Lambda the Ultimate

Ott—a tool for writing definitions of programming languages and calculi.

Ott is a tool for writing definitions of programming languages and calculi. It takes as input a definition of a language syntax and semantics, in a concise and readable ASCII notation that is close to what one would write in informal mathematics. It generates LaTeX to build a typeset version of the definition, and Coq, HOL, and Isabelle versions of the definition. Additionally, it can be run as a filter, taking a LaTeX/Coq/Isabelle/HOL source file with embedded (symbolic) terms of the defined language, parsing them and replacing them by target-system terms. For a simple example, here is an Ott source file for an untyped call-by-value lambda calculus (test10.ott), and the generated LaTeX (compiled to pdf) and (compiled to ps), Coq, Isabelle, and HOL definitions.

Most simply, the tool can be used to aid completely informal LaTeX mathematics. Here it permits the definition, and terms within proofs and exposition, to be written in a clear, editable, ASCII notation, without LaTeX noise. It generates good-quality typeset output. By parsing (and so sort-checking) this input, it quickly catches a range of simple errors, e.g. inconsistent use of judgement forms or metavariable naming conventions.

That same input can be used to generate formal definitions, for Coq, HOL, and Isabelle. It should thereby enable a smooth transition between use of informal and formal mathematics. Additionally, the tool can automatically generate definitions of functions for free variables, single and multiple substitutions, subgrammar checks (e.g. for value subgrammars), and binding auxiliary functions. (At present only a fully concrete representation of binding is supported, without quotienting by alpha equivalence.)

The distribution includes several examples, in varying levels of completeness: untyped and simply typed lambda-calculus, a calculus with ML polymorphism, the POPLmark Fsub with and without records, an ML module system taken from (Leroy, JFP 1996) and equipped with an operational semantics, and LJ, a lightweight Java fragment.

Peter Sewell and his team continue to bridge the gap between the informal and formal worlds of programming language semantics.