1,006 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
Parameterized stability and the universal property of global spectra
Extending work of Nardin, we develop a framework of parameterized semiadditivity and stability with respect to so-called atomic orbital subcategories of an indexing -category . Specializing this framework, we introduce global -categories together with genuine forms of semiadditivity and stability, yielding a higher categorical version of the notion of a Mackey 2-functor studied by Balmer-Dell'Ambrogio. As our main result, we identify the free presentable genuinely stable global -category with a natural global -category of global spectra for finite groups, in the sense of Schwede and Hausmann
Canonical Algebraic Generators in Automata Learning
Many methods for the verification of complex computer systems require the existence of a tractable mathematical abstraction of the system, often in the form of an automaton. In reality, however, such a model is hard to come up with, in particular manually. Automata learning is a technique that can automatically infer an automaton model from a system -- by observing its behaviour. The majority of automata learning algorithms is based on the so-called L* algorithm. The acceptor learned by L* has an important property: it is canonical, in the sense that, it is, up to isomorphism, the unique deterministic finite automaton of minimal size accepting a given regular language. Establishing a similar result for other classes of acceptors, often with side-effects, is of great practical importance. Non-deterministic finite automata, for instance, can be exponentially more succinct than deterministic ones, allowing verification to scale. Unfortunately, identifying a canonical size-minimal non-deterministic acceptor of a given regular language is in general not possible: it can happen that a regular language is accepted by two non-isomorphic non-deterministic finite automata of minimal size. In particular, it thus is unclear which one of the automata should be targeted by a learning algorithm. In this thesis, we further explore the issue and identify (sub-)classes of acceptors that admit canonical size-minimal representatives.
In more detail, the contributions of this thesis are three-fold.
First, we expand the automata (learning) theory of Guarded Kleene Algebra with Tests (GKAT), an efficiently decidable logic expressive enough to model simple imperative programs. In particular, we present GL*, an algorithm that learns the unique size-minimal GKAT automaton for a given deterministic language, and prove that GL* is more efficient than an existing variation of L*. We implement both algorithms in OCaml, and compare them on example programs.
Second, we present a category-theoretical framework based on generators, bialgebras, and distributive laws, which identifies, for a wide class of automata with side-effects in a monad, canonical target models for automata learning. Apart from recovering examples from the literature, we discover a new canonical acceptor of regular languages, and present a unifying minimality result.
Finally, we show that the construction underlying our framework is an instance of a more general theory. First, we see that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on a category of subobjects with respect to an epi-mono factorisation system. Second, we explore the abstract theory of generators and bases for algebras over a monad: we discuss bases for bialgebras, the product of bases, generalise the representation theory of linear maps, and compare our ideas to a coalgebra-based approach
Paranatural Category Theory
We establish and advocate for a novel branch of category theory, centered
around strong dinatural transformations (herein known as "paranatural
transformations"). Paranatural transformations generalize natural
transformations to mixed-variant difunctors, but, unlike other such
generalizations, are composable and exceptionally well-behaved. We define the
category of difunctors and paranatural transformations, prove a novel "diYoneda
Lemma" for this category, and explore some of the category-theoretic
implications.
We also develop three compelling uses for paranatural category theory:
parametric polymorphism, impredicative encodings of (co)inductive types, and
difunctor models of type theory. Paranatural transformations capture the
essence of parametricity, with their "paranaturality condition" coinciding
exactly with the "free theorem" of the corresponding polymorphic type; the
paranatural analogue of the (co)end calculus provides an elegant and general
framework for reasoning about initial algebras, terminal coalgebras,
bisimulations, and representation independence; and "diYoneda reasoning"
facilitates the lifting of Grothendieck universes into difunctor models of type
theory. We develop these topics and propose further avenues of research
Lie algebra actions on module categories for truncated shifted Yangians
We develop a theory of parabolic induction and restriction functors relating
modules over Coulomb branch algebras, in the sense of
Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's
induction and restriction functors for Cherednik algebras, but their definition
uses different tools.
