245 research outputs found
Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi
We have recently presented a general method of proving the fundamental
logical properties of Craig and Lyndon Interpolation (IPs) by induction on
derivations in a wide class of internal sequent calculi, including sequents,
hypersequents, and nested sequents. Here we adapt the method to a more general
external formalism of labelled sequents and provide sufficient criteria on the
Kripke-frame characterization of a logic that guarantee the IPs. In particular,
we show that classes of frames definable by quantifier-free Horn formulas
correspond to logics with the IPs. These criteria capture the modal cube and
the infinite family of transitive Geach logics
Universal Proof Theory: Semi-analytic Rules and Craig Interpolation
In [6], Iemhoff introduced the notion of a focused axiom and a focused rule
as the building blocks for a certain form of sequent calculus which she calls a
focused proof system. She then showed how the existence of a terminating
focused system implies the uniform interpolation property for the logic that
the calculus captures. In this paper we first generalize her focused rules to
semi-analytic rules, a dramatically powerful generalization, and then we will
show how the semi-analytic calculi consisting of these rules together with our
generalization of her focused axioms, lead to the feasible Craig interpolation
property. Using this relationship, we first present a uniform method to prove
interpolation for different logics from sub-structural logics ,
, and to their appropriate
classical and modal extensions, including the intuitionistic and classical
linear logics. Then we will use our theorem negatively, first to show that so
many sub-structural logics including \L_n, , , and and
almost all super-intutionistic logics (except at most seven of them) do not
have a semi-analytic calculus. To investigate the case that the logic actually
has the Craig interpolation property, we will first define a certain specific
type of semi-analytic calculus which we call PPF systems and we will then
present a sound and complete PPF calculus for classical logic. However, we will
show that all such PPF calculi are exponentially slower than the classical
Hilbert-style proof system (or equivalently ). We will then
present a similar exponential lower bound for a certain form of complete PPF
calculi, this time for any super-intuitionistic logic.Comment: 45 page
Uniform interpolation via nested sequents
A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g., nested sequents, hypersequents, and labelled sequents). In this paper, we turn to uniform interpolation, which is stronger than Craig interpolation. We develop a constructive method for proving uniform interpolation via nested sequents and apply it to reprove the uniform interpolation property for normal modal logics K, D, and T. While our method is proof-theoretic, the definition of uniform interpolation for nested sequents also uses semantic notions, including bisimulation modulo an atomic proposition
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 of Finite-valued Logics
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics
Extensions of K5: Proof Theory and Uniform Lyndon Interpolation
We introduce a Gentzen-style framework, called layered sequent calculi, for
modal logic K5 and its extensions KD5, K45, KD45, KB5, and S5 with the goal to
investigate the uniform Lyndon interpolation property (ULIP), which implies
both the uniform interpolation property and the Lyndon interpolation property.
We obtain complexity-optimal decision procedures for all logics and present a
constructive proof of the ULIP for K5, which to the best of our knowledge, is
the first such syntactic proof. To prove that the interpolant is correct, we
use model-theoretic methods, especially bisimulation modulo literals.Comment: 20-page conference paper + 5-page appendix with examples and proof
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
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