Verified Decision Procedures for Equivalence of Regular Expressions - - PowerPoint PPT Presentation

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Verified Decision Procedures for Equivalence of Regular Expressions - - PowerPoint PPT Presentation

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Verified Decision Procedures for Equivalence of Regular Expressions Tobias Nipkow & Dmitriy Traytel Fakult at f ur Informatik Technische Universit at M unchen Background Recent series of papers presenting such decision

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Verified Decision Procedures for Equivalence of Regular Expressions

Tobias Nipkow & Dmitriy Traytel

Fakult¨ at f¨ ur Informatik Technische Universit¨ at M¨ unchen

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Background

Recent series of papers presenting such decision procedures verified in Coq, Isabelle or Matita:

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Background

Recent series of papers presenting such decision procedures verified in Coq, Isabelle or Matita: Braibant & Pous 2010, Krauss & Nipkow 2011, Coquand & Siles 2011, Asperti 2012, Moreira et al. 2013

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Background

Recent series of papers presenting such decision procedures verified in Coq, Isabelle or Matita: Braibant & Pous 2010, Krauss & Nipkow 2011, Coquand & Siles 2011, Asperti 2012, Moreira et al. 2013 They all operate on regular expressions, not automata

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Background

Recent series of papers presenting such decision procedures verified in Coq, Isabelle or Matita: Braibant & Pous 2010, Krauss & Nipkow 2011, Coquand & Siles 2011, Asperti 2012, Moreira et al. 2013 They all operate on regular expressions, not automata They all look different but related . . .

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This talk

  • Unified framework
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This talk

  • Unified framework
  • Derivation of all previous procedures

as instances

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This talk

  • Unified framework
  • Derivation of all previous procedures

as instances

  • Verification in Isabelle
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1 The Unified Framework 2 Derivatives of Regular Expressions 3 Partial Derivatives of Regular Expressions 4 Marked regular expressions 5 Empirical Comparison

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Regular expressions

datatype α rexp = 0 | 1 | Atom α | α rexp + α rexp | α rexp · α rexp | α rexp ∗

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Regular expressions

datatype α rexp = 0 | 1 | Atom α | α rexp + α rexp | α rexp · α rexp | α rexp ∗ Semantics: L :: α rexp → α lang where α lang = α list set

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How to prove r ≡ s

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How to prove r ≡ s

1 Translate to DFAs A and B

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How to prove r ≡ s

1 Translate to DFAs A and B 2 Compare A and B

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How to prove r ≡ s

1 Translate to DFAs A and B 2 Compare A and B

  • Standard algorithm:

Minimize A and B, check isomorphism.

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How to prove r ≡ s

1 Translate to DFAs A and B 2 Compare A and B

  • Standard algorithm:

Minimize A and B, check isomorphism.

  • Easy alternative:

Check for all reachable states (p, q) of A × B that p is final iff q is final.

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Framework parameters

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Framework parameters

Type σ

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Framework parameters

Type σ Init init :: α rexp → σ

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Framework parameters

Type σ Init init :: α rexp → σ Transition δ :: α → σ → σ

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Framework parameters

Type σ Init init :: α rexp → σ Transition δ :: α → σ → σ Final fin :: σ → bool

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Framework parameters

Type σ Init init :: α rexp → σ Transition δ :: α → σ → σ Final fin :: σ → bool Language L :: σ → α lang

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Framework parameters

Type σ Init init :: α rexp → σ Transition δ :: α → σ → σ Final fin :: σ → bool Language L :: σ → α lang Assumptions: L(init(r)) = L(r)

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Framework parameters

Type σ Init init :: α rexp → σ Transition δ :: α → σ → σ Final fin :: σ → bool Language L :: σ → α lang Assumptions: L(init(r)) = L(r) L(δ x s) = {w | xw ∈ L(s)}

