Linear Bounded Composition of Tree-Walking Tree Transducers: Linear Size Increase and Complexity

Compositions of tree-walking tree transducers form a hierarchy with respect to the number of transducers in the composition. As main technical result it is proved that any such composition can be realized as a linear bounded composition, which means that the sizes of the intermediate results can be chosen to be at most linear in the size of the output tree. This has consequences for the expressiveness and complexity of the translations in the hierarchy. First, if the computed translation is a function of linear size increase, i.e., the size of the output tree is at most linear in the size of the input tree, then it can be realized by just one, deterministic, tree-walking tree transducer. For compositions of deterministic transducers it is decidable whether or not the translation is of linear size increase. Second, every composition of deterministic transducers can be computed in deterministic linear time on a RAM and in deterministic linear space on a Turing machine, measured in the sum of the sizes of the input and output tree. Similarly, every composition of nondeterministic transducers can be computed in simultaneous polynomial time and linear space on a nondeterministic Turing machine. Their output tree languages are deterministic context-sensitive, i.e., can be recognized in deterministic linear space on a Turing machine. The membership problem for compositions of nondeterministic translations is nondeterministic polynomial time and deterministic linear space. The membership problem for the composition of a nondeterministic and a deterministic tree-walking tree translation (for a nondeterministic IO macro tree translation) is log-space reducible to a context-free language, whereas the membership problem for the composition of a deterministic and a nondeterministic tree-walking tree translation (for a nondeterministic OI macro tree translation) is possibly NP-complete

Comparison-Free Polyregular Functions.

This paper introduces a new automata-theoretic class of string-to-string functions with polynomialgrowth. Several equivalent definitions are provided: a machine model which is a restricted variant ofpebble transducers, and a few inductive definitions that close the class of regular functions undercertain operations. Our motivation for studying this class comes from another characterization,which we merely mention here but prove elsewhere, based on a λ-calculus with a linear type system.As their name suggests, these comparison-free polyregular functions form a subclass of polyregularfunctions; we prove that the inclusion is strict. We also show that they are incomparable withHDT0L transductions, closed under usual function composition – but not under a certain “map”combinator – and satisfy a comparison-free version of the pebble minimization theorem.On the broader topic of polynomial growth transductions, we also consider the recently introducedlayered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that afunction can be obtained by composing such transducers together if and only if it is polyregular,and that k-layered SSTs (or k-marble transducers) are closed under “map” and equivalent to acorresponding notion of (k + 1)-layered HDT0L systems

Register Transducers Are Marble Transducers

Deterministic two-way transducers define the class of regular functions from words to words. Alur and Cerný introduced an equivalent model of transducers with registers called copyless streaming string transducers. In this paper, we drop the “copyless” restriction on these machines and show that they are equivalent to two-way transducers enhanced with the ability to drop marks, named “marbles”, on the input. We relate the maximal number of marbles used with the amount of register copies performed by the streaming string transducer. Finally, we show that the class membership problems associated with these models are decidable. Our results can be interpreted in terms of program optimization for simple recursive and iterative programs.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

Revisiting the growth of polyregular functions: output languages, weighted automata and unary inputs

Polyregular functions are the class of string-to-string functions definable by pebble transducers (an extension of finite automata) or equivalently by MSO interpretations (a logical formalism). Their output length is bounded by a polynomial in the input length: a function computed by a $k$-pebble transducer or by a $k$-dimensional MSO interpretation has growth rate $O(n^k)$. Boja\'nczyk has recently shown that the converse holds for MSO interpretations, but not for pebble transducers. We give significantly simplified proofs of those two results, extending the former to first-order interpretations by reduction to an elementary property of $\mathbb{N}$-weighted automata. For any $k$, we also prove the stronger statement that there is some quadratic polyregular function whose output language differs from that of any $k$-fold composition of macro tree transducers (and which therefore cannot be computed by any $k$-pebble transducer). In the special case of unary input alphabets, we show that $k$ pebbles suffice to compute polyregular functions of growth $O(n^k)$. This is obtained as a corollary of a basis of simple word sequences whose ultimately periodic combinations generate all polyregular functions with unary input. Finally, we study polyregular and polyblind functions between unary alphabets (i.e. integer sequences), as well as their first-order subclasses.Comment: 27 pages, not submitted ye

Pebble transducers with unary output

Boja\'nczyk recently initiated an intensive study of deterministic pebble transducers, which are two-way automata that can drop marks (named "pebbles") on their input word, and produce an output word. They describe functions from words to words. Two natural restrictions of this definition have been investigated: marble transducers by Dou\'eneau-Tabot et al., and comparison-free pebble transducers (that we rename here "blind transducers") by Nguy\^en et al. Here, we study the decidability of membership problems between the classes of functions computed by pebble, marble and blind transducers that produce a unary output. First, we show that pebble and marble transducers have the same expressive power when the outputs are unary (which is false over non-unary outputs). Then, we characterize 1-pebble transducers with unary output that describe a function computable by a blind transducer, and show that the membership problem is decidable. These results can be interpreted in terms of automated simplification of programs.Comment: 39 page

On the growth rate of polyregular functions

We consider polyregular functions, which are certain string-to-string functions that have polynomial output size. We prove that a polyregular function has output size $\mathcal O(n^k)$ if and only if it can be defined by an MSO interpretation of dimension $k$, i.e. a string-to-string transformation where every output position is interpreted, using monadic second-order logic MSO, in some $k$-tuple of input positions. We also show that this characterization does not extend to pebble transducers, another model for describing polyregular functions: we show that for every $k \in \{1,2,\ldots\}$ there is a polyregular function of quadratic output size which needs at least $k$ pebbles to be computed

Pebble Minimization of Polyregular Functions

We show that a polyregular word-to-word function is regular if and only if its output size is at most linear in its input size. Moreover a polyregular function can be realized by: a transducer with two pebbles if and only if its output has quadratic size in its input, a transducer with three pebbles if and only if its output has cubic size in its input, etc. Moreover the characterization is decidable and, given a polyregular function, one can compute a transducer realizing it with the minimal number of pebbles. We apply the result to mso interpretations from words to words. We show that mso interpretations of dimension k exactly coincide with k-pebble transductions.Comment: The main result of the article is false. Counterexamples and more can be found here: arXiv:2301.0923

String-to-String Interpretations With Polynomial-Size Output

String-to-string MSO interpretations are like Courcelle\u27s MSO transductions, except that a single output position can be represented using a tuple of input positions instead of just a single input position. In particular, the output length is polynomial in the input length, as opposed to MSO transductions, which have output of linear length. We show that string-to-string MSO interpretations are exactly the polyregular functions. The latter class has various characterisations, one of which is that it consists of the string-to-string functions recognised by pebble transducers. Our main result implies the surprising fact that string-to-string MSO interpretations are closed under composition