An in-between "implicit" and "explicit" complexity: Automata

Implicit Computational Complexity makes two aspects implicit, by manipulating programming languages rather than models of com-putation, and by internalizing the bounds rather than using external measure. We survey how automata theory contributed to complexity with a machine-dependant with implicit bounds model

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

Streamability of nested word transductions

We consider the problem of evaluating in streaming (i.e., in a single left-to-right pass) a nested word transduction with a limited amount of memory. A transduction T is said to be height bounded memory (HBM) if it can be evaluated with a memory that depends only on the size of T and on the height of the input word. We show that it is decidable in coNPTime for a nested word transduction defined by a visibly pushdown transducer (VPT), if it is HBM. In this case, the required amount of memory may depend exponentially on the height of the word. We exhibit a sufficient, decidable condition for a VPT to be evaluated with a memory that depends quadratically on the height of the word. This condition defines a class of transductions that strictly contains all determinizable VPTs

Two-Way Visibly Pushdown Automata and Transducers

Automata-logic connections are pillars of the theory of regular languages. Such connections are harder to obtain for transducers, but important results have been obtained recently for word-to-word transformations, showing that the three following models are equivalent: deterministic two-way transducers, monadic second-order (MSO) transducers, and deterministic one-way automata equipped with a finite number of registers. Nested words are words with a nesting structure, allowing to model unranked trees as their depth-first-search linearisations. In this paper, we consider transformations from nested words to words, allowing in particular to produce unranked trees if output words have a nesting structure. The model of visibly pushdown transducers allows to describe such transformations, and we propose a simple deterministic extension of this model with two-way moves that has the following properties: i) it is a simple computational model, that naturally has a good evaluation complexity; ii) it is expressive: it subsumes nested word-to-word MSO transducers, and the exact expressiveness of MSO transducers is recovered using a simple syntactic restriction; iii) it has good algorithmic/closure properties: the model is closed under composition with a unambiguous one-way letter-to-letter transducer which gives closure under regular look-around, and has a decidable equivalence problem

From Two-Way To One-Way Finite State Transducers

info:eu-repo/semantics/publishe

Two-wayness: Automata and Transducers

This PhD is about two natural extensions of Finite Automata (FA): the 2-way fa (2FA) and the 2-way transducers (2T). It is well known that 2FA s are computably equivalent to FAs, even in their nondeterministic (2nfa) variant. However, in the field of descriptional complexity, some questions remain. Raised by Sakoda and Sipser in 1978, the question of the cost of the simulation of 2NFA by 2DFA (the deterministic variant of 2FA) is still open. In this manuscript, we give an answer in a restricted case in which the nondeterministic choices of the simulated 2NFA may occur at the boundaries of the input tape only (2ONFA). We show that every 2ONFA can be simulated by a 2DFA of subexponential (but superpolynomial) size. Under the assumptions L=NL, this cost is reduced to the polynomial level. Moreover, we prove that the complementation and the simulation by a halting 2ONFA is polynomial. We also consider the anologous simulations for alternating devices. Providing a one-way write-only output tape to FAs leads to the notion of transducer. Contrary to the case of finite automata which are acceptor, 2-way transducers strictly extends the computational power of 1-way one, even in the case where both the input and output alphabets are unary. Though 1-way transducers enjoy nice properties and characterizations (algebraic, logical, etc. . . ), 2-way variants are less known, especially the nondeterministic case. In this area, this manuscript gives a new contribution: an algebraic characterization of the relations accepted by two-way transducers when both the input and output alphabets are unary. Actually, it can be reformulated as follows: each unary two-way transducer is equivalent to a sweeping (and even rotating) transducer. We also show that the assumptions made on the size of the alphabets are required, that is, sweeping transducers weakens the 2-way transducers whenever at least one of the alphabet is non-unary. On the path, we discuss on the computational power of some algebraic operations on word relations, introduced in the aim of describing the behavior of 2-way transducers or, more generally, of 2-way weighted automata. In particular, the mirror operation, consisting in reversing the input word in order to describe a right to left scan, draws our attention. Finally, we study another kind of operations, more adapted for binary word relations: the composition. We consider the transitive closure of relations. When the relation belongs to some very restricted sub-family of rational relations, we are able to compute its transitive closure and we set its complexity. This quickly becomes uncomputable when higher classes are considered