A deterministic iterated uniform finite-state transducer (for short, iufst) operates the same length-preserving transduction on several left-to-right sweeps. The first sweep occurs on the input string, while any other sweep processes the output of the previous one. We focus on constant sweep bounded iufsts. We study their descriptional power vs. deterministic finite automata, and the state cost of implementing language operations. Then, we focus on non-constant sweep bounded iufsts, showing a nonregular language hierarchy depending on sweep complexity

A. Malcher

B. Palano

C. Mereghetti

M. Kutrib

English

AIR Universita degli studi di Milano

Iterated Uniform Finite-State Transducers?
Martin Kutrib1, Andreas Malcher1, Carlo Mereghetti2, and Beatrice Palano3
1 Institut fu¨r Informatik, Universita¨t Giessen, Arndtstr. 2, 35392 Giessen, Germany
{andreas.malcher,kutrib}@informatik.uni-giessen.de
2 Dipartimento di Fisica “Aldo Pontremoli”, Universita` degli Studi di Milano
via Celoria 16, 20133 Milano, Italy, carlo.mereghetti@unimi.it
3 Dipartimento di Informatica “G. degli Antoni”, Universita` degli Studi di Milano
via Celoria 18, 20133 Milano, Italy, palano@unimi.it
Abstract. A deterministic iterated uniform finite-state transducer (for
short, iufst) operates the same length-preserving transduction on sev-
eral left-to-right sweeps. The first sweep occurs on the input string, while
any other sweep processes the output of the previous one. We focus on
constant sweep bounded iufsts. We study their descriptional power vs.
deterministic finite automata, and the state cost of implementing lan-
guage operations. Then, we focus on non-constant sweep bounded iufsts,
showing a nonregular language hierarchy depending on sweep complexity.
Keywords: Iterated transducers; State complexity; Sweep complexity.
1 Introduction
Finite-state transducers are finite automata with an output and they have been
studied at least since 1950s. A typical application of finite-state transducers is,
for example, the lexical analysis of a computer program or an XML document.
Here, the correct formatting of the input is verified, comments are removed, the
correct spelling of the commands is checked, and the sequence of input symbols
is translated into a list of tokens. The output produced is subsequently pro-
cessed by a pushdown automaton that realizes the syntactic analysis. Another
example is the use of cascading finite-state transducers. Here, one has a finite
number of transducers T1, T2, . . . , Tn, where the output of Ti is the input for
the next transducer Ti+1. Such cascades of finite-state transducers have been
used, for example, in [8] to extract information from natural language texts.
On the other hand, the Krohn-Rhodes decomposition theorem shows that every
regular language can be represented as the cascade of several finite-state trans-
ducers, each of which having a “simple” algebraic structure [10, 11]. Cascades
of deterministic pushdown transducers are investigated in [7] and it is shown
that a proper infinite hierarchy in between the deterministic context-free and
deterministic context-sensitive languages exists with respect to the number of
? Supported by Gruppo Nazionale per il Calcolo Scientifico (GNCS-INdAM). Ex-
tended version presented at the 21st Int. Conf. DCFS, July 17–19, 2019, Kosˇice,
Slovakia, [13]. Copyright c© 2019 for this paper by its authors. Use permitted under
Creative Commons License Attribution 4.0 International (CC BY 4.0)
2 M. Kutrib, A. Malcher, C. Mereghetti, B. Palano
transducers applied. All the examples considered so far have in common that the
subsequently applied transducers are, at least in principal, different transducers.
Another point of view is taken in [6, 15], where subsequently applied identical
transducers are studied. Such iterated finite-state transducers are considered as
language generating devices starting with some symbol in the initial state of the
transducer, iteratively applying in multiple sweeps the transducer to the output
produced so far, and eventually halting in an accepting state of the transducer
after a last sweep. It is an essential feature that the underlying finite-state trans-
ducer is not length-preserving. Several restrictions on finite-state transducers are
studied in [17] with respect to the question of whether the (arbitrary) iteration
of the restricted transducers is still computationally universal.
Here, we consider iterated finite-state transducers as language acceptors. Yet,
to have a simple device, the underlying transducer is considered to be a Mealy
machine [16], i.e., a deterministic length-preserving device where each input sym-
bol is translated according to its transition function into an output symbol. More
precisely, an iterated uniform finite-state transducer (iufst) works in several
sweeps on a tape which is initially the input concatenated with a right end-
marker. In every sweep the finite-state transducer starts in its initial state at the
