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Conjunctive grammars over a unary alphabet Artur Je z, Alexander Okhotin September 7, 2007 Artur Je z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 1 / 15 Conjunctive and Boolean grammars Context-free

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Conjunctive grammars over a unary alphabet

Artur Je˙ z, Alexander Okhotin September 7, 2007

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 1 / 15

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Conjunctive and Boolean grammars

Context-free grammars: Rules of the form A → α “If w is generated by α, then w is generated by A”.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 2 / 15

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Conjunctive and Boolean grammars

Context-free grammars: Rules of the form A → α “If w is generated by α, then w is generated by A”. Multiple rules for A: disjunction.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 2 / 15

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Conjunctive and Boolean grammars

Context-free grammars: Rules of the form A → α “If w is generated by α, then w is generated by A”. Multiple rules for A: disjunction. Conjunctive grammars (Okhotin, 2000) Rules of the form A → α1& . . . &αm “If w is generated by each αi, then w is generated by A”.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 2 / 15

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Conjunctive and Boolean grammars

Context-free grammars: Rules of the form A → α “If w is generated by α, then w is generated by A”. Multiple rules for A: disjunction. Conjunctive grammars (Okhotin, 2000) Rules of the form A → α1& . . . &αm “If w is generated by each αi, then w is generated by A”. Boolean grammars (Okhotin, 2003) Rules of the form A → α1& . . . &αm&¬β1& . . . &¬βn “If w is generated by each αi and by none of βj, then w is generated by A”.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 2 / 15

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Definition of conjunctive grammars

Quadruple G = (Σ, N, P, S), where S ∈ N and rules in P are A → α1& . . . &αm with A ∈ N, αi ∈ (Σ ∪ N)∗

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 3 / 15

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Definition of conjunctive grammars

Quadruple G = (Σ, N, P, S), where S ∈ N and rules in P are A → α1& . . . &αm with A ∈ N, αi ∈ (Σ ∪ N)∗ Semantics by term rewriting: ϕ(A) = ⇒ ϕ(α1& . . . &αm)

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 3 / 15

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Definition of conjunctive grammars

Quadruple G = (Σ, N, P, S), where S ∈ N and rules in P are A → α1& . . . &αm with A ∈ N, αi ∈ (Σ ∪ N)∗ Semantics by term rewriting: ϕ(A) = ⇒ ϕ(α1& . . . &αm) ϕ(w& . . . &w) = ⇒ ϕ(w)

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 3 / 15

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Definition of conjunctive grammars

Quadruple G = (Σ, N, P, S), where S ∈ N and rules in P are A → α1& . . . &αm with A ∈ N, αi ∈ (Σ ∪ N)∗ Semantics by term rewriting: ϕ(A) = ⇒ ϕ(α1& . . . &αm) ϕ(w& . . . &w) = ⇒ ϕ(w)

◮ LG(A) = {w | A =

⇒∗ w}

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 3 / 15

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Definition of conjunctive grammars

Quadruple G = (Σ, N, P, S), where S ∈ N and rules in P are A → α1& . . . &αm with A ∈ N, αi ∈ (Σ ∪ N)∗ Semantics by term rewriting: ϕ(A) = ⇒ ϕ(α1& . . . &αm) ϕ(w& . . . &w) = ⇒ ϕ(w)

◮ LG(A) = {w | A =

⇒∗ w}

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 3 / 15

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Definition of conjunctive grammars

Quadruple G = (Σ, N, P, S), where S ∈ N and rules in P are A → α1& . . . &αm with A ∈ N, αi ∈ (Σ ∪ N)∗ Semantics by term rewriting: ϕ(A) = ⇒ ϕ(α1& . . . &αm) ϕ(w& . . . &w) = ⇒ ϕ(w)

◮ LG(A) = {w | A =

⇒∗ w}

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

◮ LG(A) is the A-component of the least solution. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 3 / 15

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Properties of conjunctive and Boolean grammars

Generate {anbncn | n 0}, {wcw | w ∈ {a, b}∗}, etc., etc.

