Definitions and basic properties

Definition 3.1.1. An insertion transducer is a transducer T = (Q, A, δ, qI) whose

transitions are exactly as follows:

(i) For each stateq ∈Q and for each letter a∈A, there is exactly one state p∈Q

such that qÐÐ→a∶a pis a transition in T.

(ii) For each state q ∈ Q there is a set of letters A(q) ⊆ A such that q ÐÐ→ε∶a q is a transition in T for each a∈A(q).

A transition of type (i) described above can be regarded asT copying a letter of the input word, while a transition of type (ii) can be regarded as T inserting a letter

into the input word. Intuitively, an insertion transducer copies an input wordu and can insert certain letters into u, with the set of insertable letter depending on the state which T is in.

Definition 3.1.2. The relation generated by an insertion transducer T is called an

insertion relation, and is denoted by ≤_T .If an insertion relation is an ordering, then it is called aninsertion ordering.

An insertion relation≤_T is always reflexive sinceT can copy any word, and it is always anti-symmetric since u<_T v implies∣u∣ < ∣v∣. Hence we have the following:

Observation 3.1.3. An insertion relation is an insertion ordering if and only if it is transitive.

Sinceu<_T v implies∣u∣ < ∣v∣,an insertion relation cannot admit any infinite descend- ing chains. Hence we have the following:

Observation 3.1.4. An insertion ordering is a WQO if and only if it admits no infinite anti-chains.

We now introduce some notation which we will use throughout the section.

Notation 3.1.5. Let T = (Q, A, δ, qI) be an insertion transducer, let q ∈Q and let

u ∈A∗. Let p be the unique state such that T has a path from q to p labelled by

(u, u). Then we will writep=q⋅u.

Notation 3.1.6. LetT = (Q, A, δ, qI) be an insertion transducer and let q∈Q. We

letW(q) denote the set of words u such that T has a path from qI toq labelled by

(u, u). That is:

W(q) = {u∈A∗ ∣qI⋅u=q}.

Ifu∈W(q)then we may write A(u) in place ofA(q).

We note the following:

Then the language W(q) is regular, and is accepted by the DFA

A = (Q, A, δ′, qI,{q}) where δ′(p, a) =s whenever pÐÐ→a∶a s is a transition in T.

As stated earlier, an insertion transducer T inserts letters into a word, with the set of letters available for insertion depending on the state which T is in. With this in mind, we can characterise the insertion relation ≤_T generated by T as follows:

Observation 3.1.8. Let T = (Q, A, δ, qI) be an insertion transducer, let u, v ∈ A∗ and write u=a1. . . an. Then u≤T v if and only if there are words

v0 ∈A(ε)∗, v1 ∈A(a1)∗, v2∈A(a1a2)∗, . . . , vn∈A(u)∗ such that v =v0a1v1a2v2. . . anvn.

Examples of insertion relations

We demonstrate that some word orderings from the literature are in fact insertion relations, and then present some new examples.

Example 3.1.9. The subword ordering onA∗ is an insertion relation. Indeed, it is generated by the insertion transducer

Tw(A) = ({qI}, A, δ, qI)

whose copy transitions are given byqI a∶a

ÐÐ→qI for all a∈A and whereA(qI) =A. We

showTw(a, b)below. qI start a∶a b∶b ε∶a, b

Example 3.1.10. Aichinger et al. [1] introduced what they call the embedding

write u=a1. . . an. Thenu≤E v if there are words

v1∈ {a1}∗, v2∈ {a1, a2}∗, . . . , vn∈ {a1, . . . , an}∗

such that v =a1v1. . . anvn. Intuitively we have u ≤E v if v can be obtained from u

by inserting letters after their first occurrence inu. For instance we have ab≤E aba,

but ab/≤E bab. In their paper, Aichinger et al. showed that the embedding ordering

is a WQO. We show that the embedding ordering is an insertion relation. Indeed, it is generated by the insertion transducer

TE(A) = (P(A), A, δ,∅)

whose copy transitions are given by S ÐÐ→a∶a S∪ {a} for each S ⊆A and each a ∈A, and whereA(S) =S for eachS⊆A.For instance, the transducer TE(a, b)is given by

∅ start a b a, b a∶a b∶b a∶a ε∶a b∶b ε∶b b∶b a∶a a∶a b∶b ε∶a, b

We now present some new examples of insertion relations.

Example 3.1.11. The transducer T shown below is an insertion transducer over the alphabet {a, b}. We have A(qI) = A(t) = ∅ and A(s) = {b}. As an instance of

qI start a∶a s t b∶b ε∶b a∶a b∶b a∶a b∶b

Example 3.1.12. The transducer T shown below is an insertion transducer over the alphabet {a, b}. We have A(qI) = A(t) = {b} and A(s) = {a, b}. As an instance

of the insertion relation ≤_T generated byT, we have a≤_T bbab.

qI start s t a∶a b∶b ε∶b a∶a b∶b ε∶a, b a∶a b∶b ε∶b

Example 3.1.13. The transducer T shown below is an insertion transducer over the alphabet {a, b}. We have A(qI) = {b} and A(p) = {a}. As an instance of the

insertion relation≤_T generated byT, we have b≤_T ba.

qI start p b∶b a∶a ε∶b a∶a b∶b ε∶a