92 research outputs found
An On-the-fly Tableau-based Decision Procedure for PDL-Satisfiability
We present a tableau-based algorithm for deciding satisfiability for
propositional dynamic logic (PDL) which builds a finite rooted tree with
ancestor loops and passes extra information from children to parents to
separate good loops from bad loops during backtracking. It is easy to
implement, with potential for parallelisation, because it constructs a
pseudo-model ``on the fly'' by exploring each tableau branch independently. But
its worst-case behaviour is 2EXPTIME rather than EXPTIME. A prototype
implementation in the TWB (http://twb.rsise.anu.edu.au) is available.Comment: 26 pages, longer version of article in Methods for Modalities 2007;
improved readability of proof
Relations between Propositional Normal Modal Logics: an Overview
The modal logic literature is notorious for multiple axiomatizations of the same logic and for conflicting overloading of axiom names. Many of the interesting interderivability results are still scattered over the often hard to obtain classics. We catalogue the most interesting axioms, their numerous variants, and explore the relationships between them in terms of interderivability as both axiom (schema) and as simple formulae. In doing so we introduce the Logics Workbench (LWB, see http://lwbwww.uniba.ch:8080/LWBinfo.html), a versatile tool for proving theorems in numerous propositional (nonclassical) logics. As a side-effect we fulfill a call from the modal theorem proving community for a database of known theorem
Reasoning about obligations in Obligationes : a formal approach.
Despite the appearance of `obligation' in their name, medieval obligational dispu-
tations between an Opponent and a Respondent seem to many to be unrelated to
deontic logic. However, given that some of the example disputations found in me-
dieval texts involve Respondent reasoning about his obligations within the context
of the disputation, it is clear that some sort of deontic reasoning is involved. In this
paper, we explain how the reasoning diers from that in ordinary basic deontic logic,
and dene dynamic epistemic semantics within which the medieval obligations can
be expressed and the examples evaluated. Obligations in this framework are history-
based and closely connected to action, thus allowing for comparisons with, e.g., the
knowledge-based obligations of Pacuit, Parikh, and Cogan, and stit-theory. The con-
tributions of this paper are twofold: The introduction of a new type of obligation into
the deontic logic family, and an explanation of the precise deontic concepts involved
in obligationes
On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics
We consider two styles of proof calculi for a family of tense logics,
presented in a formalism based on nested sequents. A nested sequent can be seen
as a tree of traditional single-sided sequents. Our first style of calculi is
what we call "shallow calculi", where inference rules are only applied at the
root node in a nested sequent. Our shallow calculi are extensions of Kashima's
calculus for tense logic and share an essential characteristic with display
calculi, namely, the presence of structural rules called "display postulates".
Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable
for proof search due to the presence of display postulates and other structural
rules. The second style of calculi uses deep-inference, whereby inference rules
can be applied at any node in a nested sequent. We show that, for a range of
extensions of tense logic, the two styles of calculi are equivalent, and there
is a natural proof theoretic correspondence between display postulates and deep
inference. The deep inference calculi enjoy the subformula property and have no
display postulates or other structural rules, making them a better framework
for proof search
Reasoning about Obligations in Obligationes: A Formal Approach
Despite the appearance of `obligation' in their name, medieval obligational dispu- tations between an Opponent and a Respondent seem to many to be unrelated to deontic logic. However, given that some of the example disputations found in me- dieval texts involve Respondent reasoning about his obligations within the context of the disputation, it is clear that some sort of deontic reasoning is involved. In this paper, we explain how the reasoning diers from that in ordinary basic deontic logic, and dene dynamic epistemic semantics within which the medieval obligations can be expressed and the examples evaluated. Obligations in this framework are history- based and closely connected to action, thus allowing for comparisons with, e.g., the knowledge-based obligations of Pacuit, Parikh, and Cogan, and stit-theory. The con- tributions of this paper are twofold: The introduction of a new type of obligation into the deontic logic family, and an explanation of the precise deontic concepts involved in obligationes
A new calculus for intuitionistic Strong L\"ob logic: strong termination and cut-elimination, formalised
We provide a new sequent calculus that enjoys syntactic cut-elimination and
strongly terminating backward proof search for the intuitionistic Strong L\"ob
logic , an intuitionistic modal logic with a provability
interpretation. A novel measure on sequents is used to prove both the
termination of the naive backward proof search strategy, and the admissibility
of cut in a syntactic and direct way, leading to a straightforward
cut-elimination procedure. All proofs have been formalised in the interactive
theorem prover Coq.Comment: 21-page conference paper + 4-page appendix with proof
- …