Computability on subsets of Euclidean space I: closed and compact subsets

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique, which builds upon the techniques of Ambainis et al. [STOC 2016]. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al. [STOC 2016], to be shown in a unified manner.

https://dl.acm.org/doi/pdf/10.1145/2897518.2897644

Separations in query complexity using cheat sheets

Computability on the probability measures on the Borel sets of the unit interval

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Lois de groupes et analyseurs

Computability on continuous, lower semi-continuous and upper semi-continuous real functions

The notion of expression derivative due to Brzozowski leads to the construction of a deterministic automaton from an extended regular expression, whereas the notion of partial derivative due to Antimirov leads to the construction of a non-deterministic automaton from a simple regular expression. In this paper, we generalize Antimirov partial derivatives to regular expressions extended to complementation and intersection. For a simple regular expression with n symbols, Antimirov automaton has at most n+1 states. As far as an extended regular expression is concerned, we show that the number of states can be exponential.

/pdf/partial-derivatives-of-an-extended-regular-expression-1gotrh33pw.pdf

Partial derivatives of an extended regular expression

Acyclic automata and small expressions using multi-tilde-bar operators

The aim of this paper is to design a theoretical framework that allows us to perform the computation of regular expression derivatives through a space of generic structures. Thanks to this formalism, the main properties of regular expression derivation, such as the finiteness of the set of derivatives, need only be stated and proved one time, at the top level. Moreover, it is shown how to construct an alternating automaton associated with the derivation of a regular expression in this general framework. Finally, Brzozowski’s derivation and Antimirov’s derivation turn out to be a particular case of this general scheme and it is shown how to construct a DFA, a NFA and an AFA for both of these derivations.

/pdf/a-general-framework-for-the-derivation-of-regular-6o6cpwj1l6.pdf

A general framework for the derivation of regular expressions

Natural Numberings and Generalized Computability.

Several studies have been achieved to construct a finite automaton that recognizes the set of words that are at a bounded distance from some word of a given language. In this paper, we introduce a new family of regular operators based on a generalization of the notion of distance and we define a new family of expressions, the approximate regular expressions. We compute Brzozowski derivatives and Antimirov derivatives of such operators, which allows us to provide two recognizers for the language denoted by any approximate regular expression.

Ludovic Mignot

Papers

Partial derivatives of an extended regular expression

Acyclic automata and small expressions using multi-tilde-bar operators

A general framework for the derivation of regular expressions

Natural Numberings and Generalized Computability.

Approximate regular expressions and their derivatives