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Framework parameters

Type σ Init init :: α rexp → σ Transition δ :: α → σ → σ Final fin :: σ → bool Language L :: σ → α lang Assumptions: L(init(r)) = L(r) L(δ x s) = {w | xw ∈ L(s)} fin(s) ⇔ [] ∈ L(s)

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Equivalence checker

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Equivalence checker

eqv :: α rexp → α rexp → bool

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Equivalence checker

eqv :: α rexp → α rexp → bool eqv r s = case closure (init(r), init(s)) of Some([], ) ⇒ True | ⇒ False

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Equivalence checker

eqv :: α rexp → α rexp → bool eqv r s = case closure (init(r), init(s)) of Some([], ) ⇒ True | ⇒ False

Theorem

eqv r s = ⇒ L(r) = L(s)

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Equivalence checker

eqv :: α rexp → α rexp → bool eqv r s = case closure (init(r), init(s)) of Some([], ) ⇒ True | ⇒ False

Theorem

eqv r s = ⇒ L(r) = L(s) If the set of reachable states is finite:

Theorem

L(r) = L(s) = ⇒ eqv r s

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1 The Unified Framework 2 Derivatives of Regular Expressions 3 Partial Derivatives of Regular Expressions 4 Marked regular expressions 5 Empirical Comparison

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Derivatives (Brzozowski 1964)

d :: α → α rexp → α rexp

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Derivatives (Brzozowski 1964)

d :: α → α rexp → α rexp

  • d x r is the derivative of r wrt x
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Derivatives (Brzozowski 1964)

d :: α → α rexp → α rexp

  • d x r is the derivative of r wrt x
  • d x r = “what is left after x has been read”
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Derivatives (Brzozowski 1964)

d :: α → α rexp → α rexp

  • d x r is the derivative of r wrt x
  • d x r = “what is left after x has been read”
  • Example: d a (Atom(a) · r) = 1 · r
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Derivatives (Brzozowski 1964)

d :: α → α rexp → α rexp

  • d x r is the derivative of r wrt x
  • d x r = “what is left after x has been read”
  • Example: d a (Atom(a) · r) = 1 · r
  • Semantics is left-quotient:

L(d x r) = {w | xw ∈ L(r)}

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d x 0 = 0

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d x 0 = 0 d x 1 = 0

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d x 0 = 0 d x 1 = 0 d x (Atom y) = if x = y then 1 else 0

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d x 0 = 0 d x 1 = 0 d x (Atom y) = if x = y then 1 else 0 d x (r + s) = d x r + d x s

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d x 0 = 0 d x 1 = 0 d x (Atom y) = if x = y then 1 else 0 d x (r + s) = d x r + d x s d x (r · s) = if ε(r) then d x r · s + d x s else d x r · s

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d x 0 = 0 d x 1 = 0 d x (Atom y) = if x = y then 1 else 0 d x (r + s) = d x r + d x s d x (r · s) = if ε(r) then d x r · s + d x s else d x r · s d x (r∗) = d x r · r∗

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Regular Expression DFA

a · a∗

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Regular Expression DFA

a · a∗ 1 · a∗ a

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Regular Expression DFA

a · a∗ 1 · a∗ a 0 · a∗ + 1 · a∗ a

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Regular Expression DFA

a · a∗ 1 · a∗ a 0 · a∗ + 1 · a∗ a a

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Finiteness

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Finiteness

Let ≡ACI be the equivalence induced by ACI of +

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Finiteness

Let ≡ACI be the equivalence induced by ACI of + Theorem (Brzozowski 1964) The set {fold d w r | w ∈ Σ∗}/≡ACI is finite.

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Finiteness

Let ≡ACI be the equivalence induced by ACI of + Theorem (Brzozowski 1964) The set {fold d w r | w ∈ Σ∗}/≡ACI is finite. How large?

