Olivier Hermant

Zenon Modulo: When Achilles Outruns the Tortoise Using Deduction Modulo

We propose an extension of the tableau-based first order automated theorem prover Zenon to deduction modulo. The theory of deduction modulo is an extension of predicate calculus, which allows us to rewrite terms as well as propositions, and which is well suited for proof search in axiomatic theories, as it turns axioms into rewrite rules. We also present a heuristic to perform this latter step automatically, and assess our approach by providing some experimental results obtained on the benchmarks provided by the TPTP library, where this heuristic is able to prove difficult problems in set theory in particular. Finally, we describe an additional backend for Zenon that outputs proof certificates for Dedukti, which is a proof checker based on the λΠ-calculus modulo.

Computing Invariants with Transformers: Experimental Scalability and Accuracy

Abstract Using abstract interpretation, invariants are usually obtained by solving iteratively a system of equations linking preconditions according to program statements. However, it is also possible to abstract first the statements as transformers, and then propagate the preconditions using the transformers. The second approach is modular because procedures and loops can be abstracted once and for all, avoiding an iterative resolution over the call graph and all the control flow graphs. However, the transformer approach based on polyhedral abstract domains encurs two penalties: some invariant accuracy may be lost when computing transformers, and the execution time may increase exponentially because the dimension of a transformer is twice the dimension of a precondition. The purposes of this article are 1) to measure the benefits of the modular approach and its drawbacks in terms of execution time and accuracy using significant examples and a newly developed benchmark for loop invariant analysis, ALICe , 2) to present a new technique designed to reduce the accuracy loss when computing transformers, 3) to evaluate experimentally the accuracy gains this new technique and other previously discussed ones provide with ALICe test cases and 4) to compare the executions times and accuracies of different tools, ASPIC , ISL , PAGAI and PIPS . Our results suggest that the transformer-based approach used in PIPS , once improved with transformer lists, is as accurate as the other tools when dealing with the ALICe benchmark. Its modularity nevertheless leads to shorter execution times when dealing with nested loops and procedure calls found in real applications.

Programming Robots with Events

We introduce how to use event-based style to program robots through the INI programming language. INI features both built-in and user-defined events, a mechanism to handle various kinds of changes happening in the environment. Event handlers run in parallel either synchronously or asynchronously, and events can be reconfigured at runtime to modify their behavior when needed. We apply INI to the humanoid robot called Nao, for which we develop an object tracking program.

Polarizing Double-Negation Translations

Double-negation translations are used to encode and decode classical proofs in intuitionistic logic. We show that, in the cut-free fragment, we can simplify the translations and introduce fewer negations. To achieve this, we consider the polarization of the formulae and adapt those translation to the different connectives and quantifiers. We show that the embedding results still hold, using a customized version of the focused classical sequent calculus. We also prove the latter equivalent to more usual versions of the sequent calculus. This polarization process allows lighter embeddings, and sheds some light on the relationship between intuitionistic and classical connectives.

Using Event-Based Style for Developing M2M Applications

In this paper, we introduce how to write M2M applications by using INI, a programming language specified and implemented by ourselves that supports event-based style. With event-based programming, all M2M communication can be handled and scheduled. Programmers may use existing built-in events or define their own events. We apply our approach in a real M2M gateway, which allows gathering and exchanging information between sensors and machines in the network. The results shows that our work proposes a concise and elegant alternative and complement to industrial state-of-the-art languages such as Java or C/C++.

Orthogonality and Boolean algebras for deduction modulo

Originating from automated theorem proving, deduction modulo removes computational arguments from proofs by interleaving rewriting with the deduction process. From a proof-theoretic point of view, deduction modulo defines a generic notion of cut that applies to any first-order theory presented as a rewrite system. In such a setting, one can prove cut-elimination theorems that apply to many theories, provided they verify some generic criterion. Pre-Heyting algebras are a generalization of Heyting algebras which are used by Dowek to provide a semantic intuitionistic criterion called superconsistency for generic cut-elimination. This paper uses pre-Boolean algebras (generalizing Boolean algebras) and biorthogonality to prove a generic cut-elimination theorem for the classical sequent calculus modulo. It gives this way a novel application of reducibility candidates techniques, avoiding the use of proof-terms and simplifying the arguments.

Semantic A-translations and Super-Consistency Entail Classical Cut Elimination

We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R. We first define a syntactic and a semantic version of Friedmans A-translation, showing that it preserves the structure of pseudo-Heyting algebra, our semantic framework. Then we relate the interpretation of a theory in the A-translated algebra and its A-translation in the original algebra. This allows to show the stability of the super-consistency criterion and the cut elimination theorem.

Dynamic adaptation through event reconfiguration

We introduce a new programming language called INI, which eases the development of self-adaptive software. INI combines both rule-based and event-based programming paradigms, by allowing the definitions of rules that can be triggered by events. Besides a convenient language-level support for synchronization, the key point with INI events is that they come with a configuration layer, which is set up through input parameters passed to the events when created. Additionally, events can be stopped, reconfigured, and restarted at runtime by the program itself. Ultimately, this re-configuration can be triggered by other events, thus allowing the program to adapt to new operational environments.