Example

S → AE&BC A → aA | ε B → aBb | ε C → cC | ε E → bEc | ε

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 4 / 15

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Properties of conjunctive and Boolean grammars

Generate {anbncn | n 0}, {wcw | w ∈ {a, b}∗}, etc., etc.

Example

S → AE&BC A → aA | ε B → aBb | ε C → cC | ε E → bEc | ε LS = (LA · LE) ∩ (LB · LC) LA = {a} · LA ∪ {ε} LB = {a} · LB · {b} ∪ {ε} LC = {c} · LC ∪ {ε} LE = {b} · LE · {c} ∪ {ε}

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 4 / 15

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Properties of conjunctive and Boolean grammars

Generate {anbncn | n 0}, {wcw | w ∈ {a, b}∗}, etc., etc.

Example

S → AE&BC A → aA | ε B → aBb | ε C → cC | ε E → bEc | ε LS = (LA · LE) ∩ (LB · LC) LA = {a} · LA ∪ {ε} LB = {a} · LB · {b} ∪ {ε} LC = {c} · LC ∪ {ε} LE = {b} · LE · {c} ∪ {ε} {anbncn : n ∈ N} a∗ {anbn : n ∈ N} c∗ {bncn : n ∈ N}

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 4 / 15

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Properties of conjunctive and Boolean grammars

Generate {anbncn | n 0}, {wcw | w ∈ {a, b}∗}, etc., etc.

Example

S → AE&BC A → aA | ε B → aBb | ε C → cC | ε E → bEc | ε LS = (LA · LE) ∩ (LB · LC) LA = {a} · LA ∪ {ε} LB = {a} · LB · {b} ∪ {ε} LC = {c} · LC ∪ {ε} LE = {b} · LE · {c} ∪ {ε} {anbncn : n ∈ N} a∗ {anbn : n ∈ N} c∗ {bncn : n ∈ N} Languages contained in DTIME(n3) ∩ DSPACE(n).

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 4 / 15

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Properties of conjunctive and Boolean grammars

Generate {anbncn | n 0}, {wcw | w ∈ {a, b}∗}, etc., etc.

Example

S → AE&BC A → aA | ε B → aBb | ε C → cC | ε E → bEc | ε LS = (LA · LE) ∩ (LB · LC) LA = {a} · LA ∪ {ε} LB = {a} · LB · {b} ∪ {ε} LC = {c} · LC ∪ {ε} LE = {b} · LE · {c} ∪ {ε} {anbncn : n ∈ N} a∗ {anbn : n ∈ N} c∗ {bncn : n ∈ N} Languages contained in DTIME(n3) ∩ DSPACE(n). Linear case: equivalent to one-way real-time CA.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 4 / 15

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Properties of conjunctive and Boolean grammars

Generate {anbncn | n 0}, {wcw | w ∈ {a, b}∗}, etc., etc.

Example

S → AE&BC A → aA | ε B → aBb | ε C → cC | ε E → bEc | ε LS = (LA · LE) ∩ (LB · LC) LA = {a} · LA ∪ {ε} LB = {a} · LB · {b} ∪ {ε} LC = {c} · LC ∪ {ε} LE = {b} · LE · {c} ∪ {ε} {anbncn : n ∈ N} a∗ {anbn : n ∈ N} c∗ {bncn : n ∈ N} Languages contained in DTIME(n3) ∩ DSPACE(n). Linear case: equivalent to one-way real-time CA. Practical parsing methods: recursive descent, generalized LR.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 4 / 15

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The case of a unary alphabet

Σ = {a}.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

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The case of a unary alphabet

Σ = {a}. an ← → number n

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

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The case of a unary alphabet

Σ = {a}. an ← → number n Language ← → set of numbers

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

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The case of a unary alphabet

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X ⊞ Y = {x + y | x ∈ X, y ∈ Y}

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

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The case of a unary alphabet

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X ⊞ Y = {x + y | x ∈ X, y ∈ Y} Regular ← → ultimately periodic

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

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The case of a unary alphabet

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X ⊞ Y = {x + y | x ∈ X, y ∈ Y} Regular ← → ultimately periodic

Theorem (Bar-Hillel et al., 1961)

Every context-free language over {a} is regular.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

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The case of a unary alphabet

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X ⊞ Y = {x + y | x ∈ X, y ∈ Y} Regular ← → ultimately periodic

Theorem (Bar-Hillel et al., 1961)

Every context-free language over {a} is regular.

