Ismaël Robin Jecker

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 fork property. If the fork property is not 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.

On equivalence and uniformisation problems for finite transducers

Transductions are binary relations of finite words. For rational transductions, i.e., transductions defined by finite transducers, the inclusion, equivalence and sequential uniformisation problems are known to be undecidable. In this paper, we investigate stronger variants of inclusion, equivalence and sequential uniformisation, based on a general notion of transducer resynchronisation, and show their decidability. We also investigate the classes of finite-valued rational transductions and deterministic rational transductions, which are known to have a decidable equivalence problem. We show that sequential uniformisation is also decidable for them.

On reversible transducers

Deterministic two-way transducers define the robust class of regular functions which is, among other good properties, closed under composition. However, the best known algorithms for composing two-way transducers cause a double exponential blow-up in the size of the inputs. In this paper, we introduce a class of transducers for which the composition has polynomial complexity. It is the class of reversible transducers, for which the computation steps can be reversed deterministically. While in the one-way setting this class is not very expressive, we prove that any two-way transducer can be made reversible through a single exponential blow-up. As a consequence, we prove that the composition of two-way transducers can be done with a single exponential blow-up in the number of states. A uniformization of a relation is a function with the same domain and which is included in the original relation. Our main result actually states that we can uniformize any non-deterministic two-way transducer by a reversible transducer with a single exponential blow-up, improving the known result by de Souza which has a quadruple exponential complexity. As a side result, our construction also gives a quadratic transformation from copyless streaming string transducers to two-way transducers, improving the exponential previous bound.

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.

Degree of Sequentiality of Weighted Automata

Weighted automata WA are an important formalism to describe quantitative properties. Obtaining equivalent deterministic machines is a longstanding research problem. In this paper we consider WA with a set semantics, meaning that the semantics is given by the set of weights of accepting runs. We focus on multi-sequential WA that are defined as finite unions of sequential WA. The problem we address is to minimize the size of this union. We call this minimum the degree of sequentiality of the relation realized by the WA. For a given positive integer k, we provide multiple characterizations of relations realized by a union of k sequential WA over an infinitary finitely generated group: a Lipschitz-like machine independent property, a pattern on the automaton a new twinning property and a subclass of cost register automata. When possible, we effectively translate a WA into an equivalent union of k sequential WA. We also provide a decision procedure for our twinning property for commutative computable groups thus allowing to compute the degree of sequentiality. Last, we show that these results also hold for word transducers and that the associated decision problem is

Multi-sequential Word Relations

\textsc {Pspace}

The Complexity of Transducer Synthesis from Multi-Sequential Specifications.

-complete.

Beyond Admissibility: Dominance Between Chains of Strategies.

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.