Jean-Yves Moyen

Efficient first order functional program interpreter with time bound certifications

We demonstrate that the class of functions computed by first order functional programs over lists which terminate by multiset path ordering and admit a polynomial quasi-interpretation, is exactly the class of function computable in polynomial time. The interest of this result lies on (i) the simplicity of the conditions on programs to certify their complexity, (ii) the fact that an important class of natural programs is captured, (iii) potential applications for program optimisation.

Quasi-interpretations a way to control resources

This paper presents in a reasoned way our works on resource analysis by quasi-interpretations. The controlled resources are typically the runtime, the runspace or the size of a result in a program execution. Quasi-interpretations allow the analysis of system complexity. A quasi-interpretation is a numerical assignment, which provides an upper bound on computed functions and which is compatible with the program operational semantics. The quasi-interpretation method offers several advantages: (i) It provides hints in order to optimize an execution, (ii) it gives resource certificates, and (iii) finding quasi-interpretations is decidable for a broad class which is relevant for feasible computations. By combining the quasi-interpretation method with termination tools (here term orderings), we obtained several characterizations of complexity classes starting from Ptime and Pspace.

On Lexicographic Termination Ordering with Space Bound Certifications

We propose a method to analyse the program space complexity, based on termination orderings. This method can be implemented to certify the runspace of programs. We demonstrate that the class of functions computed by first order functional programs over free algebras which terminate by Lexicographic Path Ordering and admit a polynomial quasi-interpretation, is exactly the class of functions computable in polynomial space.

Quasi-interpretations and small space bounds

Quasi-interpretations are an useful tool to control resources usage of term rewriting systems, either time or space. They not only combine well with path orderings and provide characterizations of usual complexity classes but also give hints in order to optimize the program. Moreover, the existence of a quasi-interpretation is decidable. In this paper, we present some more characterizations of complexity classes using quasi-interpretations. We mainly focus on small space-bounded complexity classes. On one hand, by restricting quasi-interpretations to sums (that is allowing only affine quasi-interpretations), we obtain a characterization of LinSpace. On the other hand, a strong tiering discipline on programs together with quasi-interpretations yield a characterization of LogSpace. Lastly, we give two new characterizations of Pspace: in the first, the quasi-interpretation has to be strictly decreasing on each rule and in the second, some linearity constraints are added to the system but no assumption concerning the termination proof is made.

Resource control graphs

Resource Control Graphs are an abstract representation of programs. Each state of the program is abstracted by its size, and each instruction is abstracted by the effects it has on the state size whenever it is executed. The abstractions of instruction effects are then used as weights on the arcs of a programs Control Flow Graph. Termination is proved by finding decreases in a well-founded order on state-size, in line with other termination analyses, resulting in proofs similar in spirit to those produced by Size Change Termination analysis. However, the size of states may also be used to measure the amount of space consumed by the program at each point of execution. This leads to an alternative characterisation of the Non Size Increasing programs, that is, of programs that can compute without allocating new memory. This new tool is able to encompass several existing analyses and similarities with other studies, suggesting that even more analyses might be expressable in this framework, thus giving hopes for a generic tool for studying programs.

On quasi-interpretations, blind abstractions and implicit complexity

Quasi-interpretations are a technique for guaranteeing complexity bounds on first-order functional programs: in particular, with termination orderings, they give a sufficient condition for a program to be executable in polynomial time (Marion and Moyen 2000), which we call the P-criterion here. We study properties of the programs satisfying the P-criterion in order to improve the understanding of its intensional expressive power. Given a program, its blind abstraction is the non-deterministic program obtained by replacing all constructors with the same arity by a single one. A program is blindly polytime if its blind abstraction terminates in polynomial time. We show that all programs satisfying a variant of the P-criterion are in fact blindly polytime. Then we give two extensions of the P-criterion: one relaxing the termination ordering condition and the other (the bounded-value property) giving a necessary and sufficient condition for a program to be polynomial time executable, with memoisation.

Static complexity analysis of higher order programs

In the papers first part, we present a method for certifying that the values computed by a first order imperative program will be bounded by polynomials in the programs inputs. Our method does not yield concrete polynomials, but shows existence of polynomial bounds and upper bounds to their polynomial degrees. In the second part of the paper, we lift our method to allow analysis of higher order programs.

Formal proof of polynomial-time complexity with quasi-interpretations

We present a Coq library that allows for readily proving that a function is computable in polynomial time. It is based on quasi-interpretations that, in combination with termination ordering, provide a characterisation of the class fp of functions computable in polynomial time. At the heart of this formalisation is a proof of soundness and extensional completeness. Compared to the original paper proof, we had to fill a lot of not so trivial details that were left to the reader and fix a few glitches. To demonstrate the usability of our library, we apply it to the modular exponentiation.

Chains, antichains, and complements in infinite partition lattices

We consider the partition lattice