Michaël Cadilhac

BOUNDED PARIKH AUTOMATA

The Parikh finite word automaton model (PA) was introduced and studied by Klaedtke and Rues. Here, we present some expressiveness properties of a restriction of the deterministic affine PA recently introduced, and use them as a tool to show that the bounded languages recognized by PA are the same as those recognized by deterministic PA. Moreover, this class of languages is shown equal to the class of bounded languages with a semilinear iteration set.

Affine Parikh automata

The Parikh finite word automaton (PA) was introduced and studied in 2003 by Klaedtke and Rues. Natural variants of the PA arise from viewing a PA equivalently as an automaton that keeps a count of its transitions and semilinearly constrains their numbers. Here we adopt this view and define the affine PA , that extends the PA by having each transition induce an affine transformation on the PA registers, and the PA on letters , that restricts the PA by forcing any two transitions on the same letter to affect the registers equally. Then we report on the expressiveness, closure, and decidability properties of such PA variants. We note that deterministic PA are strictly weaker than deterministic reversal-bounded counter machines.

UNAMBIGUOUS CONSTRAINED AUTOMATA

The class of languages captured by Constrained Automata (CA) that are unambiguous is shown to possess more closure properties than the provably weaker class captured by deterministic CA. Problems decidable for deterministic CA are nonetheless shown to remain decidable for unambiguous CA, and testing for regularity is added to this set of decidable problems. Unambiguous CA are then shown incomparable with deterministic reversal-bounded machines in terms of expressivity, and a deterministic model equivalent to unambiguous CA is identified.

A Circuit Complexity Approach to Transductions

Low circuit complexity classes and regular languages exhibit very tight interactions that shade light on their respective expressiveness. We propose to study these interactions at a functional level, by investigating the deterministic rational transductions computable by constant-depth, polysize circuits. To this end, a circuit framework of independent interest that allows variable output length is introduced. Relying on it, there is a general characterization of the set of transductions realizable by circuits. It is then decidable whether a transduction is definable in \(\mathrm{AC}^0\) and, assuming a well-established conjecture, the same for \(\mathrm{ACC}^0\).

Unambiguous constrained automata

The class of languages captured by Constrained Automata (CA) that are unambiguous is shown to possess more closure properties than the provably weaker class captured by deterministic CA. Problems decidable for deterministic CA are nonetheless shown to remain decidable for unambiguous CA, and testing for regularity is added to this set of decidable problems. Unambiguous CA are then shown incomparable with deterministic reversal-bounded machines in terms of expressivity, and a deterministic model equivalent to unambiguous CA is identified.

Review of handbook of weighted automata, edited by Manfred Droste, Werner Kuich and Heiko Vogler

When considering a language (i.e., a set of words over a finite alphabet), one may be interested in adding some information to each word. Mappings from Σ∗ to a weighting structure S are thus of interest. When defining recognizers for those mappings, a natural algebraic structure for S emerges: that of a semiring — think of a base set with addition and multiplication, together with neutral and absorbing elements. Mappings from Σ∗ to a semiring are called formal power series; this book studies some of their classes, generalizations and applications. A natural thing to do when considering adding weights to a recognizer, say to a nondeterministic finite automaton, is to put them on its transitions. Now, what is the weight associated with a word w recognized by the automaton? (That is to say, what is the series described by the automaton?) This is where semirings become natural: for a successful run labeled w on the automaton, the weight of this run is defined as the product of the weights of the transitions. Then the weight of w is the sum of the weights of all successful runs labeled w. For instance, suppose the underlying semiring of a weighted automaton is the so-called tropical semiring 〈N ∪ {∞}, min, +,∞, 0〉, where min is the addition and + the multiplication. If each transition has weight 1, then the weight associated with a word w is the length of the shortest successful run on w (or ∞ if none exists). Weights can model numerous notions, from probability of apparition to ambiguity in a grammar, in this very framework. This book covers extensively both theory and applications of formal power series, focusing on both classical and state-of-the-art results.

Bounded Parikh Automata

The Parikh finite word automaton model (PA) was introduced and studied by Klaedtke and Ruess in 2003. Here, by means of related models, it is shown that the bounded languages recognized by PA are the same as those recognized by deterministic PA. Moreover, this class of languages is the class of bounded languages whose set of iterations is semilinear.

The Algebraic Theory of Parikh Automata

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theories of typed monoids and of rational series are used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Parikh automata languages in NC1. Affine Parikh automata, where each transition applies an affine transformation on the registers, are also considered. Relying on these characterizations, the landscape of relationships and closure properties of the classes at hand is completed, in particular over unary languages.

A Language-Theoretical Approach to Descriptive Complexity

Logical formulas are naturally decomposed into their subformulas and circuits into their layers. How are these decompositions expressed in a purely language-theoretical setting? We address that question, and in doing so, introduce a product directly on languages that parallels formula composition. This framework makes an essential use of languages of higher-dimensional words, called hyperwords, of arbitrary dimensions. It is shown here that the product thus introduced is associative over classes of languages closed under the product itself; this translates back to extra freedom in the way formulas and circuits can be decomposed.

Review of: The Golden Ratio and Fibonacci Numbers by Richard A. Dunlap

Professor Dunlap, a physicist specialized in the study of materials, presents a few tokens of the pervasiveness of the golden ratio and the Fibonacci sequence in our mathematical, physical, and biological world. Mainly focused on geometrical facts and their applications, the book contains a fantasy mix of the author’s favorite curiosities around this topic. If the book is by no means exhaustive—and it has no pretension to be—it is a nice potpourri that anyone can enjoy, while claiming to not be addressed to any particular profile.