first tape cell, is applied to the tape, and prints its output on the tape. The input
is accepted or rejected, if the transducer halts in an accepting or rejecting state.
We start our investigations of such devices having a fixed number k ≥ 1
of sweeps. Since in this case the language accepted by a k-iufst is always a
regular language, it is of interest to compare such devices with deterministic finite
automata (dfa) by investigating their descriptional complexity and the state
cost of implementing language operations. General information on descriptional
complexity can be found in the survey [12] and many results on the operational
state complexity are surveyed in [9]. Next, we consider the case when the number
of sweeps is not bounded by a fixed finite number. The resulting iufsts can thus
be considered as restricted variants of one-tape Turing machines that iteratively
sweep from left to right, starting at the first tape cell always in their initial
state. We prove that such devices can accept non-regular languages as soon
as the number of sweeps is at least the logarithm of the length of the input.
Moreover, we show a hierarchy depending on the number of sweeps.
2 Definitions and Preliminaries
An iterated uniform finite-state transducer is basically a deterministic finite-
state transducer which processes the input in multiple passes (also sweeps). In
the first pass it reads the input word followed by an endmarker and emits an
output word. In the following passes it reads the output word of the previous pass
and emits a new output word. The number of passes taken, the sweep complexity,
is given as a function of the length of the input. Since here we are interested
in weak preprocessing devices, we will consider length-preserving deterministic
finite-state transducers, also known as Mealy machines.
Formally, we define an iterated uniform finite-state transducer (iufst) as a
system T = 〈Q,Σ,∆, q0,C, δ, F+, F−〉, where Q is the set of internal states, Σ
Iterated Uniform Finite-State Transducers 3
is the set of input symbols, ∆ is the set of output symbols, q0 is the initial state,
C ∈ ∆\Σ is the endmarker, F+ ⊆ Q is the set of accepting states, F− ⊆ (Q\F+)
is the set of rejecting states, and δ : Q×(Σ∪∆)→ Q×∆ is the transition function,
which is total on (Q \ (F+ ∪F−))× (Σ ∪∆) and where the endmarker is emitted
only if it is read, i.e., no transitions δ(p, x) = (q,C) with x 6= C exist. The iufst
halts when the transition function is undefined (which may happen only for
states from F+ ∪ F−) or if it enters an accepting or rejecting state at the end of
a sweep. Since the transducer is applied in multiple passes, i.e., in any but the
initial pass it operates on the output of the previous pass, the transition function
depends on input symbols from Σ∪∆. Let v ∈ ∆∗ be the output produced by T
on input w ∈ (Σ ∪∆)∗ in a complete sweep. Then we denote v by T (w). A word
w ∈ Σ∗ is accepted by an iufst T if there is an r ≥ 1 such that w1 = T (wC),
wi+1 = T (wi), 1 ≤ i < r, and the transducer T halts on wr in an accepting state.
That is, the initial input is a word over the input alphabet Σ followed by the
endmarker, and the output computed after r − 1 iterations drives T in a final
sweep where it halts in an accepting state. Similarly, an input w ∈ Σ∗ is rejected
by T if there is an r ≥ 1 such that T halts in the final sweep on wr−1 in a
rejecting state. Note that the output of the last sweep is not used. The language
accepted by T is L(T ) = {w ∈ Σ∗ | w is accepted by T }.
Let s : N → N be a function. If any word of length n is accepted or rejected
in at most s(n) sweeps, the iufst is said to be of sweep complexity s(n). In this
case, we use the notation s(n)-iufst.
3 Iterated Transductions vs. DFAs
Iterated transduction may lead to dramatically decrease the number of states
for regular acceptance with respect to dfas. To see this, for any k ≥ 1, let us
consider the language Ek = {a, b}∗b{a, b}k−1.
Theorem 1. Any dfa for the language Ek needs at least 2k states. On the other
hand, there exists a k-iufst T accepting Ek with 3 states only.
However, some regular languages turn out to be so size-consuming that even
iterated transduction cannot help. In fact, let p be any prime number. We define
the language Lp = { am·p | m ≥ 0 }.
Theorem 2. Let k ≥ 1. Then p states are necessary and sufficient for a k-iufst
to accept Lp.
4 Descriptional Complexity
We approach in a more general way the study of the descriptional power of
k-iufst vs. dfas by providing general simulations between the two models. First,
we consider the cost of turning a k-iufst into an equivalent dfa:
Lemma 3. Let k > 0 be an integer. Every n-state k-iufst can be converted to
an equivalent dfa with at most nk states.
4 M. Kutrib, A. Malcher, C. Mereghetti, B. Palano
The result presented in Lemma 3 turns out to be optimal. To this aim, for
any n, k > 0, let us define the unary language Ln,k = { ac·nk | c ≥ 0 }.
