694 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
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Special Delivery: Programming with Mailbox Types (Extended Version)
The asynchronous and unidirectional communication model supported by
mailboxes is a key reason for the success of actor languages like Erlang and
Elixir for implementing reliable and scalable distributed systems. While many
actors may send messages to some actor, only the actor may (selectively)
receive from its mailbox. Although actors eliminate many of the issues stemming
from shared memory concurrency, they remain vulnerable to communication errors
such as protocol violations and deadlocks.
Mailbox types are a novel behavioural type system for mailboxes first
introduced for a process calculus by de'Liguoro and Padovani in 2018, which
capture the contents of a mailbox as a commutative regular expression. Due to
aliasing and nested evaluation contexts, moving from a process calculus to a
programming language is challenging.
This paper presents Pat, the first programming language design incorporating
mailbox types, and describes an algorithmic type system. We make essential use
of quasi-linear typing to tame some of the complexity introduced by aliasing.
Our algorithmic type system is necessarily co-contextual, achieved through a
novel use of backwards bidirectional typing, and we prove it sound and complete
with respect to our declarative type system. We implement a prototype type
checker, and use it to demonstrate the expressiveness of Pat on a factory
automation case study and a series of examples from the Savina actor benchmark
suite.Comment: Extended version of paper accepted to ICFP'2
On Kleene Algebra vs. Process Algebra
We try to clarify the relationship between Kleene algebra and process
algebra, based on the very recent work on Kleene algebra and process algebra.
Both for concurrent Kleene algebra (CKA) with communications and truly
concurrent process algebra APTC with Kleene star and parallel star, the
extended Milner's expansion law holds, with being primitives (atomic actions),
being the parallel composition, being the alternative composition,
being the sequential composition and the communication merge with the
background of computation. CKA and APTC are all the truly concurrent
computation models, can have the same syntax (primitives and operators), maybe
have the same or different semantics
Kleene Algebra with Dynamic Tests: Completeness and Complexity
We study versions of Kleene algebra with dynamic tests, that is, extensions
of Kleene algebra with domain and antidomain operators. We show that Kleene
algebras with tests and Propositional dynamic logic correspond to special cases
of the dynamic test framework. In particular, we establish completeness results
with respect to relational models and guarded-language models, and we show that
two prominent classes of Kleene algebras with dynamic tests have an
EXPTIME-complete equational theory
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
A Survey on Explainable Anomaly Detection
In the past two decades, most research on anomaly detection has focused on
improving the accuracy of the detection, while largely ignoring the
explainability of the corresponding methods and thus leaving the explanation of
outcomes to practitioners. As anomaly detection algorithms are increasingly
used in safety-critical domains, providing explanations for the high-stakes
decisions made in those domains has become an ethical and regulatory
requirement. Therefore, this work provides a comprehensive and structured
survey on state-of-the-art explainable anomaly detection techniques. We propose
a taxonomy based on the main aspects that characterize each explainable anomaly
detection technique, aiming to help practitioners and researchers find the
explainable anomaly detection method that best suits their needs.Comment: Paper accepted by the ACM Transactions on Knowledge Discovery from
Data (TKDD) for publication (preprint version
The Identity Problem in nilpotent groups of bounded class
Let be a unitriangular matrix group of nilpotency class at most ten. We
show that the Identity Problem (does a semigroup contain the identity matrix?)
and the Group Problem (is a semigroup a group?) are decidable in polynomial
time for finitely generated subsemigroups of . Our decidability results also
hold when is an arbitrary finitely generated nilpotent group of class at
most ten. This extends earlier work of Babai et al. on commutative matrix
groups (SODA'96) and work of Bell et al. on
(SODA'17). Furthermore, we formulate a sufficient condition for the
generalization of our results to nilpotent groups of class . For every
such , we exhibit an effective procedure that verifies this condition in
case it is true.Comment: 48 pages, title change
- …