Problem

The power of conjunctive grammars over {a}?

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

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The case of a unary alphabet

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X ⊞ Y = {x + y | x ∈ X, y ∈ Y} Regular ← → ultimately periodic

Theorem (Bar-Hillel et al., 1961)

Every context-free language over {a} is regular.

Problem

The power of conjunctive grammars over {a}? Can generate {a4n | n 0} (Je˙ z, 2007).

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

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Using positional notation

Our approach: using base-k notation.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

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Using positional notation

Our approach: using base-k notation. an ← → k-ary notation of n

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

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Using positional notation

Our approach: using base-k notation. an ← → k-ary notation of n Σk = {0, 1, . . . , k − 1}, strings in Σ∗

k \ 0Σ∗ k.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

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Using positional notation

Our approach: using base-k notation. an ← → k-ary notation of n Σk = {0, 1, . . . , k − 1}, strings in Σ∗

k \ 0Σ∗ k.

Isomorphism Σ∗

k \ 0Σ∗ k ↔ a∗.

fk(k-ary notation of n) = an

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

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Using positional notation

Our approach: using base-k notation. an ← → k-ary notation of n Σk = {0, 1, . . . , k − 1}, strings in Σ∗

k \ 0Σ∗ k.

Isomorphism Σ∗

k \ 0Σ∗ k ↔ a∗.

fk(k-ary notation of n) = an Extends to languages: fk(L) = {fk(w) | w ∈ L}

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

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SLIDE 31

Using positional notation

Our approach: using base-k notation. an ← → k-ary notation of n Σk = {0, 1, . . . , k − 1}, strings in Σ∗

k \ 0Σ∗ k.

Isomorphism Σ∗

k \ 0Σ∗ k ↔ a∗.

fk(k-ary notation of n) = an Extends to languages: fk(L) = {fk(w) | w ∈ L}

Example

f4(10∗) = {a4n | n 0}

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

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SLIDE 32

Using positional notation

Our approach: using base-k notation. an ← → k-ary notation of n Σk = {0, 1, . . . , k − 1}, strings in Σ∗

k \ 0Σ∗ k.

Isomorphism Σ∗

k \ 0Σ∗ k ↔ a∗.

fk(k-ary notation of n) = an Extends to languages: fk(L) = {fk(w) | w ∈ L}

Example

f4(10∗) = {a4n | n 0} Equations over Σ∗

k with ∩, ∪, ⊞

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

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SLIDE 33

Using positional notation

Our approach: using base-k notation. an ← → k-ary notation of n Σk = {0, 1, . . . , k − 1}, strings in Σ∗

k \ 0Σ∗ k.

Isomorphism Σ∗

k \ 0Σ∗ k ↔ a∗.

fk(k-ary notation of n) = an Extends to languages: fk(L) = {fk(w) | w ∈ L}

Example

f4(10∗) = {a4n | n 0} Equations over Σ∗

k with ∩, ∪, ⊞

Isomorphism between language equations.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

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Nonperiodic unary conjunctive languages

Example (Je˙ z, DLT 2007)

N a∗

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

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Nonperiodic unary conjunctive languages

Example (Je˙ z, DLT 2007)

N a∗ X1 = (X2⊞X2 ∩ X1⊞X3) ∪ {1} X2 = (X12⊞X2 ∩ X1⊞X1) ∪ {2} X3 = (X12⊞X12 ∩ X1⊞X2) ∪ {3} X12 = X3⊞X3 ∩ X1⊞X2

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

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SLIDE 36

Nonperiodic unary conjunctive languages

Example (Je˙ z, DLT 2007)

