Implicit complexity for coinductive data: a characterization of corecurrence

We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using productivity (fairness) as the fundamental assertion, rather than bi-simulation. The latter is expressible in terms of the former. As an application to this framework, we give an implicit characterization of corecurrence: a function is definable using corecurrence iff its productivity is provable using coinduction for formulas in which data-predicates do not occur negatively. This is an analog, albeit in weaker form, of a characterization of recurrence (i.e. primitive recursion) in [Leivant, Unipolar induction, TCS 318, 2004].Comment: In Proceedings DICE 2011, arXiv:1201.034

Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions

Recursive analysis is the most classical approach to model and discuss computations over the reals. It is usually presented using Type 2 or higher order Turing machines. Recently, it has been shown that computability classes of functions computable in recursive analysis can also be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub- and sup-classes) over the reals and algebraically defined (sub- and sup-) classes of $\R$-recursive functions à la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of functions over the reals. In particular we provide the first algebraic characterization of polynomial time computable functions over the reals. This framework opens the field of implicit complexity of functions over the reals, and also provide a new reading of some of the existing characterizations at the computability level

Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions

Accepted for publication in International Journal of Unconventional ComputingInternational audienceRecursive analysis is the most classical approach to model and discuss computations over the real numbers.Recently, it has been shown that computability classes of functions in the sense of recursive analysis can be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub- and sup-classes) over the reals and algebraically defined (sub- and sup-) classes of R-recursive functions à la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of real functions. In particular we provide the first algebraic characterization of polynomial-time computable functions over the reals. This framework opens the field of implicit complexity of analog functions, and also provides a new reading of some of the existing characterizations at the computability level

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Distances for Weighted Transition Systems: Games and Properties

We develop a general framework for reasoning about distances between transition systems with quantitative information. Taking as starting point an arbitrary distance on system traces, we show how this leads to natural definitions of a linear and a branching distance on states of such a transition system. We show that our framework generalizes and unifies a large variety of previously considered system distances, and we develop some general properties of our distances. We also show that if the trace distance admits a recursive characterization, then the corresponding branching distance can be obtained as a least fixed point to a similar recursive characterization. The central tool in our work is a theory of infinite path-building games with quantitative objectives.Comment: In Proceedings QAPL 2011, arXiv:1107.074

Combinatorics of bicubic maps with hard particles

We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting of the maps. Although these trees have no simple local characterization, we prove that their enumeration may be performed upon introducing a larger class of "admissible" trees with possibly doubly-occupied edges and summing them with appropriate signed weights. The proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non-sectile edges. The admissible trees are characterized by simple local rules, allowing eventually for an exact enumeration of planar bicubic maps with hard particles. We also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re-derive the enumeration of quartic maps with Ising spins in the present framework of admissible trees. We finally comment on a possible interpretation in terms of branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction and discussion/conclusion extended, minor corrections, references adde