1,070 research outputs found
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
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
Modular Focused Proof Systems for Intuitionistic Modal Logics
Focusing is a general technique for syntactically compartmentalizing
the non-deterministic choices in a proof system, which not only
improves proof search but also has the representational benefit of
distilling sequent proofs into synthetic normal forms. However, since
focusing is usually specified as a restriction of the sequent
calculus, the technique has not been transferred to logics that lack a
(shallow) sequent presentation, as is the case for some of the logics
of the modal cube. We have recently extended the focusing technique
to classical nested sequents, a generalization of ordinary sequents.
In this work we further extend focusing to intuitionistic nested
sequents, which can capture all the logics of the intuitionistic S5
cube in a modular fashion. We present an internal cut-elimination
procedure for the focused system which in turn is used to show its
completeness
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
From Display to Labelled Proofs for Tense Logics
We introduce an effective translation from proofs in the display calculus to proofs in the labelled calculus in the context of tense logics. We identify the labelled calculus proofs in the image of this translation as those built from labelled sequents whose underlying directed graph possesses certain properties. For the basic normal tense logic Kt, the image is shown to be the set of all proofs in the labelled calculus G3Kt
PSPACE Bounds for Rank-1 Modal Logics
For lack of general algorithmic methods that apply to wide classes of logics,
establishing a complexity bound for a given modal logic is often a laborious
task. The present work is a step towards a general theory of the complexity of
modal logics. Our main result is that all rank-1 logics enjoy a shallow model
property and thus are, under mild assumptions on the format of their
axiomatisation, in PSPACE. This leads to a unified derivation of tight
PSPACE-bounds for a number of logics including K, KD, coalition logic, graded
modal logic, majority logic, and probabilistic modal logic. Our generic
algorithm moreover finds tableau proofs that witness pleasant proof-theoretic
properties including a weak subformula property. This generality is made
possible by a coalgebraic semantics, which conveniently abstracts from the
details of a given model class and thus allows covering a broad range of logics
in a uniform way
Generic Modal Cut Elimination Applied to Conditional Logics
We develop a general criterion for cut elimination in sequent calculi for
propositional modal logics, which rests on absorption of cut, contraction,
weakening and inversion by the purely modal part of the rule system. Our
criterion applies also to a wide variety of logics outside the realm of normal
modal logic. We give extensive example instantiations of our framework to
various conditional logics. For these, we obtain fully internalised calculi
which are substantially simpler than those known in the literature, along with
leaner proofs of cut elimination and complexity. In one case, conditional logic
with modus ponens and conditional excluded middle, cut elimination and
complexity were explicitly stated as open in the literature
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