SD-Regular Transducer Expressions for Aperiodic Transformations

Abstract

FO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star-free expressions = aperiodic languages) result of Schützenberger. Our result also generalizes a lesser known characterization by Schützenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite words captured by an aperiodic deterministic two-way transducer can be described with an SDregular transducer expression (SDRTE). An SDRTE is a regular expression where Kleene stars are used in a restricted way: they can appear only on aperiodic languages which are prefix codes of bounded synchronization delay. SDRTEs are constructed from simple functions using the combinators unambiguous sum (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned. In order to construct an SDRTE associated with an aperiodic deterministic two-way transducer, (i) we concretize Schützenberger's SD=AP result, by proving that aperiodic languages are captured by SD-regular expressions which are unambiguous and stabilising; (ii) by structural induction on the unambiguous, stabilising SD-regular expressions describing the domain of the transducer, we construct SDRTEs. Finally, we also look at various formalisms equivalent to SDRTEs which use the function composition, allowing to trade the k-chained star for a 1-star.

Efficient Construction of Reversible Transducers from Regular Transducer Expressions

“…The Büchi-Elgot-Trakhtenbrot theorem was generalized by Engelfreit and Hoogeboom [10], where regular transformations were defined using two-way transducers (2DFTs) as well as by the MSO transductions of Courcelle [5]. The analogue of Kleene's theorem for transformations was proposed by [2] and [8], while [7] proved the analogue of Schützenberger's theorem [13] for transformations. In another related work, [3,4] proposes a translation from unambiguous two-way transducers to regular function expressions extending the Brzozowski and McCluskey algorithm.…”

“…Like classical regular expressions, these expressions provide a robust foundation for specifying transducer patterns in a declarative manner, and can be widely used in practical applications. An important question left open in [2], [7], [8] is the complexity of the procedure that builds the transducer from the combinator expressions. Providing efficient constructions of finite state transducers equivalent to expressions is a fundamental problem, and is often the first step of algorithmic applications, such as evaluation.…”

“…In this paper, we focus on this problem. First, we recall the combinators from [2], [7] and [8] in order of increasing difficulty. It is known that the combinators of [2] and [8] are equivalent (they both characterize regular transformations), even though the notations differ slightly.…”

“…RTE is the full class consisting of all combinators (i´ix), and its unambiguous fragment is equivalent to regular transformations (those computed by a deterministic two-way transducer, 2DFT). Note that we consider chained k-star of [7] here, even though chained 2-star suffice for expressing all regular transformations, since the idea of our construction is general.…”

Pebble Minimization of Polyregular Functions

“…There have been recurrent attempts to generalize the notions of regularity from languages to functions, leading to classes of various expressiveness, such as 1 Mealy Machines (Mealy) [35], sequential functions (Seq) [44], rational functions (Rat) [20], regular functions (Reg) [22], and polyregular functions (Poly) [7]. For all of these models, an aperiodic counterpart can be defined (respectively, AMealy, ASeq, ARat, AReg, SF), lifting the correspondence between counter-free automata, star-free languages and aperiodic monoids to the functional setting [27,4,9,14,6,7,13]. The inclusions between these classes of functions are all known to be strict, and we depicted in Figure 1 the status of the membership problem associated to these strict inclusions (that is, given a function f , decide if f can be computed by a function of a proper subclass).…”