N a∗ X1 = (X2⊞X2 ∩ X1⊞X3) ∪ {1} X2 = (X12⊞X2 ∩ X1⊞X1) ∪ {2} X3 = (X12⊞X12 ∩ X1⊞X2) ∪ {3} X12 = X3⊞X3 ∩ X1⊞X2 Y1 = (Y2Y2 ∩ Y1Y3) ∪ a Y2 = (Y12Y2 ∩ Y1Y1) ∪ aa Y3 = (Y12Y12 ∩ Y1Y2) ∪ aaa Y12 = (Y3Y3 ∩ Y1Y2)

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

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Nonperiodic unary conjunctive languages

Example (Je˙ z, DLT 2007)

N a∗ X1 = (X2⊞X2 ∩ X1⊞X3) ∪ {1} X2 = (X12⊞X2 ∩ X1⊞X1) ∪ {2} X3 = (X12⊞X12 ∩ X1⊞X2) ∪ {3} X12 = X3⊞X3 ∩ X1⊞X2 Y1 = (Y2Y2 ∩ Y1Y3) ∪ a Y2 = (Y12Y2 ∩ Y1Y1) ∪ aa Y3 = (Y12Y12 ∩ Y1Y2) ∪ aaa Y12 = (Y3Y3 ∩ Y1Y2) base 4: (10∗, 20∗, 30∗, 120∗)

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

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Nonperiodic unary conjunctive languages

Example (Je˙ z, DLT 2007)

N a∗ X1 = (X2⊞X2 ∩ X1⊞X3) ∪ {1} X2 = (X12⊞X2 ∩ X1⊞X1) ∪ {2} X3 = (X12⊞X12 ∩ X1⊞X2) ∪ {3} X12 = X3⊞X3 ∩ X1⊞X2 Y1 = (Y2Y2 ∩ Y1Y3) ∪ a Y2 = (Y12Y2 ∩ Y1Y1) ∪ aa Y3 = (Y12Y12 ∩ Y1Y2) ∪ aaa Y12 = (Y3Y3 ∩ Y1Y2) base 4: (10∗, 20∗, 30∗, 120∗)

  • {a4n}, {a2·4n}, {a3·4n}, {a6·4n}
  • Artur Je˙

z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

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SLIDE 39

Nonperiodic unary conjunctive languages

Example (Je˙ z, DLT 2007)

N a∗ X1 = (X2⊞X2 ∩ X1⊞X3) ∪ {1} X2 = (X12⊞X2 ∩ X1⊞X1) ∪ {2} X3 = (X12⊞X12 ∩ X1⊞X2) ∪ {3} X12 = X3⊞X3 ∩ X1⊞X2 Y1 = (Y2Y2 ∩ Y1Y3) ∪ a Y2 = (Y12Y2 ∩ Y1Y1) ∪ aa Y3 = (Y12Y12 ∩ Y1Y2) ∪ aaa Y12 = (Y3Y3 ∩ Y1Y2) base 4: (10∗, 20∗, 30∗, 120∗)

  • {a4n}, {a2·4n}, {a3·4n}, {a6·4n}
  • X2 ⊞ X2 = 20∗ ⊞ 20∗ = 10+ ∪ 20∗20∗

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

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SLIDE 40

Nonperiodic unary conjunctive languages

Example (Je˙ z, DLT 2007)

N a∗ X1 = (X2⊞X2 ∩ X1⊞X3) ∪ {1} X2 = (X12⊞X2 ∩ X1⊞X1) ∪ {2} X3 = (X12⊞X12 ∩ X1⊞X2) ∪ {3} X12 = X3⊞X3 ∩ X1⊞X2 Y1 = (Y2Y2 ∩ Y1Y3) ∪ a Y2 = (Y12Y2 ∩ Y1Y1) ∪ aa Y3 = (Y12Y12 ∩ Y1Y2) ∪ aaa Y12 = (Y3Y3 ∩ Y1Y2) base 4: (10∗, 20∗, 30∗, 120∗)

  • {a4n}, {a2·4n}, {a3·4n}, {a6·4n}
  • X2 ⊞ X2 = 20∗ ⊞ 20∗ = 10+ ∪ 20∗20∗

X1 ⊞ X3 = 10∗ ⊞ 30∗ = 10+ ∪ 10∗30∗ ∪ 30∗10∗,

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

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SLIDE 41

Nonperiodic unary conjunctive languages

Example (Je˙ z, DLT 2007)

