LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Transducers with Programmable Input by DNA Self-assembly 221
potentially the whole finite-state nano-machine can be reusable. The paper is
organized as follows. Section 2 describes the relationship between Wang tiles
and transducers. It describes the basic prototiles that can be used to simulate
a transducer. The composition of transducers obtained through Wang tiles is
described in Section 3. Here we also show how primitive recursive functions can
be obtained through composition of transducers. The general recursion (computable
functions) simulated with Wang tiles is presented in Section 4. DNA
implementation of transducers with TX molecules is described in Section 5 and
the use of the PX-JX2 devices for setting up the input sequence is described in
Section 5.2. We end with few concluding remarks.
2 Finite State Machines with Output: Basic Model
In this section we show the general idea of performing a computation by a
finite state machine using tiles. The actual assembly by DNA will be described
in Section 5. We briefly recall the definition of a transducer. This notion is well
known in automata theory and an introduction to transducers (Mealy machines)
can be found in [9].
A finite state machine with output or atransduceris T =(Σ,Σ ′ ,Q,δ,s0,F)
where Σ and Σ ′ are finite alphabets, Q is a finite set of states, δ is the transition
function, s0 ∈ Q is the initial state, and F ⊆ Q is the set of final or terminal
states. The alphabet Σ is the input alphabet and the alphabet Σ ′ is the output
alphabet. We denote with Σ∗ the set of all words over the alphabet Σ. This
includes the word with “no symbols”, the empty word denoted with λ. Fora
word w = a1 ···ak where ai ∈ Σ, the length of w denoted with |w| is k. Forthe
empty word λ, wehave|λ| =0.
The transition operation δ is a subset of Q × Σ × Σ ′ × Q. The elements of δ
are denoted with (q, a, a ′ ,q ′ )or(q, a) δ
↦→ (a ′ ,q ′ ) meaning that when T is in state
q and scans input symbol a, thenTchanges into state q ′ and gives output symbol
a ′ .Inthecaseofdeterministic transducers, δ is a function δ : Q × Σ → Σ ′ × Q,
i.e., at a given state reading a given input symbol, there is a unique output state
and an output symbol. Usually the states of the transducer are presented as
vertices of a graph and the transitions defined with δ are presented as directed
edges with input/output symbols as labels. If there is no edge from a vertex
q in the graph that has input label a, we assume that there is an additional
“junk” state ¯q where all such transitions end. This state is usually omitted from
the graph since it is not essential for the computation. The transducer is said
to recognize a string (or a word) w over alphabet Σ if there is a path in the
graph from the initial state s0 to a terminal state in F with input label w. The
set of words recognized by a transducer T is denoted with L(T) and is called a
language recognized by T. It is well known that finite state transducers recognize
the class of regular languages.
We concentrate on deterministic transducers. In this case the transition function
δ maps the input word w ∈ Σ∗ to a word w ′ ∈ (Σ ′ ) ∗ . So the transducer T
can be considered to be a function from L(T) to(Σ ′ ) ∗ , i.e., T : L(T) → (Σ ′ ) ∗ .