## Program verification using characteristic formulae

In my thesis, I have developed a new approach to program verification based on the notion of characteristic formula. This approach is implemented in a tool, called CFML, which can be used to establish in Coq the full functional correctness of arbitrarily complex Caml programs. I have used CFML to verify more than half of the content of Okasaki's book Purely Functional Data Structures. I have also used CFML to verify a collection of nontrivial imperative programs, including Dijkstra's shortest path algorithm, the Union-Find data structure, iterators on mutable lists, functions with local state, functions in CPS form, and recursion through the store.

The characteristic formula of a piece of code is a logical formula that describes the semantics of this code (it is a form of strongest post-condition). This characteristic formula satisfies the following property:

• it is built compositionally and automatically from the source code alone,
• it has a size linear in that of the source code,
• it can be displayed in the logic in a way that closely resembles source code (using Coq notation system),
• it can be manipulated through a small set of specialized tactics that make it unnecessary to know how the formula is constructed
• it is not just a sound, but also a complete description of the code semantics, meaning that characteristic formulae do not restrict in any way the ability to reason about the code;
• it supports modular verification, and it integrates the frame rule, which enables local reasoning.

## The concept of characteristic formula

The general concept of characteristic formula is not new: it originates in work on process calculi from the 80's (Park, 1981). In this setting, every syntactic process definition is mapped to a formula of Hennessy-Milner's logic. This mapping is such that two syntactic processes are behaviorally equivalent if and only if their associated formulae are logically equivalent. More recently, these results on process calculi were adapted to lambda-calculi by Honda, Berger and Yoshida, who derived a sound and complete Hoare logic for PCF (2006). I have turned the concept of characteristic formula into an effective approach to program verification my building formulae of linear size, expressed in a standard higher-order logic, associated with a pretty-printer, and including support for local reasoning.

It is useful to understand what the characteristic formula approach is not:

• it is not a verification condition generator (VCG): the source code needs not be annotated with invariants; instead, invariants are provided in interactive proofs;
• it is not a deep embedding: there is no inductive data type used to represent code syntax, so we avoid all technical issues related to the representation and manipulation of program syntax;
• it is not a shallow embedding: Caml functions are not represented as Coq functions, so there is no need to resort to monadic programming for avoiding the mismatch between program functions, which can be partial, and logical functions, which must be total;
• it is not a dynamic logic: characteristic formulae are not expressed in ad-hoc logic but instead in a standard higher-order logic, making it possible to leverage on existing theorem provers.

## Software and formal developments

CFML consists of two parts: a program (implemented in Caml) that parses Caml code and produces characteristic formulae in the form of Coq axioms, and a Coq library that provides all the definitions, lemmas and tactics needed for manipulating characteristic formulae. The source code of CFML and a collection of example developments (1000+ lines of Caml proved correct) can be found on the CFML webpage.

## Publications

Characteristic formulae for purely-functional programs are described in the ICFP'10 paper. The generalization to imperative programs is the matter of the ICFP'11 paper. My thesis covers both functional and imperative programs, and moreover includes detailed proofs of soundness and completeness.

Program Verification Through Characteristic Formulae
International Conference on Functional Programming (ICFP), September 2010
Characteristic Formulae for the Verification of Imperative Programs
International Conference on Functional Programming (ICFP), September 2011
Characteristic Formulae for Mechanized Program Verification
PhD thesis, December 2010

## Earlier work: use of a deep embedding

Before developing characteristic formulae, I worked on a deep embedding approach to reasoning about Caml programs using Coq. I showed that, by introducing the right definition, notation and tactics, it is actually feasible to scale up the deep embedding approach to a realistic programming language, more precisely to the pure fragment of Caml.
Interactive Verification of Call-by-Value Functional Programs Through a Deep Embedding
Report, March 2009
Nevertheless, specifications were polluted with the need to relate embedded Caml values with the corresponding Coq values, and proofs involved heavy custom tactics whose use resulted in big proof terms. Characteristic formulae appeared as a way to avoid all of the issues of the deep embedding, while retaining all of the expressivity of this approach.