FO(FD): Extending classical logic with rule-based fixpoint definitions

We introduce fixpoint definitions, a rule-based reformulation of fixpoint constructs. The logic FO(FD), an extension of classical logic with fixpoint definitions, is defined. We illustrate the relation between FO(FD) and FO(ID), which is developed as an integration of two knowledge representation paradigms. The satisfiability problem for FO(FD) is investigated by first reducing FO(FD) to difference logic and then using solvers for difference logic. These reductions are evaluated in the computation of models for FO(FD) theories representing fairness conditions and we provide potential applications of FO(FD).Comment: Presented at ICLP 2010. 16 pages, 1 figur

Call-by-Value and Call-by-Name Dual Calculi with Inductive and Coinductive Types

This paper extends the dual calculus with inductive types and coinductive types. The paper first introduces a non-deterministic dual calculus with inductive and coinductive types. Besides the same duality of the original dual calculus, it has the duality of inductive and coinductive types, that is, the duality of terms and coterms for inductive and coinductive types, and the duality of their reduction rules. Its strong normalization is also proved, which is shown by translating it into a second-order dual calculus. The strong normalization of the second-order dual calculus is proved by translating it into the second-order symmetric lambda calculus. This paper then introduces a call-by-value system and a call-by-name system of the dual calculus with inductive and coinductive types, and shows the duality of call-by-value and call-by-name, their Church-Rosser properties, and their strong normalization. Their strong normalization is proved by translating them into the non-deterministic dual calculus with inductive and coinductive types.Comment: The conference version of this paper has appeared in RTA 200

Extending Nunchaku to Dependent Type Theory

Nunchaku is a new higher-order counterexample generator based on a sequence of transformations from polymorphic higher-order logic to first-order logic. Unlike its predecessor Nitpick for Isabelle, it is designed as a stand-alone tool, with frontends for various proof assistants. In this short paper, we present some ideas to extend Nunchaku with partial support for dependent types and type classes, to make frontends for Coq and other systems based on dependent type theory more useful.Comment: In Proceedings HaTT 2016, arXiv:1606.0542

Problem solving in ID-logic with aggregates: some experiments

The goal of the LP+ project at the K.U.Leuven is to design an expressive logic, suitable for declarative knowledge representation, and to develop intelligent systems based on Logic Programming technology for solving computational problems using the declarative specifications. The ID-logic is an integration of typed classical logic and a definition logic. Different abductive solvers for this language are being developed. This paper is a report of the integration of high order aggregates into ID-logic and the consequences on the solver SLDNFA.Comment: 9 pages conference: NMR2000, special track on abductive reasonin

Logic Programming for Describing and Solving Planning Problems

A logic programming paradigm which expresses solutions to problems as stable models has recently been promoted as a declarative approach to solving various combinatorial and search problems, including planning problems. In this paradigm, all program rules are considered as constraints and solutions are stable models of the rule set. This is a rather radical departure from the standard paradigm of logic programming. In this paper we revisit abductive logic programming and argue that it allows a programming style which is as declarative as programming based on stable models. However, within abductive logic programming, one has two kinds of rules. On the one hand predicate definitions (which may depend on the abducibles) which are nothing else than standard logic programs (with their non-monotonic semantics when containing with negation); on the other hand rules which constrain the models for the abducibles. In this sense abductive logic programming is a smooth extension of the standard paradigm of logic programming, not a radical departure.Comment: 8 pages, no figures, Eighth International Workshop on Nonmonotonic Reasoning, special track on Representing Actions and Plannin

A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice

We present a Kleene realizability semantics for the intensional level of the Minimalist Foundation, for short mtt, extended with inductively generated formal topologies, Church's thesis and axiom of choice. This semantics is an extension of the one used to show consistency of the intensional level of the Minimalist Foundation with the axiom of choice and formal Church's thesis in previous work. A main novelty here is that such a semantics is formalized in a constructive theory represented by Aczel's constructive set theory CZF extended with the regular extension axiom