. Existing formalisations of (transfinite) inductive definitions in constructive mathematics are reviewed and strong correspondences with LP under least model and perfect model semantics become apparent. I point to fundamental restrictions of these existing formalisations and argue that the well-founded semantics (wfs) overcomes these problems and hence, provides a superior formalisation of the principle of inductive definition. The contribution of this study for LP is that it (re- )introduces the knowledge theoretic interpretation of LP as a logic for representing definitional knowledge. I point to fundamental differences between this knowledge theoretic interpretation of LP and the more commonly known interpretations of LP as default theories or auto-epistemic theories. The relevance is that differences in knowledge theoretic interpretation have strong impact on knowledge representation methodology and on extensions of the LP formalism, for example for representing uncertainty. Keywo...

Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iff-definitions. A second approach was to identify a unique canonical, preferred or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a non-monotonic reasoning formalism strongly related to Default Logic and Auto-epistemic Logic. These competing approaches have resulted in some confusion about the declarative meaning of logic programming. This paper investigates the problem and proposes an alternative epistemological foundation for the canonical model approach, which is not based on common sense but on a solid mathematical information principle. The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions. In particular, logic programs with negation represent non-monotone inductive definitions. It is argued that this thesis results in an alternative justification of the well-founded model as the unique intended model of the logic program. In addition, it equips logic programs with an easy to comprehend meaning

Well-known principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over well-founded sets and iterated induction. In this work, we define a logic formalizing induction over well-founded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for Non-Monotone Inductive Definitions (ID-logic). The semantics of the logic is strongly influenced by the well-founded semantics of logic programming. This paper discusses the formalisation of different forms of (non-)monotone induction by the well-founded semantics and illustrates the use of the logic for formalizing mathematical and common-sense knowledge. To model different types of induction found in mathematics, we define several subclasses of definitions, and show that they are correctly formalized by the well-founded semantics. We also present translations into classical first or second order logic. We develop modularity and totality results and demonstrate their use to analyze and simplify complex definitions. We illustrate the use of the logic for temporal reasoning. The logic formally extends Logic Programming, Abductive Logic Programming and Datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. Categories and Subject Descriptors:... [...]:... 1.

We show that the addition of name induction to the theory EETJ + (LEM -I N ) of explicit elementary types with join yields a theory proof-theoretically equivalent to ID_1.