N a∗ X1 = (X2⊞X2 ∩ X1⊞X3) ∪ {1} X2 = (X12⊞X2 ∩ X1⊞X1) ∪ {2} X3 = (X12⊞X12 ∩ X1⊞X2) ∪ {3} X12 = X3⊞X3 ∩ X1⊞X2 Y1 = (Y2Y2 ∩ Y1Y3) ∪ a Y2 = (Y12Y2 ∩ Y1Y1) ∪ aa Y3 = (Y12Y12 ∩ Y1Y2) ∪ aaa Y12 = (Y3Y3 ∩ Y1Y2) base 4: (10∗, 20∗, 30∗, 120∗)

  • {a4n}, {a2·4n}, {a3·4n}, {a6·4n}
  • X2 ⊞ X2 = 20∗ ⊞ 20∗ = 10+ ∪ 20∗20∗

X1 ⊞ X3 = 10∗ ⊞ 30∗ = 10+ ∪ 10∗30∗ ∪ 30∗10∗, (X2 ⊞ X2) ∩ (X1 ⊞ X3) = 10+.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

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More unary conjunctive languages

Theorem (Je˙ z, DLT 2007)

For any regular language R ⊆ Σ∗

k \ 0Σ∗ k,

there exists a conjunctive grammar for fk(R).

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 8 / 15

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SLIDE 43

More unary conjunctive languages

Theorem (Je˙ z, DLT 2007)

For any regular language R ⊆ Σ∗

k \ 0Σ∗ k,

there exists a conjunctive grammar for fk(R).

Note

fk(R) has linear or exponential growth.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 8 / 15

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SLIDE 44

More unary conjunctive languages

Theorem (Je˙ z, DLT 2007)

For any regular language R ⊆ Σ∗

k \ 0Σ∗ k,

there exists a conjunctive grammar for fk(R).

Note

fk(R) has linear or exponential growth.

* * *

Theorem

For every trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

there exists a conjunctive grammar for fk(L(M)).

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 8 / 15

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SLIDE 45

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where:

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 46

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 47

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 48

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 49

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

slide-50
SLIDE 50

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

slide-51
SLIDE 51

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

slide-52
SLIDE 52

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 53

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function; F ⊂ Q: final states.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 54

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function; F ⊂ Q: final states. Equivalent to linear conjunctive grammars.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 55

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function; F ⊂ Q: final states. Equivalent to linear conjunctive grammars. Closed under ∪, ∩, ∼, not closed under concatenation.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 56

Trellis automata

(one-way real-time cellular automata)

Definition

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function; F ⊂ Q: final states. Equivalent to linear conjunctive grammars. Closed under ∪, ∩, ∼, not closed under concatenation. Can recognize {wcw}, {anbncn}, {anb2n}, VALC.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

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SLIDE 57

Main lemma

Lemma

For every trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

there exists a system with ∪, ∩, ⊞ and regular constants, with least solution {1w10∗ | w ⊞ 1 ∈ L(M)}, . . . ,

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 10 / 15

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SLIDE 58

Main lemma

Lemma

For every trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

there exists a system with ∪, ∩, ⊞ and regular constants, with least solution {1w10∗ | w ⊞ 1 ∈ L(M)}, . . . , 1w10∗ represents w.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 10 / 15

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SLIDE 59

Main lemma

Lemma

For every trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

there exists a system with ∪, ∩, ⊞ and regular constants, with least solution {1w10∗ | w ⊞ 1 ∈ L(M)}, . . . , 1w10∗ represents w. Regular constants, can be changed to singleton.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 10 / 15

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SLIDE 60

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

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SLIDE 61

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}. Actually, Xq = {1w10∗ | w ⊞ 1 ∈ LM(q)}

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

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SLIDE 62

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}. Actually, Xq = {1w10∗ | w ⊞ 1 ∈ LM(q)} aub ∈ LM(q)

