284 research outputs found
Cyclic proof systems for modal fixpoint logics
This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one
Fragments and frame classes:Towards a uniform proof theory for modal fixed point logics
This thesis studies the proof theory of modal fixed point logics. In particular, we construct proof systems for various fragments of the modal mu-calculus, interpreted over various classes of frames. With an emphasis on uniform constructions and general results, we aim to bring the relatively underdeveloped proof theory of modal fixed point logics closer to the well-established proof theory of basic modal logic. We employ two main approaches. First, we seek to generalise existing methods for basic modal logic to accommodate fragments of the modal mu-calculus. We use this approach for obtaining Hilbert-style proof systems. Secondly, we adapt existing proof systems for the modal mu-calculus to various classes of frames. This approach yields proof systems which are non-well-founded, or cyclic.The thesis starts with an introduction and some mathematical preliminaries. In Chapter 3 we give hypersequent calculi for modal logic with the master modality, building on work by Ori Lahav. This is followed by an Intermezzo, where we present an abstract framework for cyclic proofs, in which we give sufficient conditions for establishing the bounded proof property. In Chapter 4 we generalise existing work on Hilbert-style proof systems for PDL to the level of the continuous modal mu-calculus. Chapter 5 contains a novel cyclic proof system for the alternation-free two-way modal mu-calculus. Finally, in Chapter 6, we present a cyclic proof system for Guarded Kleene Algebra with Tests and take a first step towards using it to establish the completeness of an algebraic counterpart
Intuitionistic S4 is decidable
In this paper we demonstrate decidability for the intuitionistic modal logic
S4 first formulated by Fischer Servi. This solves a problem that has been open
for almost thirty years since it had been posed in Simpson's PhD thesis in
1994. We obtain this result by performing proof search in a labelled deductive
system that, instead of using only one binary relation on the labels, employs
two: one corresponding to the accessibility relation of modal logic and the
other corresponding to the order relation of intuitionistic Kripke frames. Our
search algorithm outputs either a proof or a finite counter-model, thus,
additionally establishing the finite model property for intuitionistic S4,
which has been another long-standing open problem in the area.Comment: 13 pages conference paper + 26 pages appendix with examples and
proof
Categorical structures for deduction
We begin by introducing categorized judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and first-order logic as special kinds of categorized judgemental theories. We believe our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. For first-order logic we offer a deep analysis of structural rules, describing some of their properties, and putting them into context.
We then put one of the main constructions introduced, namely that of categorized judgemental dependent type theories, to the test: we frame it in the general context of categorical models for dependent types, describe a few examples, study its properties, and use it to model subtyping and as a tool to prove intrinsic properties hidden in other models.
Somehow orthogonally, then, we show a different side as to how categories can help the study of deductive systems: we transport a known model from set-based categories to enriched categories, and use the information naturally encoded into it to describe a theory of fuzzy types. We recover structural rules, observe new phenomena, and study different possible enrichments and their interpretation. We open the discussion to include different takes on the topic of definitional equality
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
Nested Sequents for Quantified Modal Logics
This paper studies nested sequents for quantified modal logics. In
particular, it considers extensions of the propositional modal logics definable
by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and
constant domains. Each calculus is proved to have good structural properties:
weakening and contraction are height-preserving admissible and cut is
(syntactically) admissible. Each calculus is shown to be equivalent to the
corresponding axiomatic system and, thus, to be sound and complete. Finally, it
is argued that the calculi are internal -- i.e., each sequent has a formula
interpretation -- whenever the existence predicate is expressible in the
language.Comment: accepted to TABLEAUX 202
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
Explorations in Subexponential Non-associative Non-commutative Linear Logic
In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, exhibiting a classical one-sided multi-succedent classical analogue of our intuitionistic system, following the exponential-free calculi of Buszkowski, and de Groote, Lamarche. A large fragment of the intuitionistic calculus is shown to embed faithfully into the classical fragment
Primal logic of information
Primal logic arose in access control; it has a remarkably efficient (linear
time) decision procedure for its entailment problem. But primal logic is a
general logic of information. In the realm of arbitrary items of information
(infons), conjunction, disjunction, and implication may seem to correspond
(set-theoretically) to union, intersection, and relative complementation. But,
while infons are closed under union, they are not closed under intersection or
relative complementation.
It turns out that there is a systematic transformation of propositional
intuitionistic calculi to the original (propositional) primal calculi; we call
it Flatting. We extend Flatting to quantifier rules, obtaining arguably the
right quantified primal logic, QPL. The QPL entailment problem is
exponential-time complete, but it is polynomial-time complete in the case, of
importance to applications (at least to access control), where the number of
quantifiers is bounded
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