SLIDE 1
Behavioural differential equations
CALCO ’11 (Winter/Bonsangue/Rutten): Context-free languages,
- Coalgebraically. . .
. . . behavioural differential equations / Brzozowski derivatives Example:
- (x)
= 1 xa = xy xb =
- (y)
On Language Equations and Grammar Coalgebras for Context-free - - PowerPoint PPT Presentation
On Language Equations and Grammar Coalgebras for Context-free - - PowerPoint PPT Presentation
On Language Equations and Grammar Coalgebras for Context-free Languages Jurriaan Rot and Joost Winter Leiden University and Centrum Wiskunde & Informatica September 2, 2013 Behavioural differential equations CALCO 11
SLIDE 2
SLIDE 3
Behavioural differential equations
CALCO ’11 (Winter/Bonsangue/Rutten): Context-free languages,
- Coalgebraically. . .
. . . behavioural differential equations / Brzozowski derivatives Example:
- (x)
= 1 xa = xy xb =
- (y)
= ya = yb = 1 . . . gives . . . x = {anbn | n ∈ N} y = {b}
SLIDE 4
◮ These systems correspond to coalgebras for the functor
2 × Pω(−∗)A. . .
SLIDE 5
◮ These systems correspond to coalgebras for the functor
2 × Pω(−∗)A. . .
◮ . . . and can be extended to (infinite) deterministic automata
by enforcing
- (x + y)
=
- (x) ∨ o(y)
(x + y)a = xa + ya
- (xy)
=
- (x) ∧ o(y)
(xy)a = xay + o(x)ya X ⊂ η
✲ Pω(X ∗)
−
✲ P(A∗)
2 × Pω(X ∗)A p′
❄
id × −A
✲ ✛
ˆ p′ 2 × P(A∗)A ∼ =
SLIDE 6
Grammars and the Greibach normal form
Context-free grammars are systems: p : X → Pω((X + A)∗) A CFG is in Greibach normal form iff p(x) ⊆ 1 + A(X + A)∗ for all x ∈ X giving an isomorphism: Pω((X + A)∗)GNF ∼ = 2 × Pω(X ∗)A Hence, grammars in GNF are 2 × Pω(−∗)A-coalgebras.
SLIDE 7
Correctness via grammar derivations
◮ CALCO 2011: coalgebraic semantics coincides with classical
semantics of context-free languages.
◮ Shown via leftmost derivations in a grammar.
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Correctness via grammar derivations
◮ CALCO 2011: coalgebraic semantics coincides with classical
semantics of context-free languages.
◮ Shown via leftmost derivations in a grammar. ◮ Context-free languages can also be seen as (least) solutions to
grammars, regarded as systems of equations.
◮ Question: can we directly relate these systems of equations to
coalgebraic semantics?
SLIDE 9
On solutions
Example: x = 1 + axb Unique solution: x = {anbn | n ∈ N}
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On solutions
Example: x = 1 + axb Unique solution: x = {anbn | n ∈ N}
◮ Such systems of equations (based on the Boolean semiring)
always have a least solution.
◮ If it corresponds to a grammar in GNF, this solution is unique.
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Formalizing solutions
Definition: a solution is any mapping s : X → P(A∗) making the following diagram commute: X s
✲ P(A∗)
Pω((X + A)∗) p
❄
[s, η]♯
✲
♯: inductive extension based on union and concatenation
SLIDE 12
Solutions and GNF
For grammars in GNF, solutions correspond to mappings making the following diagram commute: X s
✲ P(A∗)
2 × Pω(X ∗)A p′
❄
id × (s♯)A
✲ 2 × P(A∗)A
∼ =
SLIDE 13
A lemma
Coalgebraic semantics diagram: X ⊂ η
✲ Pω(X ∗)
−
✲ P(A∗)
2 × Pω(X ∗)A p′
❄
id × −A
✲ ✛
ˆ p′ 2 × P(A∗)A ∼ =
- Here − is an algebra homomorphism, or:
x + y = x ∪ y and xy = xy
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A lemma
Coalgebraic semantics diagram: X ⊂ η
✲ Pω(X ∗)
−
✲ P(A∗)
2 × Pω(X ∗)A p′
❄
id × −A
✲ ✛
ˆ p′ 2 × P(A∗)A ∼ =
- Here − is an algebra homomorphism, or:
x + y = x ∪ y and xy = xy and hence we obtain:
Lemma
(− ◦ η)♯ = −.
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A theorem
Coalgebraic semantics and classical semantics coincide:
Theorem
Given a (classical) system in GNF, − ◦ η is the unique solution.
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A proof
From the diagram X ⊂ η
✲ Pω(X ∗)
−
✲ P(A∗)
2 × Pω(X ∗)A p′
❄
id × −A
✲ ✛
ˆ p′ 2 × P(A∗)A ∼ =
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A proof
From the diagram X ⊂ η
✲ Pω(X ∗)
−
✲ P(A∗)
2 × Pω(X ∗)A p′
❄
id × −A
✲ ✛
ˆ p′ 2 × P(A∗)A ∼ =
- we obtain (by applying the lemma and deleting the diagonal arrow)
X − ◦ η
✲ P(A∗)
2 × Pω(X ∗)A p′
❄
id × ((− ◦ η)♯)A
✲ 2 × P(A∗)A
∼ =
- which is precisely the (unique) classical solution diagram for GNF.
SLIDE 18
A generalization (1)
◮ Standard generalization: formal languages ⇒ formal power
series
◮ Boolean semiring B ⇒ arbitrary semiring K ◮ generalization of P(A∗):
K A := {f : A∗ → K} (also a semiring)
SLIDE 19
A generalization (2)
By applying the following replacements we can generalize our main result to arbitrary commutative semirings: 2 ↔ K Pω(−∗) ↔ K− P(−∗) ↔ K −
SLIDE 20