431,917 research outputs found
Logic Programming with Default, Weak and Strict Negations
This paper treats logic programming with three kinds of negation: default,
weak and strict negations. A 3-valued logic model theory is discussed for logic
programs with three kinds of negation. The procedure is constructed for
negations so that a soundness of the procedure is guaranteed in terms of
3-valued logic model theory.Comment: 14 pages, to appear in Theory and Practice of Logic Programming
(TPLP
Classical Mathematics for a Constructive World
Interactive theorem provers based on dependent type theory have the
flexibility to support both constructive and classical reasoning. Constructive
reasoning is supported natively by dependent type theory and classical
reasoning is typically supported by adding additional non-constructive axioms.
However, there is another perspective that views constructive logic as an
extension of classical logic. This paper will illustrate how classical
reasoning can be supported in a practical manner inside dependent type theory
without additional axioms. We will see several examples of how classical
results can be applied to constructive mathematics. Finally, we will see how to
extend this perspective from logic to mathematics by representing classical
function spaces using a weak value monad.Comment: v2: Final copy for publicatio
Natural Factors of the Medvedev Lattice Capturing IPC
Skvortsova showed that there is a factor of the Medvedev lattice which
captures intuitionistic propositional logic (IPC). However, her factor is
unnatural in the sense that it is constructed in an ad hoc manner. We present a
more natural example of such a factor. We also show that for every non-trivial
factor of the Medvedev lattice its theory is contained in Jankov's logic, the
deductive closure of IPC plus the weak law of the excluded middle. This answers
a question by Sorbi and Terwijn
The Strength of Abstraction with Predicative Comprehension
Frege's theorem says that second-order Peano arithmetic is interpretable in
Hume's Principle and full impredicative comprehension. Hume's Principle is one
example of an abstraction principle, while another paradigmatic example is
Basic Law V from Frege's Grundgesetze. In this paper we study the strength of
abstraction principles in the presence of predicative restrictions on the
comprehension schema, and in particular we study a predicative Fregean theory
which contains all the abstraction principles whose underlying equivalence
relations can be proven to be equivalence relations in a weak background
second-order logic. We show that this predicative Fregean theory interprets
second-order Peano arithmetic.Comment: Forthcoming in Bulletin of Symbolic Logic. Slight change in title
from previous version, at request of referee
Learning Weak Constraints in Answer Set Programming
This paper contributes to the area of inductive logic programming by
presenting a new learning framework that allows the learning of weak
constraints in Answer Set Programming (ASP). The framework, called Learning
from Ordered Answer Sets, generalises our previous work on learning ASP
programs without weak constraints, by considering a new notion of examples as
ordered pairs of partial answer sets that exemplify which answer sets of a
learned hypothesis (together with a given background knowledge) are preferred
to others. In this new learning task inductive solutions are searched within a
hypothesis space of normal rules, choice rules, and hard and weak constraints.
We propose a new algorithm, ILASP2, which is sound and complete with respect to
our new learning framework. We investigate its applicability to learning
preferences in an interview scheduling problem and also demonstrate that when
restricted to the task of learning ASP programs without weak constraints,
ILASP2 can be much more efficient than our previously proposed system.Comment: To appear in Theory and Practice of Logic Programming (TPLP),
Proceedings of ICLP 201
Uniform Interpolation for Coalgebraic Fixpoint Logic
We use the connection between automata and logic to prove that a wide class
of coalgebraic fixpoint logics enjoys uniform interpolation. To this aim, first
we generalize one of the central results in coalgebraic automata theory, namely
closure under projection, which is known to hold for weak-pullback preserving
functors, to a more general class of functors, i.e.; functors with
quasi-functorial lax extensions. Then we will show that closure under
projection implies definability of the bisimulation quantifier in the language
of coalgebraic fixpoint logic, and finally we prove the uniform interpolation
theorem
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