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Finiteness

Let ≡ACI be the equivalence induced by ACI of + Theorem (Brzozowski 1964) The set {fold d w r | w ∈ Σ∗}/≡ACI is finite. How large? Brzozowski’s proof yields O(2...2n )

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Instantiation of framework

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Instantiation of framework

σ = α rexp

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Instantiation of framework

σ = α rexp init(r) = r

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Instantiation of framework

σ = α rexp init(r) = r δ x r = normACI(d x r)

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Instantiation of framework

σ = α rexp init(r) = r δ x r = normACI(d x r) fin = ε

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Instantiation of framework

σ = α rexp init(r) = r δ x r = normACI(d x r) fin = ε L = L

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Instantiation of framework

σ = α rexp init(r) = r δ x r = normACI(d x r) fin = ε L = L Finiteness:

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Instantiation of framework

σ = α rexp init(r) = r δ x r = normACI(d x r) fin = ε L = L Finiteness:

  • Not immediate from Brzozowski’s theorem
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Instantiation of framework

σ = α rexp init(r) = r δ x r = normACI(d x r) fin = ε L = L Finiteness:

  • Not immediate from Brzozowski’s theorem
  • Open for stronger normalization functions
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1 The Unified Framework 2 Derivatives of Regular Expressions 3 Partial Derivatives of Regular Expressions 4 Marked regular expressions 5 Empirical Comparison

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Antimirov 1996

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Antimirov 1996

Idea: build some of ≡ into the data structure set:

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Antimirov 1996

Idea: build some of ≡ into the data structure set: d : α → α rexp → α rexp

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Antimirov 1996

Idea: build some of ≡ into the data structure set: D : α → α rexp → α rexp set

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Antimirov 1996

Idea: build some of ≡ into the data structure set: D : α → α rexp → α rexp set d x (r + s) = d x r + d x s

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Antimirov 1996

Idea: build some of ≡ into the data structure set: D : α → α rexp → α rexp set D x (r + s) = D x r ∪ D x s

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Antimirov 1996

Idea: build some of ≡ into the data structure set: D : α → α rexp → α rexp set D x (r + s) = D x r ∪ D x s d x (r · s) = if ε(r) then d x r · s + d x s else d x r · s

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Antimirov 1996

Idea: build some of ≡ into the data structure set: D : α → α rexp → α rexp set D x (r + s) = D x r ∪ D x s D x (r · s) = if ε(r) then D x r ⊙ s ∪ D x s else D x r ⊙ s

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Antimirov 1996

Idea: build some of ≡ into the data structure set: D : α → α rexp → α rexp set D x (r + s) = D x r ∪ D x s D x (r · s) = if ε(r) then D x r ⊙ s ∪ D x s else D x r ⊙ s where {r1, . . . , rn} ⊙ s = {r1 · s, . . . , rn · s}

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Antimirov 1996

Idea: build some of ≡ into the data structure set: D : α → α rexp → α rexp set D x (r + s) = D x r ∪ D x s D x (r · s) = if ε(r) then D x r ⊙ s ∪ D x s else D x r ⊙ s . . . where {r1, . . . , rn} ⊙ s = {r1 · s, . . . , rn · s}

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Instantiation of framework

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Instantiation of framework

σ = α rexp set

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Instantiation of framework

σ = α rexp set init(r) = {r}

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Instantiation of framework

σ = α rexp set init(r) = {r} δ x R =

  • r∈R

D x r

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Instantiation of framework

σ = α rexp set init(r) = {r} δ x R =

  • r∈R

D x r fin(R) = ∃r ∈ R. ε(r)

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Instantiation of framework

σ = α rexp set init(r) = {r} δ x R =

  • r∈R

D x r fin(R) = ∃r ∈ R. ε(r) L(R) =

  • r∈R

L(r)

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Finiteness

Theorem (Antimirov 1996) Starting from a regular expression r

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Finiteness

Theorem (Antimirov 1996) Starting from a regular expression r at most |r|at + 1 regular expressions are reachable

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Finiteness

Theorem (Antimirov 1996) Starting from a regular expression r at most |r|at + 1 regular expressions are reachable where |r|at is the number of occurrences of atoms in r.

