4,376 research outputs found
Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics
We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer
and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We
give new analytic proof systems for A and use the embeddings to derive
corresponding systems for L. These include: hypersequent calculi for A and L
and terminating versions of these calculi; labelled single sequent calculi for
A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur
Grafting Hypersequents onto Nested Sequents
We introduce a new Gentzen-style framework of grafted hypersequents that
combines the formalism of nested sequents with that of hypersequents. To
illustrate the potential of the framework, we present novel calculi for the
modal logics and , as well as for extensions of the
modal logics and with the axiom for shift
reflexivity. The latter of these extensions is also known as
in the context of deontic logic. All our calculi enjoy syntactic cut
elimination and can be used in backwards proof search procedures of optimal
complexity. The tableaufication of the calculi for and
yields simplified prefixed tableau calculi for these logic
reminiscent of the simplified tableau system for , which might be
of independent interest
Automated Reasoning over Deontic Action Logics with Finite Vocabularies
In this paper we investigate further the tableaux system for a deontic action
logic we presented in previous work. This tableaux system uses atoms (of a
given boolean algebra of action terms) as labels of formulae, this allows us to
embrace parallel execution of actions and action complement, two action
operators that may present difficulties in their treatment. One of the
restrictions of this logic is that it uses vocabularies with a finite number of
actions. In this article we prove that this restriction does not affect the
coherence of the deduction system; in other words, we prove that the system is
complete with respect to language extension. We also study the computational
complexity of this extended deductive framework and we prove that the
complexity of this system is in PSPACE, which is an improvement with respect to
related systems.Comment: In Proceedings LAFM 2013, arXiv:1401.056
Theorem Proving and Partial Proof Search for Intuitionistic Propositional Logic Using a Permutation-free Calculus with Loop-Checking
this paper the history mechanism is developed in two ways and applied t
Sequent Calculus in the Topos of Trees
Nakano's "later" modality, inspired by G\"{o}del-L\"{o}b provability logic,
has been applied in type systems and program logics to capture guarded
recursion. Birkedal et al modelled this modality via the internal logic of the
topos of trees. We show that the semantics of the propositional fragment of
this logic can be given by linear converse-well-founded intuitionistic Kripke
frames, so this logic is a marriage of the intuitionistic modal logic KM and
the intermediate logic LC. We therefore call this logic
. We give a sound and cut-free complete sequent
calculus for via a strategy that decomposes
implication into its static and irreflexive components. Our calculus provides
deterministic and terminating backward proof-search, yields decidability of the
logic and the coNP-completeness of its validity problem. Our calculus and
decision procedure can be restricted to drop linearity and hence capture KM.Comment: Extended version, with full proof details, of a paper accepted to
FoSSaCS 2015 (this version edited to fix some minor typos
Space Efficiency of Propositional Knowledge Representation Formalisms
We investigate the space efficiency of a Propositional Knowledge
Representation (PKR) formalism. Intuitively, the space efficiency of a
formalism F in representing a certain piece of knowledge A, is the size of the
shortest formula of F that represents A. In this paper we assume that knowledge
is either a set of propositional interpretations (models) or a set of
propositional formulae (theorems). We provide a formal way of talking about the
relative ability of PKR formalisms to compactly represent a set of models or a
set of theorems. We introduce two new compactness measures, the corresponding
classes, and show that the relative space efficiency of a PKR formalism in
representing models/theorems is directly related to such classes. In
particular, we consider formalisms for nonmonotonic reasoning, such as
circumscription and default logic, as well as belief revision operators and the
stable model semantics for logic programs with negation. One interesting result
is that formalisms with the same time complexity do not necessarily belong to
the same space efficiency class
On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems
This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus
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