Lila Kari, Shinnosuke Seki UWORCS 2010lkari/pdfs/Schema for... · Lila Kari, Shinnosuke Seki...

Parallel insertion and deletion schemaLanguage equations

References

Schema for parallel insertion and deletion

Lila Kari, Shinnosuke Seki

Department of Computer Science, University of Western Ontario

UWORCS 2010April 12th, 2010

Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion


References

Notation

Σ alphabet

Σ∗ the set of all words over Σ

u, v ,w words

L, L1, L2, L3 given languages

R ,R1,R2,R3 given regular languages

X ,Y unknown variables

+ sum of sets

Lc complement of L, i.e., Lc = Σ∗ \ L

2L power set of L



References

p-schema-based insertionp-schema-based deletionProperties of p-schema-based operations

Parallel operations

Example ([Kari91])

Parallel insertion ⇐ is defined as follows: for a wordu = a1a2 · · · an(ai ∈ Σ) and a language L,

u ⇐ L = La1La2L · · · Lan−1LanL.

Question

How to control parallel insertion (where to insert L)?



References


p-schema-based insertion

Let F = {(u1, u2, . . . , un−1, un) | n ≥ 1, u1, . . . , un ∈ Σ∗}.We call F ⊆ F a p-schema because it can specify how toparallel-insert a language L into a word u.

Definition

For a p-schema F , insertion �F based on F is defined as:

u �F L =⋃

n≥0,u=u1···un,

(u1,...,un)∈F

u1Lu2L · · · un−1Lun.

Example

Let F = {(λ, ab, c), (a, bc), (abc)}. Then

abc �F L = LabLc ∪ aLbc ∪ abc .



References


Various instances of p-schema-based insertion I

Syntactic and Semantic p-schema-based insertions

Operation p-schema F

catenation L1L2 Σ∗ × {λ}insertion {xL2y | xy ∈ L1} Σ∗ × Σ∗

parallel insertion L1 ⇐ L2⋃

n≥0{λ} × Σ × · · · × Σ︸︷︷︸

n times

×{λ}

inserting exactly 2 L’s Σ∗ × Σ∗ × Σ∗

(x , y)-contextual insertion Σ∗x × yΣ∗



References


Various instances of p-schema-based insertion II

Example

Parallel insertion next to b ∈ Σ is defined as follows: foru = a1a2 · · · an(ai ∈ Σ) and a language L,

u ⇐a L = a1L1a2L2 · · · an−1Ln−1anLn,

where

Li =

{

L if ai = b

{λ} otherwise.

The p-schema F = {(u1, . . . , un) | n ≥ 1, u1, . . . , un ∈ (Σ \ {b})∗b}implements parallel insertion next to b.



References


p-schema-based deletion

Definition

For a p-schema G , deletion �G based on G is defined as:

w �G L = {u1 · · · un | n ≥ 1, x1, . . . , xn−1 ∈ L,

(u1, . . . , un) ∈ G ,w = u1x1u2x2 · · · un−1xn−1un}.

Example

Let F = {(λ, a, c), (ba, bc), (babbc)} and L = b∗. Then

babbc �F L = {ac , babc , babbc}.



References


p-schema-based insertion/deletion over languages

Let us extend p-schema-based operations into operations betweenlanguages as follows:

L1 �F L2 =⋃

u∈L1

u �F L2

L1 �G L2 =⋃

w∈L1

w �G L2.



References


Distributivity I

By this extension, the following distributivity hold.

(L1 + L′1) �F L2 = (L1 �F L2) + (L′

1 �F L2)

L1 �F+G L2 = (L1 �F L2) + (L1 �G L2).

These distributivity hold also for p-schema-based deletion.In contrast, we can say no more than

L1 �F (L2 + L′2) ⊇ (L1 �F L2) + (L1 �F L′

2),

L1 �F (L2 + L′2) ⊇ (L1 �F L2) + (L1 �F L′

2),



References


Distributivity II

Example

Let Aeven = {a2m | m ≥ 0}, Aodd = {a2m+1 | m ≥ 0}, andF = Σ∗ + Σ∗ × Σ∗ × Σ∗.For L1 = L2 = Aeven,

L1 �F Aeven = L1 �F Aodd = L2

However a3 ∈ L1 �F (Aeven + Aodd), which is not in L2.



References

Solving L1 ffF X = L3Solving L1 F X = L3

Objectives

Question

Is it decidable whether language equations of the following forms:

1 X �F L2 = L3 and X �F L2 = L3

2 L1 �X L2 = L3 and L1 �X L2 = L3

3 L1 �F X = L3 and L1 �F X = L3

have a solution or not.



References


Existence of maximum solutions

If X �F L2 = L3 has two solutions L1, L′1, then

(L1 + L′1) �F L2 = (L1 �F L2) + (L′

1 �F L2)

= L3 + L3 = L3.

This first equality holds due to the distributivity. This holds alsofor

1 X �F L2 = L3,

2 L1 �X L2 = L3,

3 L1 �X L2 = L3.

This means that these equations have the maximum solutions (thesum of all of their solutions).Methods to solve such language equations have beenwell-established by Kari [Kari94].



References


Different approach to solve language equations I

On the contrary, L1 �F X = L3 or L1 �F X = L3 may not havethe maximum solution.Let us propose another approach to solving these equations basedon the notion of syntactic congruence.