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

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SLIDE 63

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}. Actually, Xq = {1w10∗ | w ⊞ 1 ∈ LM(q)} aub ∈ LM(q) ⇔

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

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SLIDE 64

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}. Actually, Xq = {1w10∗ | w ⊞ 1 ∈ LM(q)} aub ∈ LM(q) ⇔ ∃q ′, q ′′ : δ(q ′, q ′′) = q,

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

slide-65
SLIDE 65

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}. Actually, Xq = {1w10∗ | w ⊞ 1 ∈ LM(q)} aub ∈ LM(q) ⇔ ∃q ′, q ′′ : δ(q ′, q ′′) = q, au ∈ LM(q ′),

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

slide-66
SLIDE 66

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}. Actually, Xq = {1w10∗ | w ⊞ 1 ∈ LM(q)} aub ∈ LM(q) ⇔ ∃q ′, q ′′ : δ(q ′, q ′′) = q, au ∈ LM(q ′), ub ∈ LM(q ′′).

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

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SLIDE 67

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}. Actually, Xq = {1w10∗ | w ⊞ 1 ∈ LM(q)} aub ∈ LM(q) ⇔ ∃q ′, q ′′ : δ(q ′, q ′′) = q, au ∈ LM(q ′), ub ∈ LM(q ′′). Let 1au10∗ ⊆ Xq ′, 1ub10∗ ⊆ Xq ′′. Xq =

  • q ′,q ′′:δ(q ′,q ′′)=q

a,b∈Σk

ρb(Xq ′)∩λa(Xq ′′)

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

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SLIDE 68

The construction

Set of variables {Xq | q ∈ Q}, representing {LM(q) | q ∈ Q}. Actually, Xq = {1w10∗ | w ⊞ 1 ∈ LM(q)} aub ∈ LM(q) ⇔ ∃q ′, q ′′ : δ(q ′, q ′′) = q, au ∈ LM(q ′), ub ∈ LM(q ′′). Let 1au10∗ ⊆ Xq ′, 1ub10∗ ⊆ Xq ′′. Xq =

