Louis Rustenholz

Louis Rustenholz, Manuel V. Hermenegildo, Pedro Lopez-Garcia et al.

We present the theory underpinning a complexity analysis tool (under development) that aims at automating tedious parts of the analysis of complex algorithms originating from the field of automated reasoning. Examples are given by super-exponential quantifier elimination procedures in real and integer arithmetics. Our tool implements the following pipeline. * Together with the algorithm to be analysed, the user (expert, e.g. the algorithm designer) can provide key metrics to track, and lemmas to improve the analysis. In pen-and-paper proofs, these correspond to the "non-tedious" and "creative" parts of the complexity analysis, that require human ingenuity. * The second step consists in the extraction of (generalised) recurrence equations. Here, we rely on a novel higher-order abstract interpretation technique, built on the concept of operator semantics. It enables (optimal) abstract compilation of symbolic programs to different kinds of purely numerical recursive representations, such as recurrence equations on interval-valued functions or numerical logic programs. * Finally, our tool solves the recurrence equations. We propose to go beyond the direct usage of computer algebra systems (CAS), and use pre/postfixpoint-based techniques to discover and verify candidate bounds on the solution. This approach makes use, in turn, of recent progress in SMT solvers, and can also be improved by techniques originating in termination analysis research.

John P. Gallagher, Manuel Hermenegildo, José Morales et al.

The concepts of big-step and small-step derivations are familiar from the operational semantics of programming languages. These concepts are applicable in the more general setting of Horn clause derivations. We prove equivalence between big-step derivations and two versions of small-step derivations for Horn clauses. By specialising interpreters for these derivation strategies, any set of Horn clauses can be transformed into a provably equivalent set of clauses that inherits the behaviour of a given (big- or small-step) Horn clause interpreter. As a special case of this transformation, big-step semantics for any programming language, expressed directly as Horn clauses, can be transformed into equivalent small-step semantics. Experiments with a variety of programming languages are reported.

Louis Rustenholz, Maximiliano Klemen, Miguel Ángel Carreira-Perpiñán et al.

Automatic static cost analysis infers information about the resources used by programs without actually running them with concrete data, and presents such information as functions of input data sizes. Most of the analysis tools for logic programs (and many for other languages), as CiaoPP, are based on setting up recurrence relations representing (bounds on) the computational cost of predicates, and solving them to find closed-form functions. Such recurrence solving is a bottleneck in current tools: many of the recurrences that arise during the analysis cannot be solved with state-of-the-art solvers, including Computer Algebra Systems (CASs), so that specific methods for different classes of recurrences need to be developed. We address such a challenge by developing a novel, general approach for solving arbitrary, constrained recurrence relations, that uses machine-learning (sparse-linear and symbolic) regression techniques to guess a candidate closed-form function, and a combination of an SMT-solver and a CAS to check if it is actually a solution of the recurrence. Our prototype implementation and its experimental evaluation within the context of the CiaoPP system show quite promising results. Overall, for the considered benchmarks, our approach outperforms state-of-the-art cost analyzers and recurrence solvers, and solves recurrences that cannot be solved by them. Under consideration in Theory and Practice of Logic Programming (TPLP).

Eric Goubault, Sébastien Palumby, Sylvie Putot et al.

This paper studies the problem of range analysis for feedforward neural networks, which is a basic primitive for applications such as robustness of neural networks, compliance to specifications and reachability analysis of neural-network feedback systems. Our approach focuses on ReLU (rectified linear unit) feedforward neural nets that present specific difficulties: approaches that exploit derivatives do not apply in general, the number of patterns of neuron activations can be quite large even for small networks, and convex approximations are generally too coarse. In this paper, we employ set-based methods and abstract interpretation that have been very successful in coping with similar difficulties in classical program verification. We present an approach that abstracts ReLU feedforward neural networks using tropical polyhedra. We show that tropical polyhedra can efficiently abstract ReLU activation function, while being able to control the loss of precision due to linear computations. We show how the connection between ReLU networks and tropical rational functions can provide approaches for range analysis of ReLU neural networks.