Definition

For a language L, the syntactic congruence ≡L is an equivalencerelation defined as: for u, v ∈ Σ∗,

u ≡L vdef⇐⇒ ∀x , y ∈ Σ∗, xuy ∈ L ⇐⇒ xvy ∈ L

Definition

For a word w ∈ Σ∗, a set [w ]≡L= {u ∈ Σ∗ | u ≡L w} is called an

equivalence class with w as its representative.



References


Different approach to solve language equations II

Σ∗/ ≡L: the set of all equivalence classes w.r.t. ≡L (quotientset)

index of ≡L: the size of Σ∗/ ≡L.

By taking a representative from every class, we can constructa complete system of representatives of Σ∗/ ≡L.

Theorem ([Rabin and Scott, 1959])

The index of ≡L is finite iff L is regular.

Theorem

If L is regular, then so are equivalence classes in Σ∗/ ≡L.



References


Solving L1 �F X = L3 I

Lemma

Let L1, L3 ⊆ Σ∗. Then for any w ∈ Σ∗ and L2 ⊆ Σ∗,

(L1 �F ({w}+L2))∩Lc3 6= ∅ ⇐⇒ (L1 �F ([w ]≡Lc

3+L2))∩Lc

3 6= ∅.

Assume that u = u1u2u3u4 ∈ L1, (u1, u2, u3, u4) ∈ F ,w1,w2 ∈ [w ]≡L

, and v ∈ L2. Then

u1w1u2vu3w2u4 ∈ Lc3 ⇐⇒ u1wu2vu3w2u4 ∈ Lc

3

⇐⇒ u1wu2vu3wu4 ∈ Lc3.

For a language L, we say that a solution of L1 �F X = L3 issyntactic w.r.t. L if it is a union of equivalence classes in Σ∗/ ≡L.



References


Solving L1 �F X = L3 II

Proposition

For languages L1, L3, L1 �F X = L3 has a solution iff it has a

syntactic solution w.r.t. L3.

To decide whether L1 �F X = L3 has a solution, therefore, itsuffices to check whether it has a syntactic solution w.r.t. L3.Recall that if L3 is regular, then

there exist at most finite numbers of syntactic solutions, (theindex of ≡L3 is finite)

such syntactic solutions are regular, and

solely determined by L3



References


Solving L1 �F X = L3 III

Proposition

For regular languages R1,R3 and a regular p-schema F , it is

decidable whether R1 �F X = R3 has a solution.

Note that all maximal solutions of L1 �F X = L3 are syntacticw.r.t. L3.

Theorem

For regular languages R1,R3 and a regular p-schema F , the set of

all maximal solutions of R1 �F X = R3 is computable.



References


Solving X �F Y = L3 and L1 �X Y = L3

Remember that syntactic solutions of L1 �F Y = L3 are solelydetermined by L3.

Theorem

For a regular language R3 and p-schema F , it is decidable whether

X �F Y = R3 has a solution.

Proof.

N.B. |Σ∗/ ≡R3 | is finite. So for each candidate Rc of syntacticsolutions, let us check whether X �F Rc = R3 has a solution.

Theorem

For regular languages R1,R3, it is decidable whether

R1 �X Y = R3 has a solution.



References


Solving L1 �F X = L3

Lemma

Let L1 ⊆ Σ∗. For any word w ∈ Σ∗ and a language L2 ⊆ Σ∗,

L1 �F ({w} + L2) = L1 �F ([w ]≡L1+ L2).

In other words, a representative w can do whatever elements of[w ]≡L1

can do for �F . Let R(L1) be a complete system of

representatives of Σ∗/ ≡L1 . Then 2R(L1) characterizes the wholeset of solutions of L1 �F X = L3 as shown below.



References


Syntactic and representative solutions

Definition

A solution of L1 �F X = L3 is syntactic w.r.t. L1 if it is a unionof equivalence classes in Σ∗/ ≡L1 .

Definition

A solution of L1 �F X = L3 is representative w.r.t. R(L1) if it isa subset of R(L1).



References


Maximal solutions of L1 �F X = L3

Proposition

For languages L1, L3, the equation L1 �F X = L3 has a solution

iff it has a syntactic solution w.r.t. L1.

Proposition

For a regular language R1, a DCFL L3, and a regular p-schema F ,

it is decidable whether R1 �F X = L3 has a solution or not.

Maximal solutions of L1 �F X = L3 are syntactic w.r.t. L1.

Theorem

For a regular language R1, a DCFL L3, and a regular p-schema F ,

we can compute the set of all maximal solutions of R1 �F X = L3.



References


Solving X �F Y = L3 and L1 �X Y = L3

Recall that syntactic solutions of L1 �F Y = L3 is determined byL1 (not L3).

Theorem

For regular languages R1,R3, it is decidable whether

R1 �X Y = R3 has a solution.

Open problem

Is it decidable whether X �F Y = R3 for a regular language R3

and p-schema F?



References

References I

[Kari91] L. Kari,On insertion and deletion in formal languages,Ph.D. Thesis, University of Turku, 1991.

[Kari94] L. Kari,On language equations with invertible operations,Theoretical Computer Science, 132 (1994) 129-150.

[Rabin and Scott, 1959] M. Rabin and D. Scott,Finite automata and their decision problems,em IBM Journal of Research and Development, 3 (1959)114-125.


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