Section: New Results

Tree Structured Representation of Symbolic Temporal Data

Participant : Florent Jacquemard.

In traditional music notation, in particular in the languages used for the notation of mixed music such as Antescofo DSL, the durations are not expressed by numerical quantities but by symbols representing successive subdivisions of a reference time value (the beat). For this reason, trees data structures are commonly used for the symbolic representation of rhythms in computer aided composition softwares such as OpenMusic (developed at Ircam).

Following this idea, we have been working on using several tree automata techniques for the challenging and long-standing problem of automatic transcription of rhythm (in traditional music notations) from symbolic input data (symbolic traces with timestamps in ms, like e.g. in MIDI format). To summarize, the main problem in rhythm transcription is to find an acceptable balance between timing precision (the goal is to minimize the loss obtained by transformation of ms timing values into fractions of beats) and the complexity of the notation obtained. The relative importance of these two measures may vary largely according to the user (composer), his workflow, and the musical style considered. It is therefore important to be able to control this balance during the transcription process, in order to adapt to the case of users. In traditional approaches, the transcription is done by an alignement of the input trace on a grid, and the two measures (precision of the grid and complexity) are either defined by parameters fixed a priori or hardcoded e.g. for a precise musical style and composition workflow. During two internships co-supervised by Jean Bresson (Ircam, main developer of OpenMusic) and Florent Jacquemard, we have been studying more flexible new approaches, based on computations on the tree representation of rhythms.

Pierre Donat-Bouillud (L3 ENS Rennes) [29] has worked on an approach by transformation of trees following some rewrite rules. The general idea is to start with a complex tree representing timings very close to the input data, and to simplify it by rewriting until an acceptable level of complexity is reached. The rewrite rules are either generic (defining an equational theory of rhythm notation) or user defined (defining approximations). This approach has been implemented in an OpenMusic library.

Adrien Maire (M1 ENS Cachan) has studied another very promising approach based on stochastic tree automata learning in an interactive authoring scenario. The generated automaton is supposed to represent (by the weighted tree langage it defines) the expected complexity of rhythm notations (i.e. the user's "style").

Moreover, we have following other work on several classes of tree recognizers and tree transformations which could be of interest in this context. With Luis Barguñó, Carlos Creus, Guillem Godoy, and Camille Vacher, [11] we define a class of ranked tree automata called TABG generalizing both the tree automata with local brother tests of Bogaert and Tison [37] and with global equality and disequality constraints (TAGED) of Filiot et al. [39] . TABG can test for equality and disequality modulo a given flat equational theory between brother subterms and between subterms whose positions are defined by the states reached during a computation. In particular, TABG can check that all the subterms reaching a given state are distinct. This constraint is related to monadic key constraints for XML documents, meaning that every two distinct positions of a given type have different values. We have proven decidability of the emptiness problem for TABG. This solves, in particular, the open question of decidability of emptiness for TAGED. We further extended our result by allowing global arithmetic constraints for counting the number of occurrences of some state or the number of different equivalence classes of subterms (modulo a given flat equational theory) reaching some state during a computation. We also adapt the model to unranked ordered terms. As a consequence of our results for TABG, we prove the decidability of a fragment of the monadic second order logic on trees extended with predicates for equality and disequality between subtrees, and cardinality.

With Michaël Rusinowitch (EPI Cassis), we have introduced in [25] an extension of unranked tree automata called bi-dimensional context-free hedge automata. The languages they define are context free in two dimensions: in the the sequence of successors of a node and also along paths. This formalism is useful for the static type-checking of tree transformations such as XML updates defined in the W3C XQuery Update Facility. We have developed with the same author in the past years a general framework for the verification of unranked (XML) tree transformations based on tree automata techniques. It has been presented this year in an invited keynote [16] . We have also presented with Emmanuel Filiot and Sophie Tison a survey on tree automata with constraints [33] during a Dagstuhl Seminar (number 13192) on tree transducers and formal methods.