11 research outputs found
Intensional properties of polygraphs
We present polygraphic programs, a subclass of Albert Burroni's polygraphs,
as a computational model, showing how these objects can be seen as first-order
functional programs. We prove that the model is Turing complete. We use
polygraphic interpretations, a termination proof method introduced by the
second author, to characterize polygraphic programs that compute in polynomial
time. We conclude with a characterization of polynomial time functions and
non-deterministic polynomial time functions.Comment: Proceedings of TERMGRAPH 2007, Electronic Notes in Computer Science
(to appear), 12 pages, minor changes from previous versio
Higher-dimensional categories with finite derivation type
We study convergent (terminating and confluent) presentations of
n-categories. Using the notion of polygraph (or computad), we introduce the
homotopical property of finite derivation type for n-categories, generalizing
the one introduced by Squier for word rewriting systems. We characterize this
property by using the notion of critical branching. In particular, we define
sufficient conditions for an n-category to have finite derivation type. Through
examples, we present several techniques based on derivations of 2-categories to
study convergent presentations by 3-polygraphs
A Rewriting Coherence Theorem with Applications in Homotopy Type Theory
Higher-dimensional rewriting systems are tools to analyse the structure of
formally reducing terms to normal forms, as well as comparing the different
reduction paths that lead to those normal forms. This higher structure can be
captured by finding a homotopy basis for the rewriting system. We show that the
basic notions of confluence and wellfoundedness are sufficient to recursively
build such a homotopy basis, with a construction reminiscent of an argument by
Craig C. Squier. We then go on to translate this construction to the setting of
homotopy type theory, where managing equalities between paths is important in
order to construct functions which are coherent with respect to higher
dimensions. Eventually, we apply the result to approximate a series of open
questions in homotopy type theory, such as the characterisation of the homotopy
groups of the free group on a set and the pushout of 1-types.
This paper expands on our previous conference contribution "Coherence via
Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the
language of higher-dimensional rewriting.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2001.0765
A rewriting coherence theorem with applications in homotopy type theory
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higher-dimensional rewriting
Dependency-based analyses for function words – Introducing the polygraphic approach
International audienceThis paper scrutinizes various dependency based representations of the syntax of function words, such as prepositions. The focus is on the underlying formal object used to encode the linguistic analyses and its relation to the corresponding linguistic theory. The poly-graph structure is introduced: it consists of a generalization of the concept of graph that allows edges to be vertices of other edges. Such a structure is used to encode dependency-based analyses that are founded on two kinds of morphosyn-tactic criteria: presence constraints and distributional constraints
A rewriting coherence theorem with applications in homotopy type theory
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higher-dimensional rewriting
Homotopical Algebra and Higher Structures (hybrid meeting)
Homotopical algebra and higher category theory play an increasingly important role in pure mathematics, and higher methods
have seen tremendous development in the last couple of decades. The talks delivered at the workshop described some of the latest progress in this area
and applications to various problems of algebra, geometry, and combinatorics