  • q ′,q ′′:δ(q ′,q ′′)=q

a,b∈Σk

ρb(Xq ′)∩λa(Xq ′′) λa(1w10k) = 1aw10k ρb(1w10k) = 1wb10k−1

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

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SLIDE 69

How does ρ look like

The equations for ρj: ρj(X) =

  • j ′
  • (X ∩ 1Σ∗

kj ′10∗⊞10∗) ∩ 1Σ∗ kj ′20∗

⊞(j − 2)10∗

  • ∩ 1Σ∗

kj10∗

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

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SLIDE 70

How does ρ look like

The equations for ρj: ρj(X) =

  • j ′
  • (X ∩ 1Σ∗

kj ′10∗⊞10∗) ∩ 1Σ∗ kj ′20∗

⊞(j − 2)10∗

  • ∩ 1Σ∗

kj10∗

Word Operation New word

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

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SLIDE 71

How does ρ look like

The equations for ρj: ρj(X) =

  • j ′
  • (X ∩ 1Σ∗

kj ′10∗⊞10∗) ∩ 1Σ∗ kj ′20∗

⊞(j − 2)10∗

  • ∩ 1Σ∗

kj10∗

Word Operation New word 1w10ℓ

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

slide-72
SLIDE 72

How does ρ look like

The equations for ρj: ρj(X) =

  • j ′
  • (X ∩ 1Σ∗

kj ′10∗⊞10∗) ∩ 1Σ∗ kj ′20∗

⊞(j − 2)10∗

  • ∩ 1Σ∗

kj10∗

Word Operation New word 1w10ℓ = 1w ′j ′10ℓ

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

slide-73
SLIDE 73

How does ρ look like

The equations for ρj: ρj(X) =

  • j ′
  • (X ∩ 1Σ∗

kj ′10∗⊞10∗) ∩ 1Σ∗ kj ′20∗

⊞(j − 2)10∗

  • ∩ 1Σ∗

kj10∗

Word Operation New word 1w10ℓ = 1w ′j ′10ℓ ∩1Σ∗

kj ′10∗

1w ′j ′10ℓ

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

slide-74
SLIDE 74

How does ρ look like

The equations for ρj: ρj(X) =

  • j ′
  • (X ∩ 1Σ∗

kj ′10∗⊞10∗) ∩ 1Σ∗ kj ′20∗

⊞(j − 2)10∗

  • ∩ 1Σ∗

kj10∗

Word Operation New word 1w10ℓ = 1w ′j ′10ℓ ∩1Σ∗

kj ′10∗

1w ′j ′10ℓ 1w ′j ′10ℓ ⊞10∗ ∩ 1Σ∗

kj ′20∗

1w ′j ′20ℓ

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

slide-75
SLIDE 75

How does ρ look like

The equations for ρj: ρj(X) =

  • j ′
  • (X ∩ 1Σ∗

kj ′10∗⊞10∗) ∩ 1Σ∗ kj ′20∗

⊞(j − 2)10∗

  • ∩ 1Σ∗

kj10∗

Word Operation New word 1w10ℓ = 1w ′j ′10ℓ ∩1Σ∗

kj ′10∗

1w ′j ′10ℓ 1w ′j ′10ℓ ⊞10∗ ∩ 1Σ∗

kj ′20∗

1w ′j ′20ℓ 1w ′j ′20ℓ ⊞(j − 2)10∗ ∩ 1Σ∗

kj10∗

1w ′j ′j10ℓ−1

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

slide-76
SLIDE 76

How does ρ look like

The equations for ρj: ρj(X) =

  • j ′
  • (X ∩ 1Σ∗

kj ′10∗⊞10∗) ∩ 1Σ∗ kj ′20∗

⊞(j − 2)10∗

  • ∩ 1Σ∗

kj10∗

Word Operation New word 1w10ℓ = 1w ′j ′10ℓ ∩1Σ∗

kj ′10∗

1w ′j ′10ℓ 1w ′j ′10ℓ ⊞10∗ ∩ 1Σ∗

kj ′20∗

1w ′j ′20ℓ 1w ′j ′20ℓ ⊞(j − 2)10∗ ∩ 1Σ∗

kj10∗

1w ′j ′j10ℓ−1 1w ′j ′j10ℓ−1

  • j ′

1wj10ℓ−1

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

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SLIDE 77

Undecidable properties

Proposition

“Given conjunctive grammar G over {a}, determine whether L(G) = ∅”

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

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SLIDE 78

Undecidable properties

Proposition

“Given conjunctive grammar G over {a}, determine whether L(G) = ∅” — undecidable.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

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SLIDE 79

Undecidable properties

Proposition

“Given conjunctive grammar G over {a}, determine whether L(G) = ∅” — undecidable. Turing machine T recognizes X;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

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SLIDE 80

Undecidable properties

Proposition

“Given conjunctive grammar G over {a}, determine whether L(G) = ∅” — undecidable. Turing machine T recognizes X; Trellis automaton M for VALC(T);

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

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SLIDE 81

Undecidable properties

Proposition

“Given conjunctive grammar G over {a}, determine whether L(G) = ∅” — undecidable. Turing machine T recognizes X; Trellis automaton M for VALC(T); Conjunctive grammar G for fk(VALC(T));

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

slide-82
SLIDE 82

Undecidable properties

Proposition

“Given conjunctive grammar G over {a}, determine whether L(G) = ∅” — undecidable. Turing machine T recognizes X; Trellis automaton M for VALC(T); Conjunctive grammar G for fk(VALC(T)); L(G) = ∅ ⇔ L(T) = ∅

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

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SLIDE 83

Undecidable properties

Proposition

“Given conjunctive grammar G over {a}, determine whether L(G) = ∅” — undecidable. Turing machine T recognizes X; Trellis automaton M for VALC(T); Conjunctive grammar G for fk(VALC(T)); L(G) = ∅ ⇔ L(T) = ∅

Theorem

For every fixed conjunctive L0 ⊆ a∗, the problem “Given conjunctive grammar G over {a}, determine whether L(G) = L0” — undecidable.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

