21 research outputs found
Homotopy Bisimilarity for Higher-Dimensional Automata
We introduce a new category of higher-dimensional automata in which the morphisms are functional homotopy simulations, i.e. functional simulations up to concurrency of independent events. For this, we use unfoldings of higher-dimensional automata into higher-dimensional trees. Using a notion of open maps in this category, we define homotopy bisimilarity. We show that homotopy bisimilarity is equivalent to a straight-forward generalization of standard bisimilarity to higher dimensions, and that it is finer than split bisimilarity and incomparable with history-preserving bisimilarity
Weak equivalence of higher-dimensional automata
This paper introduces a notion of equivalence for higher-dimensional
automata, called weak equivalence. Weak equivalence focuses mainly on a
traditional trace language and a new homology language, which captures the
overall independence structure of an HDA. It is shown that weak equivalence is
compatible with both the tensor product and the coproduct of HDAs and that,
under certain conditions, HDAs may be reduced to weakly equivalent smaller ones
by merging and collapsing cubes.This research was partially supported by FCT (Fundacao para a Ciencia e a Tecnologia, Portugal) through project UID/MAT/00013/2013
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science
The Directed Homotopy Hypothesis
The homotopy hypothesis was originally stated by Grothendieck: topological spaces should be "equivalent" to (weak) infinite-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g., for rewriting, distributed systems, (homotopy) type theory etc.
But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis\u27 directed topological spaces should be "equivalent" to a weak form of topologically enriched categories, still very close to (infinite,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology
Greatest HITs: Higher Inductive Types in Coinductive Definitions via Induction under Clocks
Guarded recursion is a powerful modal approach to recursion that can be seen
as an abstract form of step-indexing. It is currently used extensively in
separation logic to model programming languages with advanced features by
solving domain equations also with negative occurrences. In its multi-clocked
version, guarded recursion can also be used to program with and reason about
coinductive types, encoding the productivity condition required for recursive
definitions in types. This paper presents the first type theory combining
multi-clocked guarded recursion with the features of Cubical Type Theory, as
well as a denotational semantics. Using the combination of Higher Inductive
Types (HITs) and guarded recursion allows for simple programming and reasoning
about coinductive types that are traditionally hard to represent in type
theory, such as the type of finitely branching labelled transition systems. For
example, our results imply that bisimilarity for these imply path equality, and
so proofs can be transported along bisimilarity proofs. Among our technical
contributions is a new principle of induction under clocks. This allows
universal quantification over clocks to commute with HITs up to equivalence of
types, and is crucial for the encoding of coinductive types. Such commutativity
requirements have been formulated for inductive types as axioms in previous
type theories with multi-clocked guarded recursion, but our present formulation
as an induction principle allows for the formulation of general computation
rules.Comment: 29 page