Paulin Jacobé de Naurois

The complexity of semilinear problems in succinct representation

Abstract.We prove completeness results for twenty-three problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the Blum–Shub–Smale additive model of computation. If, in contrast, the circuit is constant-free, then the completeness results are for the Turing model of computation. One such result, the PNP[log]-completeness of deciding Zariski irreducibility, exhibits for the first time a problem with a geometric nature complete in this class.

Computability over an arbitrary structure: sequential and parallel polynomial time

We provide several machine-independent characterizations of deterministic complexity classes in the model of computation proposed by L. Blum, M. Shub and S. Smale. We provide a characterization of partial recursive functions over any arbitrary structure. We show that polynomial time computable functions over any arbitrary structure can be characterized in term of safe recursive functions. We show that polynomial parallel time decision problems over any arbitrary structure can be characterized in terms of safe recursive functions with substitutions.

Implicit Complexity over an Arbitrary Structure: Sequential and Parallel Polynomial Time

We provide several machine-independent characterizations of deterministic complexity classes in the model of computation proposed by L. Blum, M. Shub and S. Smale. We provide a characterization of partial recursive functions over any arbitrary structure. We show that polynomial time over an arbitrary structure can be characterized in terms of safe recursion. We show that polynomial parallel time over an arbitrary structure can be characterized in terms of safe recursion with substitutions.

Correctness of linear logic proof structures is NL-complete

We provide new correctness criteria for all fragments (multiplicative, exponential, additive) of linear logic. We use these criteria for proving that deciding the correctness of a linear logic proof structure is NL-complete.

Implicit complexity over an arbitrary structure: quantifier alternations

We provide machine-independent characterizations of some complexity classes, over an arbitrary structure, in the model of computation proposed by L. Blum, M. Shub, and S. Smale. We show that the levels of the polynomial hierarchy correspond to safe recursion with predicative minimization and the levels of the digital polynomial hierarchy to safe recursion with digital predicative minimization. Also, we show that polynomial alternating time corresponds to safe recursion with predicative substitutions and that digital polynomial alternating time corresponds to safe recursion with digital predicative substitutions.

Correctness of multiplicative (and exponential) proof structures is NL-complete

We provide a new correctness criterion for unit-free MLL proof structures and MELL proof structures with units. We prove that deciding the correctness of a MLL and of a MELL proof structure is NL- complete. We also prove that deciding the correctness of an intuitionistic multiplicative essential net is NL-complete.

A measure of space for computing over the reals

We propose a new complexity measure of space for the BSS model of computation. We define LOGSPACEW and PSPACEW complexity classes over the reals. We prove that LOGSPACEW is included in

Tailoring Recursion to Characterize Non-Deterministic Complexity Classes Over Arbitrary Structures

{\sf NC}^2_{\mathbb{R}} \cap {\sf P}_W

Rewriting systems for reachability in vector addition systems with pairs

, i.e. is small enough for being relevant. We prove that the Real Circuit Decision Problem is Pℝ-complete under LOGSPACEW reductions, i.e. that LOGSPACEW is large enough for containing natural algorithms. We also prove that PSPACEW is included in PARℝ.

Parallel Time and Quantifier Prefixes

We provide machine-independent characterizations of some complexity classes, over an arbitrary structure, in the model of computation proposed by L. Blum, M. Shub and S. Smale. We show that the levels of the polynomial hierarchy correspond to safe recursion with predicative minimization. The levels of the digital polynomial hierarchy correspond to safe recursion with digital predicative minimization. Also, we show that polynomial alternating time corresponds to safe recursion with predicative substitutions and that digital polynomial alternating time corresponds to safe recursion with digital predicative substitutions.