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
Parallel insertion and deletion schemaLanguage equations
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
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
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)?
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
p-schema-based insertionp-schema-based deletionProperties of p-schema-based operations
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 .
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
p-schema-based insertionp-schema-based deletionProperties of p-schema-based operations
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∗
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
p-schema-based insertionp-schema-based deletionProperties of p-schema-based operations
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
p-schema-based insertionp-schema-based deletionProperties of p-schema-based operations
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}.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
p-schema-based insertionp-schema-based deletionProperties of p-schema-based operations
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
p-schema-based insertionp-schema-based deletionProperties of p-schema-based operations
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),
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
p-schema-based insertionp-schema-based deletionProperties of p-schema-based operations
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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].
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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).
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
References
Solving L1 ffF X = L3Solving L1 F X = L3
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?
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
Parallel insertion and deletion schemaLanguage equations
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.
Lila Kari, Shinnosuke Seki Schema for parallel insertion and deletion
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