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SLIDE 84

Unbounded growth

Theorem

∀ r. e. set X = {gX(n) | n 1}, with gXր ∃ conjunctive grammar G with L(G) = {agG(n) | n 1}:

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

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SLIDE 85

Unbounded growth

Theorem

∀ r. e. set X = {gX(n) | n 1}, with gXր ∃ conjunctive grammar G with L(G) = {agG(n) | n 1}: gG(n) > gX(n) (∀n 1)

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

slide-86
SLIDE 86

Unbounded growth

Theorem

∀ r. e. set X = {gX(n) | n 1}, with gXր ∃ conjunctive grammar G with L(G) = {agG(n) | n 1}: gG(n) > gX(n) (∀n 1) Turing machine T recognizes X;

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

slide-87
SLIDE 87

Unbounded growth

Theorem

∀ r. e. set X = {gX(n) | n 1}, with gXր ∃ conjunctive grammar G with L(G) = {agG(n) | n 1}: gG(n) > gX(n) (∀n 1) Turing machine T recognizes X; Trellis automaton M for VALC(T);

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

slide-88
SLIDE 88

Unbounded growth

Theorem

∀ r. e. set X = {gX(n) | n 1}, with gXր ∃ conjunctive grammar G with L(G) = {agG(n) | n 1}: gG(n) > gX(n) (∀n 1) Turing machine T recognizes X; Trellis automaton M for VALC(T); Conjunctive grammar G for fk(VALC(T));

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

slide-89
SLIDE 89

Unbounded growth

Theorem

∀ r. e. set X = {gX(n) | n 1}, with gXր ∃ conjunctive grammar G with L(G) = {agG(n) | n 1}: gG(n) > gX(n) (∀n 1) Turing machine T recognizes X; Trellis automaton M for VALC(T); Conjunctive grammar G for fk(VALC(T)); gG(n) > gX(n).

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

slide-90
SLIDE 90

Unbounded growth

Theorem

∀ r. e. set X = {gX(n) | n 1}, with gXր ∃ conjunctive grammar G with L(G) = {agG(n) | n 1}: gG(n) > gX(n) (∀n 1) Turing machine T recognizes X; Trellis automaton M for VALC(T); Conjunctive grammar G for fk(VALC(T)); gG(n) > gX(n).

Remark

Polynomial growth can be achieved.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

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SLIDE 91

Conclusion

Conjunctive grammar: CFG with intersection.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-92
SLIDE 92

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-93
SLIDE 93

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-94
SLIDE 94

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-95
SLIDE 95

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. ◮ Positional notation. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-96
SLIDE 96

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. ◮ Positional notation.

Unary notation of VALC(T) is conjunctive.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-97
SLIDE 97

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. ◮ Positional notation.

Unary notation of VALC(T) is conjunctive.

◮ Undecidability. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-98
SLIDE 98

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. ◮ Positional notation.

Unary notation of VALC(T) is conjunctive.

◮ Undecidability. ◮ Growth not recursively bounded. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-99
SLIDE 99

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. ◮ Positional notation.

Unary notation of VALC(T) is conjunctive.

◮ Undecidability. ◮ Growth not recursively bounded.

Research problems.

Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-100
SLIDE 100

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. ◮ Positional notation.

Unary notation of VALC(T) is conjunctive.

◮ Undecidability. ◮ Growth not recursively bounded.

Research problems.

◮ Closure under complementation. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-101
SLIDE 101

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. ◮ Positional notation.

Unary notation of VALC(T) is conjunctive.

◮ Undecidability. ◮ Growth not recursively bounded.

Research problems.

◮ Closure under complementation. ◮ Separation from DSPACE(n). Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

slide-102
SLIDE 102

Conclusion

Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages.

◮ {afk(w) | w ∈ L(M)} for any TA M. ◮ Equations over sets of integers. ◮ Positional notation.

Unary notation of VALC(T) is conjunctive.

◮ Undecidability. ◮ Growth not recursively bounded.

Research problems.

◮ Closure under complementation. ◮ Separation from DSPACE(n). ◮ Automaton representation. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15