Algorithmic Properties of Transducers

In this thesis, we consider three fundamental problems of transducers theory. The containment problem asks, given two transducers,whether the relation defined by the first is included into the relation defined by the second. The equivalence problem asks, given two transducers,whether they define the same relation. Finally, the sequential uniformisation problem,corresponding to the synthesis problem in the setting of transducers,asks, given a transducer, whether it is possible to deterministically pick an output correspondingto each input of its domain. These three decision problems are undecidable in general. As a first step, we consider different manners of recovering the decidability of the three problems considered.First, we characterise a family of classes of transducers, called controlled by effective languages, for which the containment and equivalence problems are decidable. Second, we add structural constraints to the problems considered: for instance, instead of only asking that two transducers define the same relation, we require that this relation is defined by both transducers in a similar way. This `similarity' is formalised through the notion of delay,used to measure the difference between the output production of two transducers. This allows us to introduce stronger decidable versions of our three decision problems, which we use to prove the decidability of the original problems in the setting of finite-valued transducers. In the second part, we study extensions of the automaton model,together with the adaptation of the sequential uniformisation problems to these new settings.Weighted automata are automata which,along each transition, output a weight in Z. Then, whereas a transducer preserves all the output mapped to a given input, weighted automata only preserve the maximal weight. In this setting, the sequential uniformisation problem turns into the determinisation problem: given a weighted automaton, is it possible to deterministically pick the maximal output mapped to each input? The decidability of this problem is open.The notion of delay allows us to devise a complete semi-algorithm deciding it. Finally, we consider two-way transducers, that are allowed to move back and forth over the input tape. These transducers enjoy good properties with respect to the sequential uniformisation problem: every transducer admits a sequential two-way uniformiser. We strengthen this result by showing that every transducer admits a reversible two-way uniformiser, i.e. a uniformiser that is both sequential and cosequential (backward sequential).Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe

Multi-Sequential Word Relations

Rational relations are binary relations of finite words that are realised by non-deterministic finite state transducers (NFT). A multi-sequential relation is a rational relation which is equal to a finite union of (graphs) of partial sequential functions, i.e. functions realised by input-deterministic transducers. The particular case of multi-sequential functions was studied by Choffrut and Schützenberger who proved that given a rational function (as a transducer), it is decidable whether it is multi-sequential. Their procedure is based on an effective characterisation of unambiguous transducers that do not define multi-sequential functions, that we call the fork property. In this paper, we show that the fork property also characterises the class of transducers that do not define multi-sequential relations. Moreover, we prove that the fork property can be decided in PTime. This leads to a PTime procedure which, given a transducer, decides whether it defines a multi-sequential relation.SCOPUS: cp.jinfo:eu-repo/semantics/publishe

Aperiodic String Transducers

Regular string-to-string functions enjoy a nice triple characterization through deterministic two-way transducers (2DFT), streaming string transducers (SST) and MSO definable functions. This result has recently been lifted to FO definable functions, with equivalent representations by means of aperiodic 2DFT and aperiodic 1-bounded SST, extending a well-known result on regular languages. In this paper, we give three direct transformations: i) from 1-bounded SST to 2DFT, ii) from 2DFT to copyless SST, and iii) from k-bounded to 1-bounded SST. We give the complexity of each construction and also prove that they preserve the aperiodicity of transducers. As corollaries, we obtain that FO definable string-to-string functions are equivalent to SST whose transition monoid is finite and aperiodic, and to aperiodic copyless SST.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

On Equivalence and Uniformisation Problems for Finite Transducers (ICALP'16)

info:eu-repo/semantics/publishe

On delay and regret determinization of max-plus automata

Decidability of the determinization problem for weighted automata over the semiring (ℤ∪{-∞}, max; +), WA for short, is a long-standing open question. We propose two ways of approaching it by constraining the search space of deterministic WA: k-delay and r-regret. A WA N is k-delay determinizable if there exists a deterministic automaton D that defines the same function as N and for all words α in the language of N, the accepting run of D on α is always at most k-away from a maximal accepting run of N on α. That is, along all prefixes of the same length, the absolute difference between the running sums of weights of the two runs is at most k. A WA N is r-regret determinizable if for all words α in its language, its non-determinism can be resolved on the fly to construct a run of N such that the absolute difference between its value and the value assigned to α by N is at most r. We show that a WA is determinizable if and only if it is k-delay determinizable for some k. Hence deciding the existence of some k is as difficult as the general determinization problem. When k and r are given as input, the k-delay and r-regret determinization problems are shown to be EXPTIME-complete. We also show that determining whether a WA is r-regret determinizable for some r is in EXPTIME.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

On delay and regret determinization of max-plus automata

Decidability of the determinization problem for weighted automata over the semiring (ℤ∪{-∞}, max; +), WA for short, is a long-standing open question. We propose two ways of approaching it by constraining the search space of deterministic WA: k-delay and r-regret. A WA N is k-delay determinizable if there exists a deterministic automaton D that defines the same function as N and for all words α in the language of N, the accepting run of D on α is always at most k-away from a maximal accepting run of N on α. That is, along all prefixes of the same length, the absolute difference between the running sums of weights of the two runs is at most k. A WA N is r-regret determinizable if for all words α in its language, its non-determinism can be resolved on the fly to construct a run of N such that the absolute difference between its value and the value assigned to α by N is at most r. We show that a WA is determinizable if and only if it is k-delay determinizable for some k. Hence deciding the existence of some k is as difficult as the general determinization problem. When k and r are given as input, the k-delay and r-regret determinization problems are shown to be EXPTIME-complete. We also show that determining whether a WA is r-regret determinizable for some r is in EXPTIME.SCOPUS: cp.pinfo:eu-repo/semantics/publishe