23 research outputs found
A general theory of syntax with bindings
In this thesis we give a general theory of syntax with bindings. We address the problem from a mathematical point of view and at the same time we give a formalization, in the Isabelle/HOL proof assistant.
Our theory uses explicit names for variables, and then deals with alpha-equivalence classes, remaining intuitive and close to informal mathematics, although being fully formalized and sound in classical high-order logic. In this sense it can be regarded as a generalization of nominal logic. Our end product can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. We provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition
principles, obeying Barendregt’s convention.
We present our work as a thinking process that starts from some desiderata, and then evolves in different formalization stages for the general theory. We start by taking a “universal algebra” approach, modelling syntaxes via algebraic-style binding signatures, which we employ in a substantial case study on formal reasoning: Church-Rosser and standardization theorems for lamda-calculus. This solution proves itself too restrictive, so we refine it into a more flexible one, which constitutes the main original contribution of this thesis: We construct a universe of functors on sets that handle bindings on a general, flexible and modular level.
Our functors do not commit to any a priori syntactic format, cater for codatatypes in addition to datatypes, and are supported by powerful definition and reasoning principles. They generalize the bounded natural functors (BNFs), which have been previously implemented in Isabelle/HOL to support (co)datatypes
On Well-Founded and Recursive Coalgebras
This paper studies fundamental questions concerning category-theoretic models
of induction and recursion. We are concerned with the relationship between
well-founded and recursive coalgebras for an endofunctor. For monomorphism
preserving endofunctors on complete and well-powered categories every coalgebra
has a well-founded part, and we provide a new, shorter proof that this is the
coreflection in the category of all well-founded coalgebras. We present a new
more general proof of Taylor's General Recursion Theorem that every
well-founded coalgebra is recursive, and we study under which hypothesis the
converse holds. In addition, we present a new equivalent characterization of
well-foundedness: a coalgebra is well-founded iff it admits a
coalgebra-to-algebra morphism to the initial algebra
A Completeness Theorem for Probabilistic Regular Expressions
We introduce Probabilistic Regular Expressions (PRE), a probabilistic
analogue of regular expressions denoting probabilistic languages in which every
word is assigned a probability of being generated. We present and prove the
completeness of an inference system for reasoning about probabilistic language
equivalence of PRE based on Salomaa's axiomatisation of Kleene Algebra