675 research outputs found
A Labelled Sequent Calculus for BBI: Proof Theory and Proof Search
We present a labelled sequent calculus for Boolean BI, a classical variant of
O'Hearn and Pym's logic of Bunched Implication. The calculus is simple, sound,
complete, and enjoys cut-elimination. We show that all the structural rules in
our proof system, including those rules that manipulate labels, can be
localised around applications of certain logical rules, thereby localising the
handling of these rules in proof search. Based on this, we demonstrate a free
variable calculus that deals with the structural rules lazily in a constraint
system. A heuristic method to solve the constraints is proposed in the end,
with some experimental results
Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents
We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants
Tool support for reasoning in display calculi
We present a tool for reasoning in and about propositional sequent calculi.
One aim is to support reasoning in calculi that contain a hundred rules or
more, so that even relatively small pen and paper derivations become tedious
and error prone. As an example, we implement the display calculus D.EAK of
dynamic epistemic logic. Second, we provide embeddings of the calculus in the
theorem prover Isabelle for formalising proofs about D.EAK. As a case study we
show that the solution of the muddy children puzzle is derivable for any number
of muddy children. Third, there is a set of meta-tools, that allows us to adapt
the tool for a wide variety of user defined calculi
Sequent calculi and interpolation for non-normal modal and deonticlogics
G3-style sequent calculi for the logics in the cube of non-normal modal
logics and for their deontic extensions are studied. For each calculus we prove
that weakening and contraction are height-preserving admissible, and we give a
syntactic proof of the admissibility of cut. This implies that the subformula
property holds and that derivability can be decided by a terminating proof
search whose complexity is in PSPACE. These calculi are shown to be equivalent
to the axiomatic ones and, therefore, they are sound and complete with respect
to neighbourhood semantics. Finally, it is given a Maehara-style proof of
Craig's interpolation theorem for most of the logics considered
Hypersequent Calculi for S5: The Methods of Cut Elimination
S5 is one of the most important modal logic with nice syntactic, semantic and algebraic properties. In spite of that, a successful (i.e. cut-free) formalization of S5 on the ground of standard sequent calculus (SC) was problematic and led to the invention of numerous nonstandard, generalized forms of SC. One of the most interesting framework which was very often used for this aim is that of hypersequent calculi (HC). The paper is a survey of HC for S5 proposed by Pottinger, Avron, Restall, Poggiolesi, Lahav and Kurokawa. We are particularly interested in examining different methods which were used for proving the eliminability/admissibility of cut in these systems and present our own variant of a system which admits relatively simple proof of cut elimination
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