1,789 research outputs found
Kleene Algebra with Converse
International audienceThe equational theory generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure was studied by Bernátsky, Bloom, Ésik, and Stefanescu in 1995. We reformulate some of their proofs in syntactic and elementary terms, and we provide a new algorithm to decide the corresponding theory. This algorithm is both simpler and more efficient; it relies on an alternative automata construction, that allows us to prove that the considered equational theory lies in the complexity class PSPACE. Specific regular languages appear at various places in the proofs. Those proofs were made tractable by considering appropriate automata recognising those languages, and exploiting symmetries in those automata
On Tools for Completeness of Kleene Algebra with Hypotheses
In the literature on Kleene algebra, a number of variants have been proposed
which impose additional structure specified by a theory, such as Kleene algebra
with tests (KAT) and the recent Kleene algebra with observations (KAO), or make
specific assumptions about certain constants, as for instance in NetKAT. Many
of these variants fit within the unifying perspective offered by Kleene algebra
with hypotheses, which comes with a canonical language model constructed from a
given set of hypotheses. For the case of KAT, this model corresponds to the
familiar interpretation of expressions as languages of guarded strings. A
relevant question therefore is whether Kleene algebra together with a given set
of hypotheses is complete with respect to its canonical language model. In this
paper, we revisit, combine and extend existing results on this question to
obtain tools for proving completeness in a modular way. We showcase these tools
by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and
we prove completeness for new variants of KAT: KAT extended with a constant for
the full relation, KAT extended with a converse operation, and a version of KAT
where the collection of tests only forms a distributive lattice
Kleene algebra with domain
We propose Kleene algebra with domain (KAD), an extension of Kleene algebra
with two equational axioms for a domain and a codomain operation, respectively.
KAD considerably augments the expressiveness of Kleene algebra, in particular
for the specification and analysis of state transition systems. We develop the
basic calculus, discuss some related theories and present the most important
models of KAD. We demonstrate applicability by two examples: First, an
algebraic reconstruction of Noethericity and well-foundedness; second, an
algebraic reconstruction of propositional Hoare logic.Comment: 40 page
Kleene Algebras, Regular Languages and Substructural Logics
We introduce the two substructural propositional logics KL, KL+, which use
disjunction, fusion and a unary, (quasi-)exponential connective. For both we
prove strong completeness with respect to the interpretation in Kleene algebras
and a variant thereof. We also prove strong completeness for language models,
where each logic comes with a different interpretation. We show that for both
logics the cut rule is admissible and both have a decidable consequence
relation.Comment: In Proceedings GandALF 2014, arXiv:1408.556
- …