Report on the design of a Galculator (original) (raw)

Galculator': functional prototype of a Galois-connection based proof assistant

2008

Galculator is the name of the prototype of a proof assistant of a special brand: it is solely based on the algebra of Galois connections. When combined with the pointfree transform and tactics such as the indirect equality principle, Galois connections offer a very powerful, generic device to tackle the complexity of proofs in program verification. The paper describes the architecture of the current Galculator prototype, which is implemented in Haskell in order to steer types as much as possible. The prospect of integrating the Galculator with other proof assistants such as e.g. Coq is also discussed.

An equality theorem prover based on grammar rewriting

1996

ABSTRACT We present the implementation of a term rewriting procedure based on congruence closure. The procedure can be used with arbitrary equational theories. It uses context free grammars to represent equivalence classes of terms. This representation is motivated by the need to handle equational theories where confluence cannot be achieved under traditional term rewriting. Context free grammars provide concise representation of arbitrary-sized equivalence classes of terms.

Automated termination proofs for haskell by term rewriting

ACM Transactions on Programming Languages and Systems, 2011

There are many powerful techniques for automated termination analysis of term rewriting. However, up to now they have hardly been used for real programming languages. We present a new approach which permits the application of existing techniques from term rewriting to prove termination of most functions defined in Haskell programs. In particular, we show how termination techniques for ordinary rewriting can be used to handle those features of Haskell which are missing in term rewriting (e.g., lazy evaluation, polymorphic types, and higher-order functions). We implemented our results in the termination prover AProVE and successfully evaluated them on existing Haskell-libraries.

An Emacs-Interface for Type-Directed Support for Constructing Proofs and Programs.

ENTCS, 2006

This paper presents an emacs interface for an interactive editor for proofs and programs. The interface allows for editing proofs in the same way as we write programs, but in addition offers commands that help term construction. It differs from most other proof editors for its support for direct construction of terms rather than tactics for building them.

Zenon: An Extensible Automated Theorem Prover Producing Checkable Proofs

Lecture Notes in Computer Science, 2007

We present Zenon, an automated theorem prover for first order classical logic (with equality), based on the tableau method. Zenon is intended to be the dedicated prover of the Focal environment, an objectoriented algebraic specification and proof system, which is able to produce OCaml code for execution and Coq code for certification. Zenon can directly generate Coq proofs (proof scripts or proof terms), which can be reinserted in the Coq specifications produced by Focal. Zenon can also be extended, which makes specific (and possibly local) automation possible in Focal.

A Generic Theorem Prover Construction Framework

Formal software engineering methodologies provide a vast array of languages for specifying correctness properties and associated tools for property verification. Unfortunately, the implementation of each verification tool requires an early commitment to a particular methodology and language, in terms of both high-level semantic concerns and low-level syntactic representation of properties and proofs. In this paper, we present Prufrock, a novel approach to constructing automated reasoning systems, which abstracts semantic concerns over entire classes of potential object languages. Prufrock is a modular, generic prover framework written in Haskell taking advantage of its type class system. It consists of a set of independent logic modules defining the semantics required for proof over entire classes of abstract syntaxes using generic programming techniques. The fundamental contribution of Prufrock is that any object language may be used for specifying and verifying properties, as long as it provides a semantics consistent with the logic modules required for a proof. The implementation details of the reasoning system thus remain independent from the structure of the specification language. This facilitates large-scale reuse of logics as well as tacticals and proofs themselves when constructing or retargeting automated reasoning tools. At the same time, Prufrock aids in closing the gap between an object language and verification of objects written in it by operating on its abstract syntax directly rather than transforming it into a separate theorem proving language. * This material is based upon work supported by the United States National Science Foundation under Grant Nos. 0209193 and 0350425 † This work was completed while a student in the System-Level Design Group at The University of Kansas properties, and axioms used in the verification process. Inference and proof are carried out by manipulating this syntax in a logically sound fashion. Unfortunately, this approach can be a serious hindrance to modularity and reuse due to the tight coupling between abstract syntax and the proof techniques used to manipulate it.

Combining Logic and Algebraic Techniques for Program Verification in Theorema

Second International Symposium on Leveraging Applications of Formal Methods, Verification and Validation (isola 2006), 2006

We study and implement concrete methods for the verification of both imperative as well as functional programs in the frame of the Theorema system. The distinctive features of our approach consist in the automatic generation of loop invariants (by using combinatorial and algebraic techniques), and the generation of verification conditions as first-order logical formulae which do not refer to a specific model of computation.

Automated termination proofs for logic programs by term rewriting

ACM Transactions on …, 2009

There are two kinds of approaches for termination analysis of logic programs: "transformational" and "direct" ones. Direct approaches prove termination directly on the basis of the logic program. Transformational approaches transform a logic program into a term rewrite system (TRS) and then analyze termination of the resulting TRS instead. Thus, transformational approaches make all methods previously developed for TRSs available for logic programs as well. However, the applicability of most existing transformations is quite restricted, as they can only be used for certain subclasses of logic programs. (Most of them are restricted to well-moded programs.) In this paper we improve these transformations such that they become applicable for any definite logic program. To simulate the behavior of logic programs by TRSs, we slightly modify the notion of rewriting by permitting infinite terms. We show that our transformation results in TRSs which are indeed suitable for automated termination analysis. In contrast to most other methods for termination of logic programs, our technique is also sound for logic programming without occur check, which is typically used in practice. We implemented our approach in the termination prover AProVE and successfully evaluated it on a large collection of examples.

Defining and Reasoning About Recursive Functions: A Practical Tool for the Coq Proof Assistant

Lecture Notes in Computer Science, 2006

We present a practical tool for defining and proving properties of recursive functions in the Coq proof assistant. The tool generates from pseudo-code the graph of the intended function as an inductive relation. Then it proves that the relation actually represents a function, which is by construction the function that we are trying to define. Then, we generate induction and inversion principles, and a fixpoint equation for proving other properties of the function. Our tool builds upon stateof-the-art techniques for defining recursive functions, and can also be used to generate executable functions from inductive descriptions of their graph. We illustrate the benefits of our tool on two case studies.