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

A formalized general theory of syntax with bindings

We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying numbers of inputs, quotiented to alpha-equivalence and sorted according to a binding signature. The theory includes a rich collection of properties of the standard operators on terms, such as substitution and freshness. It also includes induction and recursion principles and support for semantic interpretation, all tailored for smooth interaction with the bindings and the standard operators

A formalized general theory of syntax with bindings

We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying numbers of inputs, quotiented to alpha-equivalence and sorted according to a binding signature. The theory includes a rich collection of properties of the standard operators on terms, such as substitution and freshness. It also includes induction and recursion principles and support for semantic interpretation, all tailored for smooth interaction with the bindings and the standard operators

Bindings as bounded natural functors

We present a general framework for specifying and reasoning about syntax with bindings. Abstract binder types are modeled using a universe of functors on sets, subject to a number of operations that can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. Despite not committing to any syntactic format, the framework is “concrete” enough to provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition principles. This work is compatible with classical higher-order logic and has been formalized in the proof assistant Isabelle/HOL

Nominal Unification of Higher Order Expressions with Recursive Let

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems.Comment: Pre-proceedings paper presented at the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh, Scotland UK, 6-8 September 2016 (arXiv:1608.02534

Bindings as bounded natural functors

We present a general framework for specifying and reasoning about syntax with bindings. Abstract binder types are modeled using a universe of functors on sets, subject to a number of operations that can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. Despite not committing to any syntactic format, the framework is “concrete” enough to provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition principles. This work is compatible with classical higher-order logic and has been formalized in the proof assistant Isabelle/HOL

A formalized general theory of syntax with bindings: extended version

We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying numbers of inputs, quotiented to alpha-equivalence and sorted according to a binding signature. The theory contains a rich collection of properties of the standard operators on terms, including substitution, swapping and freshness—namely, there are lemmas showing how each of the operators interacts with all the others and with the syntactic constructors. The theory also features induction and recursion principles and support for semantic interpretation, all tailored for smooth interaction with the bindings and the standard operators