385 research outputs found
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
From G\"odel's Incompleteness Theorem to the completeness of bot beliefs (Extended abstract)
Hilbert and Ackermann asked for a method to consistently extend incomplete
theories to complete theories. G\"odel essentially proved that any theory
capable of encoding its own statements and their proofs contains statements
that are true but not provable. Hilbert did not accept that G\"odel's
construction answered his question, and in his late writings and lectures,
G\"odel agreed that it did not, since theories can be completed incrementally,
by adding axioms to prove ever more true statements, as science normally does,
with completeness as the vanishing point. This pragmatic view of validity is
familiar not only to scientists who conjecture test hypotheses but also to real
estate agents and other dealers, who conjure claims, albeit invalid, as
necessary to close a deal, confident that they will be able to conjure other
claims, albeit invalid, sufficient to make the first claims valid. We study the
underlying logical process and describe the trajectories leading to testable
but unfalsifiable theories to which bots and other automated learners are
likely to converge.Comment: 19 pages, 13 figures; version updates: changed one word in the title,
expanded Introduction, improved presentation, tidied up some diagram
Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors
Two graphs and are homomorphism indistinguishable over a class of
graphs if for all graphs the number of
homomorphisms from to is equal to the number of homomorphisms from
to . Many natural equivalence relations comparing graphs such as (quantum)
isomorphism, spectral, and logical equivalences can be characterised as
homomorphism indistinguishability relations over certain graph classes.
Abstracting from the wealth of such instances, we show in this paper that
equivalences w.r.t. any self-complementarity logic admitting a characterisation
as homomorphism indistinguishability relation can be characterised by
homomorphism indistinguishability over a minor-closed graph class.
Self-complementarity is a mild property satisfied by most well-studied logics.
This result follows from a correspondence between closure properties of a graph
class and preservation properties of its homomorphism indistinguishability
relation.
Furthermore, we classify all graph classes which are in a sense finite
(essentially profinite) and satisfy the maximality condition of being
homomorphism distinguishing closed, i.e. adding any graph to the class strictly
refines its homomorphism indistinguishability relation. Thereby, we answer
various question raised by Roberson (2022) on general properties of the
homomorphism distinguishing closure.Comment: 26 pages, 1 figure, 1 tabl
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
The Way We Were: Structural Operational Semantics Research in Perspective
This position paper on the (meta-)theory of Structural Operational Semantic
(SOS) is motivated by the following two questions: (1) Is the (meta-)theory of
SOS dying out as a research field? (2) If so, is it possible to rejuvenate this
field with a redefined purpose?
In this article, we will consider possible answers to those questions by
first analysing the history of the EXPRESS/SOS workshops and the data
concerning the authors and the presentations featured in the editions of those
workshops as well as their subject matters.
The results of our quantitative and qualitative analyses all indicate a
diminishing interest in the theory of SOS as a field of research. Even though
`all good things must come to an end', we strive to finish this position paper
on an upbeat note by addressing our second motivating question with some
optimism. To this end, we use our personal reflections and an analysis of
recent trends in two of the flagship conferences in the field of Programming
Languages (namely POPL and PDLI) to draw some conclusions on possible future
directions that may rejuvenate research on the (meta-)theory of SOS. We hope
that our musings will entice members of the research community to breathe new
life into a field of research that has been kind to three of the authors of
this article.Comment: In Proceedings EXPRESS/SOS2023, arXiv:2309.0578
Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras
De Finetti theorems tell us that if we expect the likelihood of outcomes to
be independent of their order, then these sequences of outcomes could be
equivalently generated by drawing an experiment at random from a distribution,
and repeating it over and over. In particular, the quantum de Finetti theorem
says that exchangeable sequences of quantum states are always represented by
distributions over a single state produced over and over. The main result of
this paper is that this quantum de Finetti construction has a universal
property as a categorical limit. This allows us to pass canonically between
categorical treatments of finite dimensional quantum theory and the infinite
dimensional. The treatment here is through understanding properties of
(co)limits with respect to the contravariant functor which takes a C*-algebra
describing a physical system to its convex, compact space of states, and
through discussion of the Radon probability monad. We also show that the same
categorical analysis also justifies a continuous de Finetti theorem for
classical probability.Comment: In Proceedings QPL 2022, arXiv:2311.0837
Shades of Iteration: from Elgot to Kleene
Notions of iteration range from the arguably most general Elgot iteration to
a very specific Kleene iteration. The fundamental nature of Elgot iteration has
been extensively explored by Bloom and Esik in the form of iteration theories,
while Kleene iteration became extremely popular as an integral part of
(untyped) formalisms, such as automata theory, regular expressions and Kleene
algebra. Here, we establish a formal connection between Elgot iteration and
Kleene iteration in the form of Elgot monads and Kleene monads, respectively.
We also introduce a novel class of while-monads, which like Kleene monads admit
a relatively simple description in algebraic terms. Like Elgot monads,
while-monads cover a large variety of models that meaningfully support
while-loops, but may fail the Kleene algebra laws, or even fail to support a
Kleen iteration operator altogether.Comment: Extended version of the accepted one for "Recent Trends in Algebraic
Development Techniques - 26th IFIP WG 1.3 International Workshop, WADT 2022
Explicit Hopcroft's Trick in Categorical Partition Refinement
Algorithms for partition refinement are actively studied for a variety of
systems, often with the optimisation called Hopcroft's trick. However, the
low-level description of those algorithms in the literature often obscures the
essence of Hopcroft's trick. Our contribution is twofold. Firstly, we present a
novel formulation of Hopcroft's trick in terms of general trees with weights.
This clean and explicit formulation -- we call it Hopcroft's inequality -- is
crucially used in our second contribution, namely a general partition
refinement algorithm that is \emph{functor-generic} (i.e. it works for a
variety of systems such as (non-)deterministic automata and Markov chains).
Here we build on recent works on coalgebraic partition refinement but depart
from them with the use of fibrations. In particular, our fibrational notion of
-partitioning exposes a concrete tree structure to which Hopcroft's
inequality readily applies. It is notable that our fibrational framework
accommodates such algorithmic analysis on the categorical level of abstraction
Construction and application of adjusted higher gauge theories
This thesis investigates several aspects of nonabelian higher gauge theories, which appear
in many areas of physics, notably string theory and gauged supergravity. We show that
nonabelian higher gauge theory admits a consistent classical nonperturbative formulation
insofar as a higher nonabelian parallel transport exists consistently, without requiring
certain curvature components (fake curvature) to vanish.
Next, we explore examples of nonabelian higher gauge theories that naturally appear
in high-energy physics. Using a generalisation of L∞-algebras called EL∞-algebras, we
show that tensor hierarchies of gauged supergravity naturally admit a formulation in terms
of higher nonabelian gauge theories. Furthermore, toroidal compactifications of string
theory exhibiting T-duality also naturally contain higher gauge symmetry, which explain
several features of nongeometric compactifications (Q- and R-fluxes)
Promonads and String Diagrams for Effectful Categories
Premonoidal and Freyd categories are both generalized by non-cartesian Freyd
categories: effectful categories. We construct string diagrams for effectful
categories in terms of the string diagrams for a monoidal category with a
freely added object. We show that effectful categories are pseudomonoids in a
monoidal bicategory of promonads with a suitable tensor product.Comment: In Proceedings ACT 2022, arXiv:2307.1551
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