22 research outputs found
Fixed Points Theorems for Non-Transitive Relations
In this paper, we develop an Isabelle/HOL library of order-theoretic
fixed-point theorems. We keep our formalization as general as possible: we
reprove several well-known results about complete orders, often with only
antisymmetry or attractivity, a mild condition implied by either antisymmetry
or transitivity. In particular, we generalize various theorems ensuring the
existence of a quasi-fixed point of monotone maps over complete relations, and
show that the set of (quasi-)fixed points is itself complete. This result
generalizes and strengthens theorems of Knaster-Tarski, Bourbaki-Witt, Kleene,
Markowsky, Pataraia, Mashburn, Bhatta-George, and Stouti-Maaden
A Coalgebraic View on Reachability
Coalgebras for an endofunctor provide a category-theoretic framework for
modeling a wide range of state-based systems of various types. We provide an
iterative construction of the reachable part of a given pointed coalgebra that
is inspired by and resembles the standard breadth-first search procedure to
compute the reachable part of a graph. We also study coalgebras in Kleisli
categories: for a functor extending a functor on the base category, we show
that the reachable part of a given pointed coalgebra can be computed in that
base category
A Categorical Framework for Program Semantics and Semantic Abstraction
Categorical semantics of type theories are often characterized as
structure-preserving functors. This is because in category theory both the
syntax and the domain of interpretation are uniformly treated as structured
categories, so that we can express interpretations as structure-preserving
functors between them. This mathematical characterization of semantics makes it
convenient to manipulate and to reason about relationships between
interpretations. Motivated by this success of functorial semantics, we address
the question of finding a functorial analogue in abstract interpretation, a
general framework for comparing semantics, so that we can bring similar
benefits of functorial semantics to semantic abstractions used in abstract
interpretation. Major differences concern the notion of interpretation that is
being considered. Indeed, conventional semantics are value-based whereas
abstract interpretation typically deals with more complex properties. In this
paper, we propose a functorial approach to abstract interpretation and study
associated fundamental concepts therein. In our approach, interpretations are
expressed as oplax functors in the category of posets, and abstraction
relations between interpretations are expressed as lax natural transformations
representing concretizations. We present examples of these formal concepts from
monadic semantics of programming languages and discuss soundness.Comment: MFPS 202
Fixed Points Theorems for Non-Transitive Relations
In this paper, we develop an Isabelle/HOL library of order-theoretic
fixed-point theorems. We keep our formalization as general as possible: we
reprove several well-known results about complete orders, often with only
antisymmetry or attractivity, a mild condition implied by either antisymmetry
or transitivity. In particular, we generalize various theorems ensuring the
existence of a quasi-fixed point of monotone maps over complete relations, and
show that the set of (quasi-)fixed points is itself complete. This result
generalizes and strengthens theorems of Knaster-Tarski, Bourbaki-Witt, Kleene,
Markowsky, Pataraia, Mashburn, Bhatta-George, and Stouti-Maaden
Formal Verification of Safety Architectures for Automated Driving
Safety architectures play a crucial role in the safety assurance of automated
driving vehicles (ADVs). They can be used as safety envelopes of black-box ADV
controllers, and for graceful degradation from one ODD to another. Building on
our previous work on the formalization of responsibility-sensitive safety
(RSS), we introduce a novel program logic that accommodates assume-guarantee
reasoning and fallback-like constructs. This allows us to formally define and
prove the safety of existing and novel safety architectures. We apply the logic
to a pull over scenario and experimentally evaluate the resulting safety
architecture.Comment: In proceedings of 2023 IEEE Intelligent Vehicles Symposium (IV), 8
pages, 5 figure