1,576 research outputs found
Constructive Cardinality
We describe a set of necessary conditions that are useful for generating propagation algorithms for the cardinality operator as well as for over-constrained problems with preferences. Constructive disjunction as well as the entailments rules originally proposed for the cardinality operator can be seen as simple cases of these necessary conditions. In addition these necessary conditions have the advantage of providing more pruning
Logic Programming as Constructivism
The features of logic programming that
seem unconventional from the viewpoint of classical logic
can be explained in terms of constructivistic logic. We
motivate and propose a constructivistic proof theory of
non-Horn logic programming. Then, we apply this formalization
for establishing results of practical interest.
First, we show that 'stratification can be motivated in a
simple and intuitive way. Relying on similar motivations,
we introduce the larger classes of 'loosely stratified' and
'constructively consistent' programs. Second, we give a
formal basis for introducing quantifiers into queries and
logic programs by defining 'constructively domain
independent* formulas. Third, we extend the Generalized
Magic Sets procedure to loosely stratified and constructively
consistent programs, by relying on a 'conditional
fixpoini procedure
Type-elimination-based reasoning for the description logic SHIQbs using decision diagrams and disjunctive datalog
We propose a novel, type-elimination-based method for reasoning in the
description logic SHIQbs including DL-safe rules. To this end, we first
establish a knowledge compilation method converting the terminological part of
an ALCIb knowledge base into an ordered binary decision diagram (OBDD) which
represents a canonical model. This OBDD can in turn be transformed into
disjunctive Datalog and merged with the assertional part of the knowledge base
in order to perform combined reasoning. In order to leverage our technique for
full SHIQbs, we provide a stepwise reduction from SHIQbs to ALCIb that
preserves satisfiability and entailment of positive and negative ground facts.
The proposed technique is shown to be worst case optimal w.r.t. combined and
data complexity and easily admits extensions with ground conjunctive queries.Comment: 38 pages, 3 figures, camera ready version of paper accepted for
publication in Logical Methods in Computer Scienc
On the logical structure of choice and bar induction principles
We develop an approach to choice principles and their contrapositive
bar-induction principles as extensionality schemes connecting an "intensional"
or "effective" view of respectively ill-and well-foundedness properties to an
"extensional" or "ideal" view of these properties. After classifying and
analysing the relations between different intensional definitions of
ill-foundedness and well-foundedness, we introduce, for a domain , a
codomain and a "filter" on finite approximations of functions from
to , a generalised form GDC of the axiom of dependent choice and
dually a generalised bar induction principle GBI such that:
GDC intuitionistically captures the strength of
the general axiom of choice expressed as when is a
filter that derives point-wise from a relation on without
introducing further constraints,
the Boolean Prime Filter Theorem / Ultrafilter Theorem if is
the two-element set (for a constructive definition of prime
filter),
the axiom of dependent choice if ,
Weak K{\"o}nig's Lemma if and (up
to weak classical reasoning)
GBI intuitionistically captures the strength of
G{\"o}del's completeness theorem in the form validity implies
provability for entailment relations if ,
bar induction when ,
the Weak Fan Theorem when and .
Contrastingly, even though GDC and GBI smoothly capture
several variants of choice and bar induction, some instances are inconsistent,
e.g. when is and is .Comment: LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun
2021, Rome / Virtual, Ital
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