaction of environment and human genetic endowment. Over the past two decades, a novel approach to this interaction has been developed by automata theorists and mathematicians. Within this approach, the impact of experience on the emerging competence of an organism is construed as a species of functional dependency. Using methods proper to the theory of computation, this construal has yielded mathematical insights that occasionally permit speculation about the acquisition of competence to be cast in sharper terms than heretofore. This cluster of related mathematical results has come to be known as (formal) learning theory. 1

The Logic of Provability

This chapter is dedicated to the memory of George Boolos. From the start of the subject until his death on 27 May 1996 he was the prime inspirer of the work in the logic of provability.

Provability logics for relative interpretability

In this paper the system IL for relative interpretability described in Visser (1988) is studied.1 In IL formulae A ⊳ B (read: A interprets B) are added to the provability logic L. The intended interpretation of a formula A ⊳ B in an (arithmetical) theory T is: T + B is relatively interpretable in T + A. The system has been shown to be sound with respect to such arithmetical interpretations (Svejdar 1983, Montagna 1984, Visser 1986, 1988P).

Characterization of strongly equivalent logic programs in intermediate logics

The non-classical, nonmonotonic inference relation associated with the answer set semantics for logic programs gives rise to a relationship of strong equivalence between logical programs that can be verified in 3-valued Godel logic, G3, the strongest non-classical intermediate propositional logic (Lifschitz et al., 2001). In this paper we will show that KC (the logic obtained by adding axiom

Angluin's theorem for indexed families of r.e. sets and applications

\neg A\vee\neg\neg A

Preference, priorities and belief

to intuitionistic logic), is the weakest intermediate logic for which strongly equivalent logic programs, in a language allowing negations, are logically equivalent.

A simplification of a completeness proof of Guaspari and Solovay

We extend Angluin s theorem to char acterize identi ability of indexed families of r e languages as opposed to indexed families of recursive languages We also prove some variants characterizing conservativity and two other similar restrictions paralleling Zeug mann Lange and Kapur s results for indexed families of recursive languages

On the proof of Solovay's theorem

In this paper we consider preference over objects. We show how this preference can be derived from priorities, properties of these objects, a concept which is initially from optimality theory. We do this both in the case when an agent has complete information and in the case when an agent only has beliefs about the properties. After the single agent case we also consider the multi-agent case. In each of these cases, we construct preference logics, some of them extending the standard logic of belief. This leads to interesting connections between preference and beliefs. We strengthen the usual completeness results for logics of this kind to representation theorems. The representation theorems describe the reasoning that is valid for preference relations that have been obtained from priorities. In the multi-agent case, these representation theorems are strengthened to the special cases of cooperative and competitive agents. We study preference change with regard to changes of the priority sequence, and change of beliefs. We apply the dynamic epistemic logic approach, and in consequence reduction axioms are presented. We conclude with some possible directions for future work.

Computations in fragments of intuitionistic propositional logic

The modal completeness proofs of Guaspari and Solovay (1979) for their systems R and R− are improved and the relationship between R and R− is clarified.

Intuitionistic implication without disjunction

Solovays 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof be replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic (the so-called Rosser logic of Gauspari-Solovay, 1979) so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in particular IΔ0+EXP. The method is adapted to obtain a similar completeness result for the Rosser logic.