Lemma 4. The language Ln,k is accepted by an n-state k-iufst, while any
equivalent dfa needs at least nk states.
Let us now focus on the opposite simulation, that is, dfas by k-iufst.
Lemma 5. For n ≥ 3, every n-state dfa can be converted into an equivalent
n-state k-iufst, with k ≥ 1.
Lemma 5 holds for at least three states. For dfas with less than three states,
we can choose either to maintain the same number of states and perform one
sweep only, or to add a new state and perform k sweeps. One may expect that the
size of a k-iufst may always be decreased by increasing the number of sweeps.
However by Theorem 2, the construction provided by Lemma 5 is optimal.
4.1 States versus Sweeps
The possibility of trading states for input sweeps and vice versa has been inves-
tigated in the literature for several models of computations (see, e.g., [5, 14]).
Theorem 2 shows there are languages for which additional sweeps do not help
to decrease the number of states at all. On the other hand, the next theorem
provides a gradual reduction of the number of sweeps and the necessary states
for the language Ek = {a, b}∗b{a, b}k−1 of Theorem 1.
Theorem 6. Let ` ≥ 1, k = 2` be integers. Then the useful sweep range for
the language Ek is from 1 to 2`. Moreover, for any i ∈ {0, 1, . . . , ` − 1} there
is a 2i-iufst accepting Ek with 22
`−i
states, and for i = ` there is a 2i-iufst
accepting Ek with 3 states.
5 The State Cost of Language Operations on k-IUFSTs
As naturally done for many models of computation accepting regular languages
(see, e.g., [1–4]), we now analyze the state complexity of language operations on
k-iufsts. To this aim, we asssume that their transition functions are always de-
fined, so that acceptance/rejection takes place on the endmarker only, at the jth
sweep, for some 1 ≤ j ≤ k. If not, transition functions can be easily completed by
adding at most two states which store the accept or reject outcome obtained in
the middle of the input, while reaching the endmarker. In what follows, L and LR
denote, respectively, the complement and reversal of a language L.
Theorem 7. Let m,n, k ≥ 1 be integers, T1 be an m-state k-iufst and T2
be an n-state k-iufst. Then m · n states are sufficient for a k-iufst to ac-
cept L(T1) ∩ L(T2).
The cost in Theorem 7 is optimal, as proved in the following
Iterated Uniform Finite-State Transducers 5
Theorem 8. Let k ≥ 1 be an integer. There exist infinitely many integers
m,n > 1 such that an m-state k-iufst T1 and an n-state k-iufst T2 can be
built, for which m ·n states are necessary to accept L(T1)∩L(T2) on a k-iufst.
Complementing languages does not increase the size of k-iufst:
Theorem 9. Let k, n ≥ 1 be integers and T be an n-state k-iufst. Then n states
are sufficient and necessary in the worst case for a k-iufst to accept L(T ).
The above two theorems enable us to establish the optimal cost of imple-
menting intersection:
Theorem 10. Let m,n, k ≥ 1 be integers, T1 be an m-state k-iufst and T2 be
an n-state k-iufst. Then m · n states are sufficient and necessary in the worst
case for a k-iufst to accept L(T1) ∪ L(T2).
Concerning the cost of performing reversal on k-iufst, we get
Theorem 11. Let k, n ≥ 1 be integers and T be an n-state k-iufst. Then 2nk
states are sufficient and at least 2
nk
k2k states are necessary in the worst case for
a k-iufst to accept L(T )R.
6 Hierarchy of Non-Constant Sweep Complexities
Since any constant sweep bounded iufst accepts a regular language, the first
natural question is the following: “How many sweeps are necessary to cross the
edge to non-regularity?” The answer is contained in the following theorem (by lg,
we denote the logarithm to base 2):
Proposition 12. Let s(n) ∈ o(lg n). The language accepted by any s(n)-iufst
is regular.
In fact, the gap between constant and ‘useful’ non-constant sweep complex-
ities ends at a logarithmic level. The witness language given by the following
lemma is even unary and non-context-free.
Lemma 13. The non-context-free unary language Luexpo = { a2k | k ≥ 0 } is
accepted by a six-state s(n)-iufst with s(n) ∈ O(lg n).
Next, we extend the sweep complexity hierarchy beyond the logarithm. To
this end, we consider sweep complexities of order O(
√
n), and define the language
Lcpsq = {w$w#am−1#am−2 · · · #a1# | m ≥ 0, w ∈ {a, b}m−1 } for which we get
Theorem 14. The language Lcpsq is accepted by an s(n)-iufst with s(n) ∈
O(
√
n), while it cannot be accepted by any s(n)-iufst with s(n) ∈ o(√n).
Finally, the sweep complexity hierarchy can be further extended beyond the
square root up to sweep complexities of order O(n). In fact, we define the lan-
guage Lcpc = {w$mw | w ∈ {a, b}∗,m ≥ 1 } for which we prove
Theorem 15. The language Lcpc is accepted by an s(n)-iufst with s(n) ∈
O(n), while it cannot be accepted by any s(n)-iufst with s(n) ∈ o(n).
Acknowledgements. The authors wish to thank the anonymous referees.
6 M. Kutrib, A. Malcher, C. Mereghetti, B. Palano
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