Multi-sequential Word Relations

Rational relations are binary relations of finite words that are realised by non-deterministic finite state transducers (NFT). A particular kind of rational relations is the sequential functions. Sequential functions are the functions that can be realised by input-deterministic transducers. Some rational functions are not sequential. However, based on a property on transducers called the twinning property, it is decidable in PTime whether a rational function given by an NFT is sequential. In this paper, we investigate the generalisation of this result to multi-sequential relations, i.e. relations that are equal to a finite union of sequential functions. We show that given an NFT, it is decidable in PTime whether the relation it defines is multi-sequential, based on a property called the weak twinning property. If the weak twinning property is satisfied, we give a procedure that effectively constructs a finite set of input-deterministic transducers whose union defines the relation. This procedure generalises to arbitrary NFT the determinisation procedure of functional NFT. Finite transducers extend finite automata with output words on transitions. Any successful computation (called run) of a transducer defines an output word obtained by concatenating, from left to right, the words occurring along the transitions of that computation. Since transitions are non-deterministic in general, there might be several successful runs on the same input word u, and hence several output words associated with u. Therefore, finite transducers can define binary relations of finite words, the so-called class of rational relations [7,4]. Unlike finite automata, the equivalence problem, i.e. whether two transducers define the same relation, is undecidable [9]. This has motivated the study of different subclasses of rational relations, and their associated definability problems: given a finite transducer T , does the relation JT K it defines belong to a given class C of relations? We survey the most important known subclasses of rational relations. Rational Functions An important subclass of rational relations is the class of rational functions. It enjoys decidable equivalence and moreover, it is decidable whether a transducer is functional, i.e. defines a function. This latter result was first shown by Schützenberger with polynomial space complexity [14] and the complexity has been refined to polynomial time in [10,3]. A subclass of rational functions which enjoys good algorithmic properties with respect to evaluation is the class of sequential functions. Sequential functions are those functions defined by finite transducers whose underlying input automaton is deterministic (called sequential transducers). Some rational functions are not sequential. E.g., over the alphabet Σ = {a, b}, the function fswap ar X iv :1 50 4. 03 86 4v 1 [ cs .F L ] 1 5 A pr 2 01 5 mapping any word of the form uσ to σu, where u ∈ Σ∗ and σ ∈ Σ, is rational but not sequential, because finite transducers process input words from left-toright, and therefore any transducer implementing that function should guess non-deterministically the last symbol of uσ. Given a functional transducer, it is decidable whether it defines a sequential function [5], based on a structural property of finite transducers called the twinning property. This property can be decided in PTime [3] and therefore, deciding whether a functional transducer defines a sequential function is in PTime. If the twinning property holds, one can “determinise” the original transducer into an equivalent sequential transducer. It turns out that many examples of rational functions from the literature which are not sequential are almost sequential, in the sense that they are equal to a finite union of sequential functions. Such functions are called multi-sequential. For instance, the function fswap is multi-sequential, as fswap = fa ∪ fb, where fa is the partial sequential function mapping all words of the form ua to au (and similarly for fb). Some rational functions are not multi-sequential, such as functions that are iterations of non-sequential functions. E.g., the function mapping u1#u2# . . .#un to fswap(u1) . . . fswap(un) for some separator #, is not multi-sequential. Multi-sequential functions have been considered by Choffrut and Schützenberger in [6], where it is shown that multi-sequentiality for functional transducers is a decidable problem. Contribution In this paper, we investigate multi-sequential relations, i.e. relations that are equal to a finite union of sequential functions. Our main result shows that, given a finite transducer, it is decidable in PTime whether the relation it defines is multi-sequential. Our procedure is based on a simple characterisation of multi-sequential relations by means of a structural property, called the weak twinning property (WTP), on finite transducers. We show that a finite transducer defines a multi-sequential relation iff it satisfies the WTP. We define a “determinisation” procedure of finite transducers satisfying the WTP, that decomposes them into finite unions of sequential transducers. Finally, we also investigate the computational properties of multi-sequential relations and show, that, for a natural computational model for word relations, multi-sequential relations correspond to the relations that can be evaluated with constant memory. Related Works As already mentioned, multi-sequential functions were considered in [6]. Our weak twinning property is close to the characterisation of multisequential functions of [6], which is based on analysing families of branching paths in transducers. Thanks to the notion of delay between output words, our property is simpler and can be decided in PTime, for arbitrary (not necessarily functional) transducers. Compared to [6], our decomposition procedure is more constructive. It extends the known determinisation procedure of functional transducers, to multi-sequential relations, and applies directly on arbitrary finite state transducers (in [6], the functional transducers are assumed to be unambiguous, but removing ambiguity is worst-case exponential). Finite-valued rational relations are rational relations such that any input word has at most k images by the relation, for a fixed constant k. Finite-