After this general definition, we focus on quiver gauge theories attached to
a quiver . The induction and restriction functors allow us to define a
categorical action of the corresponding symmetric Kac-Moody algebra
on category for these Coulomb branch
algebras. When is of Dynkin type, the Coulomb branch algebras are
truncated shifted Yangians and quantize generalized affine Grassmannian slices.
Thus, we regard our action as a categorification of the geometric Satake
correspondence.
To establish this categorical action, we define a new class of "flavoured"
KLRW algebras, which are similar to the diagrammatic algebras originally
constructed by the second author for the purpose of tensor product
categorification. We prove an equivalence between the category of
Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a
flavoured KLRW algebra. This equivalence relates the categorical action by
induction and restriction functors to the usual categorical action on modules
over a KLRW algebra.Comment: 66 pages, version 2: many corrections, improved treatment of GK
dimension, 71 page
The representation theory of the increasing monoid
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras
Multi-graded Featherweight Java
Resource-aware type systems statically approximate not only the expected
result type of a program, but also the way external resources are used, e.g.,
how many times the value of a variable is needed. We extend the type system of
Featherweight Java to be resource-aware, parametrically on an arbitrary grade
algebra modeling a specific usage of resources. We prove that this type system
is sound with respect to a resource-aware version of reduction, that is, a
well-typed program has a reduction sequence which does not get stuck due to
resource consumption. Moreover, we show that the available grades can be
heterogeneous, that is, obtained by combining grades of different kinds, via a
minimal collection of homomorphisms from one kind to another. Finally, we show
how grade algebras and homomorphisms can be specified as Java classes, so that
grade annotations in types can be written in the language itself
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Normal Form Bisimulations By Value
Normal form bisimilarities are a natural form of program equivalence resting
on open terms, first introduced by Sangiorgi in call-by-name. The literature
contains a normal form bisimilarity for Plotkin's call-by-value
-calculus, Lassen's \emph{enf bisimilarity}, which validates all of
Moggi's monadic laws and can be extended to validate . It does not
validate, however, other relevant principles, such as the identification of
meaningless terms -- validated instead by Sangiorgi's bisimilarity -- or the
commutation of \letexps. These shortcomings are due to issues with open terms
of Plotkin's calculus. We introduce a new call-by-value normal form
bisimilarity, deemed \emph{net bisimilarity}, closer in spirit to Sangiorgi's
and satisfying the additional principles. We develop it on top of an existing
formalism designed for dealing with open terms in call-by-value. It turns out
that enf and net bisimilarities are \emph{incomparable}, as net bisimilarity
does not validate Moggi's laws nor . Moreover, there is no easy way to
merge them. To better understand the situation, we provide an analysis of the
rich range of possible call-by-value normal form bisimilarities, relating them
to Ehrhard's relational model.Comment: Rewritten version (deleted toy similarity and explained proof method
on naive similarity) -- Submitted to POPL2
Representations of Partition Categories
We explain a new approach to the representation theory of the partition category based on a reformulation of the definition of the Jucys-Murphy elements introduced originally by Halverson and Ram and developed further by Enyang. Our reformulation involves a new graphical monoidal category, the affine partition category, which is defined here as a certain monoidal subcategory of Khovanov's Heisenberg category. We use the Jucys-Murphy elements to constructsome special projective functors, then apply these functors to give self-contained proofs of results of Comes and
Ostrik on blocks of Deligne’s category \REP(S_t). We then study a restriction functor \REP(S_t)\to\REP(S_{t-1}) and prove a conjecture of Comes and Ostrik involving this functor. Finally, we use the restriction functor to verify
a criterion of Benson, Etingof, and Ostrik, thereby identifying the abelian envelope of \REP(S_t) with the Ringel dual of the category of locally finite-dimensional \Par_t-modules.
This dissertation includes published co-authored material
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