Carles Creus

The Emptiness Problem for Tree Automata with Global Constraints

We define tree automata with global constraints (TAGC), generalizing the class of tree automata with global equality and disequality constraints (TAGED). TAGC can test for equality and disequality between subterms whose positions are defined by the states reached during a computation. In particular, TAGC 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 prove decidability of the emptiness problem for TAGC. This solves, in particular, the open question of decidability of emptiness for TAGED. We further extend our result by allowing global arithmetic constraints for counting the number of occurrences of some state or the number of different subterms reaching some state during a computation. We also allow local equality and disequality tests between sibling positions and the extension to unranked ordered trees. As a consequence of our results for TAGC, 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.

Decidable Classes of Tree Automata Mixing Local and Global Constraints Modulo Flat Theories

We define a class of ranked tree automata TABG generalizing both the tree automata with local brother tests of Bogaert and Tison (1992) and with global equality and disequality constraints (TAGED) of Filiot et al. (2007). 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 prove decidability of the emptiness problem for TABG. This solves, in particular, the open question of decidability of emptiness for TAGED. We further extend 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.

Technical Section: R4: Realistic rain rendering in realtime

Realistic rain simulation is a challenging problem due to the variety of different phenomena to consider. In this paper we propose a new rain rendering algorithm that extends present state of the art in the field, achieving real-time rendering of rain streaks and splashes with complex illumination effects, along with fog, halos and light glows as hints of the participating media. Our algorithm creates particles in the scene using an artist-defined storm distribution (e.g., provided as a 2D cloud distribution). Unlike previous algorithms, no restrictions are imposed on the rain area dimension or shape. Our technique adaptively samples the storm area to simulate rain particles only in the relevant regions and only around the observer. Particle simulation is executed entirely in the graphics hardware, by placing the particles at their updated coordinates at each time-step, also checking for collisions with the scene. To render the rain streaks, we use precomputed images and combine them to achieve complex illumination effects. Several optimizations are introduced to render realistic rain with virtually millions of falling rain droplets.

One-context Unification with STG-Compressed Terms is in NP

One-context unification is an extension of first-order term unification in which a variable of arity one standing for a context may occur in the input terms. This problem arises in areas like program analysis, term rewriting and XML processing and is known to be solvable in nondeterministic polynomial time. We prove that this problem can be solved in nondeterministic polynomial time also when the input is compressed using Singleton Tree Grammars (STGs). STGs are a grammar-based compression method for terms that generalizes the directed acyclic graph representation. They have been recently considered as an efficient in-memory representation for large terms, since several operations on terms can be performed efficiently on their STG representation without a prior decompression.

Automatic Evaluation of Context-Free Grammars (System Description)

We implement an online judge for context-free grammars. Our system contains a list of problems describing formal languages, and asking for grammars generating them. A submitted proposal grammar receives a verdict of acceptance or rejection depending on whether the judge determines that it is equivalent to the reference solution grammar provided by the problem setter. Since equivalence of context-free grammars is an undecidable problem, we consider a maximum length l and only test equivalence of the generated languages up to words of length l. This length restriction is very often sufficient for the well-meant submissions. Since this restricted problem is still NP-complete, we design and implement methods based on hashing, SAT, and automata that perform well in practice.

Learning theory through videos: a teaching experience in a theoretical course based on self-learning videos and problem-solving sessions

