## Project-Team : calligramme

## Section: New Results

## Implicit Complexity of Computation

### Complexity in the Blum, Shub, Smale model of computation

In order to model discrete time computation over real numbers, Blum, Shub and Smale have introduced in 1989 a new model of computation, referred as the BSS machine or sometimes the real Turing machine. In this model, the complexity of a computational problem is given by the number of elementary arithmetical operations and comparisons needed to solve this problem, independently of the underlying representation of real numbers. This makes it very different from the recursive analysis model, and occurs to be a very natural way to describe computational problems over real numbers. The BSS model of computation has been later on extended to the notion of computation over arbitrary logical structure. Since classical complexity theory occurs to be the restriction of this general model to boolean structures, it gives a new insight on previous questions on classical complexity theory and its links with logic.

Paulin de Naurois' PhD thesis [7], jointly supervised by Olivier Bournez and Jean-Yves Marion, is focused on the study of complexity in this model. It provides new completeness results for geometrical problems when the arbitrary structure is specialized to the set of real numbers with addition and order. The range of these completeness results covers many of the most important complexity classes over this setting, which is one of the three major ones for complexity theory over arbitrary structures. He also provides several machine-independent characterizations of complexity classes over arbitrary structures. Also included is the extension of some results by Grädel, Gurevich and Meer in descriptive complexity, characterizing deterministic and non deterministic polynomial time decision problems in terms of logics over metafinite structures. In addition there can be found the extension of some results in implicit complexity by Bellantoni and Cook [34], that characterizes functions computable in sequential deterministic polynomial time, and by Leivant and Marion [45], that characterizes functions computable in parallel deterministic polynomial time in terms of algebras of recursive functions. In addition, there are characterizations of functions computable within the polynomial hierarchy, extending a result by Bellantoni [33], and in polynomial alternating time. These results have been published in 2003. [35][36]. Other related publications are [18][8].

### Implicit Complexity

New results have been obtained this year, from which the main ones are the following. First of all, G. Bonfante, J.-Y. Marion and J.-Y. Moyen have shown that the synthesis of quasi-interpretation is decidable, more precisely, it can be done in double exponential time with respect to the size of the program. This is of major importance as one may automatize the process of certification. In mobile computing, the quasi-interpretation may be sent as a certificate with the code, such a certificate can be verified by the receiving device. For more restricted classes of quasi-interpretations, R. Péchoux has shown that synthesis of multilinear quasi-interpretations is NP-complete. This result slightly improves on Amadio's work because it applies to QI over real numbers and not over rationals.

Next, G. Bonfante, J.-Y Marion and J.-Y. Moyen have refined the complexity hierarchy, by introducing some new constraints on programs. This give rise to various complexity classes, among them: Logspace, Linspace, Ptime, Pspace.