Internship report - Streaming String Transducers

In formal language theory, two very different models sometimes turn out to describe the same class of languages. This usually shows that there is a fundamental concept described by those models. A well-known example is the class of regular languages, which can be characterized by logic (monadic second order (MSO) logic), algebra (syntactic monoids), and many computational models (automata). In particular, it was shown by Rabin and Scott [11] that two-way finite state automata are equivalent to finite state automata, even in presence of non-determinism. However, these results do not hold when, instead of languages, we consider transductions, that is, relations from strings to strings. Indeed, two-way finite state transducers are strictly more powerful than their one-way counterparts. The non-deterministic versions of these models are also strictly more powerful than the deterministic ones. Nonetheless, some notable equalities have been proven between those classes of transductions: in [8], Engelfriet and Hoogeboom have shown that two-way deterministic generalized sequential machines (2dgsm) define the same class of transductions as deterministic MSO string transducers (dmsos). Recently, Alur has described a new model for defining string transductions [1], deterministic streaming string transducers (dsst), which is equally expressive as 2dgsm and MSO logic [4]. An interesting feature of this model compared to 2dgsm is that it only does a one-way pass through its input, using a finite number of string variables in order to compute the output. Moreover, when we consider the non-deterministic models, nsst are shown to be as expressive as non-deterministic MSO logic [3], whereas they are incomparable to 2ngsm. Alur has also extended his model to string-to-tree and tree-to-tree transductions, with interesting results in terms of expressiveness and decidability [2]. During this internship, I focused on the deterministic versions of those models. More precisely, I looked at the equivalence between dsst and 2dgsm in terms of resource usage, and tried to exhibit a relation between the number of variables of a dsst and the number of back-and-forth moves of an equivalent 2dgsm. Section 2 introduces the definitions of the different models. Section 3 proves some expressiveness results between dsst, dmsos and 2dgsm. Section 4 deals with relations between the resources of dsst and 2dgsm.