In this paper we describe a teaching experience applied to a theoretical course tought in a computer science degree. The main feature of our experiment is the introduction of videos specifically designed for self-learning as part of the learning process. Master classes are replaced by working sessions in which the involvement of students gains prominence. The teacher explains almost nothing in class. Instead, most of the time is devoted to the presentation of solutions to exercises assigned to students in advance. All presentations are done by students, and the teacher only intervenes in order to complete explanations and correct mistakes. The result of our experiment is promising from several perspectives. The exam results are better with the new approach. The students learn to learn on their own and take better advantage of the time in class. The work load is uniformly distributed along the course. The new approach also benefits the teacher since he/she spends considerably less time preparing theory lectures, and gets continuous feedback to better follow the students developement. The videos are valuable in themselves and have been made publicly available. In fact, our students prefer them to a master class. They can pause, rewind and replay the video, take a rest, and postpone the lecture if necessary. Moreover, the interest for these videos goes beyond our university boundaries: according to the visits procedence and posted comments, they are being used by students from other countries.In this paper we describe a teaching experience applied to a theoretical course thought in a computer science degree. The main feature of our experiment is the introduction of videos specifically designed for self-learning as part of the learning process. Master classes are replaced by working sessions in which the involvement of students gains prominence. The teacher explains almost nothing in class. Instead, most of the time is devoted to the presentation of solutions to exercises assigned to students in advance. All presentations are done by students, and the teacher only intervenes in order to complete explanations and correct mistakes. The result of our experiment is promising from several perspectives. The exam results are better with the new approach. The students learn to learn on their own and take better advantage of the time in class. The work load is uniformly distributed along the course. The new approach also benefits the teacher since he/she spends considerably less time preparing theory lectures, and gets continuous feedback to better follow the students

Tree Automata with Height Constraints between Brothers

We define the tree automata with height constraints between brothers (TACBB H ). Constraints of equalities and inequalities between heights of siblings that restrict the applicability of the rules are allowed in TACBB H . These constraints allow to express natural tree languages like complete or balanced (like AVL) trees. We prove decidability of emptiness and finiteness for TACBB H , and also for a more general class that additionally allows to combine equality and disequality constraints between brothers.

Emptiness and Finiteness for Tree Automata with Global Reflexive Disequality Constraints

In recent years, several extensions of tree automata have been considered. Most of them are related with the capability of testing equality or disequality of certain subterms of the term evaluated by the automaton. In particular, tree automata with global constraints are able to test equality and disequality of subterms depending on the state to which they are evaluated. The emptiness problem is known decidable for this kind of automata, but with a non-elementary time complexity, and the finiteness problem remains unknown. In this paper, we consider the particular case of tree automata with global constraints when the constraint is a conjunction of disequalities between states, and the disequality predicate is forced to be reflexive. This restriction is significant in the context of XML definitions with monadic key constraints. We prove that emptiness and finiteness are decidable in triple exponential time for this kind of automata.

Non-linear Rewrite Closure and Weak Normalization

A rewrite closure is an extension of a term rewrite system with new rules, usually deduced by transitivity. Rewrite closures have the nice property that all rewrite derivations can be transformed into derivations of a simple form. This property has been useful for proving decidability results in term rewriting. Unfortunately, when the term rewrite system is not linear, the construction of a rewrite closure is quite challenging. In this paper, we construct a rewrite closure for term rewrite systems that satisfy two properties: the right-hand side term in each rewrite rule contains no repeated variable (right-linear) and contains no variable at depth greater than one (right-shallow). The left-hand side term is unrestricted, and in particular, it may be non-linear. As a consequence of the rewrite closure construction, we are able to prove decidability of the weak normalization problem for right-linear right-shallow term rewrite systems. Proving this result also requires tree automata theory. We use the fact that right-shallow right-linear term rewrite systems are regularity preserving. Moreover, their set of normal forms can be represented with a tree automaton with disequality constraints, and emptiness of this kind of automata, as well as its generalization to reduction automata, is decidable.

The HOM Problem is EXPTIME-Complete

We define a new class of tree automata with constraints and prove decidability of the emptiness problem for this class in exponential time. As a consequence, we obtain several EXPTIME-completeness results for problems on images of regular tree languages under tree homomorphisms, like set inclusion, regularity (HOM problem), and finiteness of set difference. Our result also has implications in term rewriting, since the set of reducible terms of a term rewrite system can be described as the image of a tree homomorphism. In particular, we prove that inclusion of sets of normal forms of term rewrite systems can be decided in exponential time. Analogous consequences arise in the context of XML typechecking, since types are defined by tree automata and some type transformations are homomorphic.