Strategies to scan pictures with automata based on Wang tiles

Wang automata are devices for picture language recognition recently introduced by us, which characterize the class REC of recognizable picture languages. Thus, Wang automata are equivalent to tiling systems or online tessellation acceptors, and are based like Wang systems on labeled Wang tiles. The present work focus on scanning strategies, to prove that the ones Wang automata are based on are those following four kinds of movements: boustrophedonic, ``L-like'', ``U-like'', and spirals

Deterministic recognizability of picture languages with Wang automata

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to ApplicationsInternational audienceWe present a model of automaton for picture language recognition, called Wang automaton, which is based on labeled Wang tiles. Wang automata combine features of both online tessellation acceptors and 4-way automata: as in online tessellation acceptors, computation assigns states to each picture position; as in 4-way automata, the input head visits the picture moving from one pixel to an adjacent one, according to some scanning strategy. Wang automata recognize the class REC, i.e. they are equivalent to tiling systems or online tessellation acceptors, and hence strictly more powerful than 4-way automata. We also introduce a natural notion of determinism for Wang automata, and study the resulting class, extending the more traditional approach of diagonal-based determinism, used e. g. by deterministic tiling systems. In particular, we prove that the concept of row (or column) ambiguity defines the class of languages recognized by Wang automata directed by boustrophedonic scanning strategies

Pattern statistics and Vandermonde matrices

In this paper we determine some limit distributions of pattern statistics in rational stochastic models. We present a general approach to analyze these statistics in rational models having an arbitrary number of strongly connected components. We explicitly establish the limit distributions in most significant cases; they are characterized by a family of unimodal density functions defined by means of confluent Vandermonde matrices

Frequency of symbol occurrences in simple non-primitive stochastic models

We study the random variable Y-n representing the number of occurrences of a given symbol in a word of length n generated at random. The stochastic model we assume is a simple non-ergodic model defined by the product of two primitive rational formal series, which form two distinct ergodic components. We obtain asymptotic evaluations for the mean and the variance of Y-n and its limit distribution. It turns out that there are two main cases: if one component is dominant and non-degenerate we get a Gaussian limit distribution; if the two components are equipotent and have different leading terms of the mean, we get a uniform limit distribution. Other particular limit distributions are obtained in the case of a degenerate dominant component and in the equipotent case when the leading terms of the expectation values are equal

Frequency of symbol occurrences in bicomponent stochastic models

We give asymptotic estimates of the frequency of occurrences of a symbol in a random word Generated by any bicomponent stochastic model. More precisely, we consider the random variable Y-n representing the number of occurrences of a given symbol in a word of length n generated at random; the stochastic model is defined by a rational formal series r having a linear representation with two primitive components. This model includes the case when r is the product or the sum of two primitive rational formal series. We obtain asymptotic evaluations for the mean value and the variance of Yn and its limit distribution

The Number of Convex Permutominoes

Permutominoes are polyominoes defined by suitable pairs of permutations. In this paper we provide a formula to count the number of convex permutominoes of given perimeter. To this aim we define the transform of a generic pair of permutations, we characterize the transform of any pair defining a convex permutomino, and we solve the counting problem in the transformed space

Situated Learning with Bebras Tasklets

A Bebras short task, a tasklet, is designed to provide a source for exploring a computational thinking concept: at the end of the contest it could be used as a starting point to delve deeper into a computing topic. In this paper we report an experience which aims at taking full advantage of the potential of Bebras tasklets. A math teacher asked her pupils to act as Bebras \u201ctrainers\u201d for younger mates. The pupils, in pairs, were assigned to design and prepare a tangible game inspired by a Bebras tasklet, devised for the younger pupils to practice. They also had to explain the game to the younger pupils, make them play and support them in solving it. In carrying out this assignment the pupils acting as trainers had to deeply explore the Bebras tasklet and face its computational thinking challenge, and also practiced soft skills as collaborating with peers towards a common goal, adapting language and communicative style to engage with younger mates, devising and designing a tangible object, and planning its creation. The experience proved that using Bebras tasklets as the social and cultural context for situated learning of computational thinking competencies is indeed quite productive

Logic Characterization of Invisibly Structured Languages: The Case of Floyd Languages

Operator precedence grammars define a classical Boolean and deterministic context-free language family (called Floyd languages or FLs). FLs have been shown to strictly include the well-known Visibly Pushdown Languages, and enjoy the same nice closure properties. In this paper we provide a complete characterization of FLs in terms of a suitable Monadic Second-Order Logic. Traditional approaches to logic characterization of formal languages refer explicitly to the structures over which they are interpreted - e.g, trees or graphs - or to strings that are isomorphic to the structure, as in parenthesis languages. In the case of FLs, instead, the syntactic structure of input strings is “invisible” and must be reconstructed through parsing. This requires that logic formulae encode some typical context-free parsing actions, such as shift-reduce ones