290 research outputs found
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
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
Proof Theory for Intuitionistic Strong L\"ob Logic
This paper introduces two sequent calculi for intuitionistic strong L\"ob
logic : a terminating sequent calculus based
on the terminating sequent calculus for intuitionistic
propositional logic and an extension of the
standard cut-free sequent calculus without structural rules for
. One of the main results is a syntactic proof of the
cut-elimination theorem for . In addition, equivalences
between the sequent calculi and Hilbert systems for are
established. It is known from the literature that is complete
with respect to the class of intuitionistic modal Kripke models in which the
modal relation is transitive, conversely well-founded and a subset of the
intuitionistic relation. Here a constructive proof of this fact is obtained by
using a countermodel construction based on a variant of . The
paper thus contains two proofs of cut-elimination, a semantic and a syntactic
proof.Comment: 29 pages, 4 figures, submitted to the Special Volume of the Workshop
Proofs! held in Paris in 201
Proof Theory for Lax Logic
In this paper some proof theory for propositional Lax Logic is developed. A
cut free terminating sequent calculus is introduced for the logic, and based on
that calculus it is shown that the logic has uniform interpolation.
Furthermore, a separate, simple proof of interpolation is provided that also
uses the sequent calculus. From the literature it is known that Lax Logic has
interpolation, but all known proofs use models rather than proof systems
Intuitionistic Gödel-Löb Logic, à la Simpson:Labelled Systems and Birelational Semantics
We derive an intuitionistic version of Gödel-Löb modal logic (GL) in the style of Simpson, via proof theoretic techniques. We recover a labelled system, ℓIGL, by restricting a non-wellfounded labelled system for GL to have only one formula on the right. The latter is obtained using techniques from cyclic proof theory, sidestepping the barrier that GL’s usual frame condition (converse well-foundedness) is not first-order definable. While existing intuitionistic versions of GL are typically defined over only the box (and not the diamond), our presentation includes both modalities. Our main result is that ℓIGL coincides with a corresponding semantic condition in birelational semantics: the composition of the modal relation and the intuitionistic relation is conversely well-founded. We call the resulting logic IGL. While the soundness direction is proved using standard ideas, the completeness direction is more complex and necessitates a detour through several intermediate characterisations of IGL
Intuitionistic G\"odel-L\"ob logic, \`a la Simpson: labelled systems and birelational semantics
We derive an intuitionistic version of G\"odel-L\"ob modal logic ()
in the style of Simpson, via proof theoretic techniques. We recover a labelled
system, , by restricting a non-wellfounded labelled system for
to have only one formula on the right. The latter is obtained using
techniques from cyclic proof theory, sidestepping the barrier that 's
usual frame condition (converse well-foundedness) is not first-order definable.
While existing intuitionistic versions of are typically defined over
only the box (and not the diamond), our presentation includes both modalities.
Our main result is that coincides with a corresponding
semantic condition in birelational semantics: the composition of the modal
relation and the intuitionistic relation is conversely well-founded. We call
the resulting logic . While the soundness direction is proved using
standard ideas, the completeness direction is more complex and necessitates a
detour through several intermediate characterisations of .Comment: 25 pages including 8 pages appendix, 4 figure
Ecumenical modal logic
The discussion about how to put together Gentzen's systems for classical and
intuitionistic logic in a single unified system is back in fashion. Indeed,
recently Prawitz and others have been discussing the so called Ecumenical
Systems, where connectives from these logics can co-exist in peace. In Prawitz'
system, the classical logician and the intuitionistic logician would share the
universal quantifier, conjunction, negation, and the constant for the absurd,
but they would each have their own existential quantifier, disjunction, and
implication, with different meanings. Prawitz' main idea is that these
different meanings are given by a semantical framework that can be accepted by
both parties. In a recent work, Ecumenical sequent calculi and a nested system
were presented, and some very interesting proof theoretical properties of the
systems were established. In this work we extend Prawitz' Ecumenical idea to
alethic K-modalities
Towards a Proof Theory of G\"odel Modal Logics
Analytic proof calculi are introduced for box and diamond fragments of basic
modal fuzzy logics that combine the Kripke semantics of modal logic K with the
many-valued semantics of G\"odel logic. The calculi are used to establish
completeness and complexity results for these fragments
Two loop detection mechanisms: a comparison
In order to compare two loop detection mechanisms we describe two calculi for theorem proving in intuitionistic propositional logic. We call them both MJ Hist, and distinguish between them by description as `Swiss' or `Scottish'. These calculi combine in different ways the ideas on focused proof search of Herbelin and Dyckhoff & Pinto with the work of Heuerding emphet al on loop detection. The Scottish calculus detects loops earlier than the Swiss calculus but at the expense of modest extra storage in the history. A comparison of the two approaches is then given, both on a theoretic and on an implementational level
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