1,636 research outputs found
Kleene algebra with domain
We propose Kleene algebra with domain (KAD), an extension of Kleene algebra
with two equational axioms for a domain and a codomain operation, respectively.
KAD considerably augments the expressiveness of Kleene algebra, in particular
for the specification and analysis of state transition systems. We develop the
basic calculus, discuss some related theories and present the most important
models of KAD. We demonstrate applicability by two examples: First, an
algebraic reconstruction of Noethericity and well-foundedness; second, an
algebraic reconstruction of propositional Hoare logic.Comment: 40 page
On the number of types in sparse graphs
We prove that for every class of graphs which is nowhere dense,
as defined by Nesetril and Ossona de Mendez, and for every first order formula
, whenever one draws a graph and a
subset of its nodes , the number of subsets of which are of
the form
for some valuation of in is bounded by
, for every . This provides
optimal bounds on the VC-density of first-order definable set systems in
nowhere dense graph classes.
We also give two new proofs of upper bounds on quantities in nowhere dense
classes which are relevant for their logical treatment. Firstly, we provide a
new proof of the fact that nowhere dense classes are uniformly quasi-wide,
implying explicit, polynomial upper bounds on the functions relating the two
notions. Secondly, we give a new combinatorial proof of the result of Adler and
Adler stating that every nowhere dense class of graphs is stable. In contrast
to the previous proofs of the above results, our proofs are completely
finitistic and constructive, and yield explicit and computable upper bounds on
quantities related to uniform quasi-wideness (margins) and stability (ladder
indices)
On Local Testability in the Non-Signaling Setting
Non-signaling strategies are a generalization of quantum strategies that have been studied in physics for decades, and have recently found applications in theoretical computer science. These applications motivate the study of local-to-global phenomena for non-signaling functions.
We prove that low-degree testing in the non-signaling setting is possible, assuming that the locality of the non-signaling function exceeds a threshold. We additionally show that if the locality is below the threshold then the test fails spectacularly, in that there exists a non-signaling function which passes the test with probability 1 and yet is maximally far from being low-degree.
Along the way, we present general results about the local testability of linear codes in the non-signaling setting. These include formulating natural definitions that capture the condition that a non-signaling function "belongs" to a given code, and characterizing the sets of local constraints that imply membership in the code. We prove these results by formulating a logical inference system for linear constraints on non-signaling functions that is complete and sound
The Meaning of Memory Safety
We give a rigorous characterization of what it means for a programming
language to be memory safe, capturing the intuition that memory safety supports
local reasoning about state. We formalize this principle in two ways. First, we
show how a small memory-safe language validates a noninterference property: a
program can neither affect nor be affected by unreachable parts of the state.
Second, we extend separation logic, a proof system for heap-manipulating
programs, with a memory-safe variant of its frame rule. The new rule is
stronger because it applies even when parts of the program are buggy or
malicious, but also weaker because it demands a stricter form of separation
between parts of the program state. We also consider a number of pragmatically
motivated variations on memory safety and the reasoning principles they
support. As an application of our characterization, we evaluate the security of
a previously proposed dynamic monitor for memory safety of heap-allocated data.Comment: POST'18 final versio
Stochastic Relational Presheaves and Dynamic Logic for Contextuality
Presheaf models provide a formulation of labelled transition systems that is
useful for, among other things, modelling concurrent computation. This paper
aims to extend such models further to represent stochastic dynamics such as
shown in quantum systems. After reviewing what presheaf models represent and
what certain operations on them mean in terms of notions such as internal and
external choices, composition of systems, and so on, I will show how to extend
those models and ideas by combining them with ideas from other
category-theoretic approaches to relational models and to stochastic processes.
It turns out that my extension yields a transitional formulation of
sheaf-theoretic structures that Abramsky and Brandenburger proposed to
characterize non-locality and contextuality. An alternative characterization of
contextuality will then be given in terms of a dynamic modal logic of the
models I put forward.Comment: In Proceedings QPL 2014, arXiv:1412.810
Descriptive complexity for pictures languages
This paper deals with logical characterizations of picture languages of any dimension by syntactical fragments of existential second-order logic. Two classical classes of picture languages are studied:
- the class of "recognizable" picture languages, i.e. projections of languages defined by local constraints (or tilings): it is known as the most robust class extending the class of regular languages to any dimension;
- the class of picture languages recognized on "nondeterministic cellular automata in linear time" : cellular automata are the simplest and most natural model of parallel computation and linear time is the minimal time-bounded class allowing synchronization of nondeterministic cellular automata.
We uniformly generalize to any dimension the characterization by Giammarresi et al. (1996) of the class of "recognizable" picture languages in existential monadic second-order logic.
We state several logical characterizations of the class of picture languages recognized in linear time on nondeterministic cellular automata. They are the first machine-independent characterizations of complexity classes of cellular automata.
Our characterizations are essentially deduced from normalization results we prove for first-order and existential second-order logics over pictures. They are obtained in a general and uniform framework that allows to extend them to other "regular" structures
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