4,059 research outputs found
On the isomorphism problem of concept algebras
Weakly dicomplemented lattices are bounded lattices equipped with two unary
operations to encode a negation on {\it concepts}. They have been introduced to
capture the equational theory of concept algebras \cite{Wi00}. They generalize
Boolean algebras. Concept algebras are concept lattices, thus complete
lattices, with a weak negation and a weak opposition. A special case of the
representation problem for weakly dicomplemented lattices, posed in
\cite{Kw04}, is whether complete {\wdl}s are isomorphic to concept algebras. In
this contribution we give a negative answer to this question (Theorem
\ref{T:main}). We also provide a new proof of a well known result due to M.H.
Stone \cite{St36}, saying that {\em each Boolean algebra is a field of sets}
(Corollary \ref{C:Stone}). Before these, we prove that the boundedness
condition on the initial definition of {\wdl}s (Definition \ref{D:wdl}) is
superfluous (Theorem \ref{T:wcl}, see also \cite{Kw09}).Comment: 15 page
Finite Distributive Concept Algebras
Concept algebras are concept lattices enriched by a weak negation and a weak opposition. In Ganter and Kwuida (Contrib. Gen. Algebra, 14:63-72, 2004) we gave a contextual description of the lattice of weak negations on a finite lattice. In this contribution1 we use this description to give a characterization of finite distributive concept algebra
A Polynomial Translation of Logic Programs with Nested Expressions into Disjunctive Logic Programs: Preliminary Report
Nested logic programs have recently been introduced in order to allow for
arbitrarily nested formulas in the heads and the bodies of logic program rules
under the answer sets semantics. Nested expressions can be formed using
conjunction, disjunction, as well as the negation as failure operator in an
unrestricted fashion. This provides a very flexible and compact framework for
knowledge representation and reasoning. Previous results show that nested logic
programs can be transformed into standard (unnested) disjunctive logic programs
in an elementary way, applying the negation as failure operator to body
literals only. This is of great practical relevance since it allows us to
evaluate nested logic programs by means of off-the-shelf disjunctive logic
programming systems, like DLV. However, it turns out that this straightforward
transformation results in an exponential blow-up in the worst-case, despite the
fact that complexity results indicate that there is a polynomial translation
among both formalisms. In this paper, we take up this challenge and provide a
polynomial translation of logic programs with nested expressions into
disjunctive logic programs. Moreover, we show that this translation is modular
and (strongly) faithful. We have implemented both the straightforward as well
as our advanced transformation; the resulting compiler serves as a front-end to
DLV and is publicly available on the Web.Comment: 10 pages; published in Proceedings of the 9th International Workshop
on Non-Monotonic Reasonin
The Fan Theorem, its strong negation, and the determinacy of games
IIn the context of a weak formal theory called Basic Intuitionistic
Mathematics , we study Brouwer's Fan Theorem and a strong
negation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem). We
prove that the Fan Theorem is equivalent to contrapositions of a number of
intuitionistically accepted axioms of countable choice and that Kleene's
Alternative is equivalent to strong negations of these statements. We also
discuss finite and infinite games and introduce a constructively useful notion
of determinacy. We prove that the Fan Theorem is equivalent to the
Intuitionistic Determinacy Theorem, saying that every subset of Cantor space
is, in our constructively meaningful sense, determinate, and show that Kleene's
Alternative is equivalent to a strong negation of a special case of this
theorem. We then consider a uniform intermediate value theorem and a
compactness theorem for classical propositional logic, and prove that the Fan
Theorem is equivalent to each of these theorems and that Kleene's Alternative
is equivalent to strong negations of them. We end with a note on a possibly
important statement, provable from principles accepted by Brouwer, that one
might call a Strong Fan Theorem.Comment: arXiv admin note: text overlap with arXiv:1106.273
Efficient Normalization of Linear Temporal Logic
In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem
stating that every formula of Past LTL (the extension of LTL with past
operators) is equivalent to a formula of the form , where
and contain only past operators. Some years later, Chang,
Manna, and Pnueli built on this result to derive a similar normal form for LTL.
Both normalization procedures have a non-elementary worst-case blow-up, and
follow an involved path from formulas to counter-free automata to star-free
regular expressions and back to formulas. We improve on both points. We present
direct and purely syntactic normalization procedures for LTL, yielding a normal
form very similar to the one by Chang, Manna, and Pnueli, that exhibit only a
single exponential blow-up. As an application, we derive a simple algorithm to
translate LTL into deterministic Rabin automata. The algorithm normalizes the
formula, translates it into a special very weak alternating automaton, and
applies a simple determinization procedure, valid only for these special
automata.Comment: Submitted to J. ACM. arXiv admin note: text overlap with
arXiv:2304.08872, arXiv:2005.0047
- …