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Finiteness

Theorem (Antimirov 1996) Starting from a regular expression r at most |r|at + 1 regular expressions are reachable where |r|at is the number of occurrences of atoms in r. = ⇒ 2|r|at+1 sets of regular expressions reachable

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1 The Unified Framework 2 Derivatives of Regular Expressions 3 Partial Derivatives of Regular Expressions 4 Marked regular expressions 5 Empirical Comparison

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History

McNaughton & Yamada 1960, Glushkov 1961:

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History

McNaughton & Yamada 1960, Glushkov 1961:

  • Translation of regular expression to N/DFA
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History

McNaughton & Yamada 1960, Glushkov 1961:

  • Translation of regular expression to N/DFA
  • Atoms in regular expression are indexed, eg

a1 · a2 + a3 · b1

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History

McNaughton & Yamada 1960, Glushkov 1961:

  • Translation of regular expression to N/DFA
  • Atoms in regular expression are indexed, eg

a1 · a2 + a3 · b1

  • States are (sets of) indexed atoms, eg {a1, a3}
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History

McNaughton & Yamada 1960, Glushkov 1961:

  • Translation of regular expression to N/DFA
  • Atoms in regular expression are indexed, eg

a1 · a2 + a3 · b1

  • States are (sets of) indexed atoms, eg {a1, a3}

Functional implementation by Fischer, Huch & Wilke [ICFP 2009]:

  • Replace sets of positions

by marked regular expressions: Atom(bool, α)

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History

McNaughton & Yamada 1960, Glushkov 1961:

  • Translation of regular expression to N/DFA
  • Atoms in regular expression are indexed, eg

a1 · a2 + a3 · b1

  • States are (sets of) indexed atoms, eg {a1, a3}

Functional implementation by Fischer, Huch & Wilke [ICFP 2009]:

  • Replace sets of positions

by marked regular expressions: Atom(bool, α)

  • Only matching, not ≡, no proofs
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Example: (a · a + a · b)∗

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Example: (a · a + a · b)∗

q0

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Example: (a · a + a · b)∗

q0 (a · a + a · b)∗ a

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Example: (a · a + a · b)∗

q0 (a · a + a · b)∗ a (a · a + a · b)∗ a

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Example: (a · a + a · b)∗

q0 (a · a + a · b)∗ a (a · a + a · b)∗ a a

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Example: (a · a + a · b)∗

q0 (a · a + a · b)∗ a (a · a + a · b)∗ a a (a · a + a · b)∗ b

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Example: (a · a + a · b)∗

q0 (a · a + a · b)∗ a (a · a + a · b)∗ a a (a · a + a · b)∗ b a

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Instantiation of framework

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Instantiation of framework

σ = bool × (bool × α) rexp

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Instantiation of framework

σ = bool × (bool × α) rexp init(r) = (True, map (λa. (False, a)) r)

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Instantiation of framework

σ = bool × (bool × α) rexp init(r) = (True, map (λa. (False, a)) r) δ x (m, r) = (False, read x (follow m r))

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Instantiation of framework

σ = bool × (bool × α) rexp init(r) = (True, map (λa. (False, a)) r) δ x (m, r) = (False, read x (follow m r)) fin(m, r) = . . . L(m, r) = . . .

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Conceptually, the marks in McNaugton/Glushkov/Fisher are after the atoms

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Marked regular expressions II

Asperti [ITP 2012]:

  • Verified ≡-checker via marked rexp in Matita
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Marked regular expressions II

Asperti [ITP 2012]:

  • Verified ≡-checker via marked rexp in Matita
  • Says he has formalised McNaughton & Yamada
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Marked regular expressions II

Asperti [ITP 2012]:

  • Verified ≡-checker via marked rexp in Matita
  • Says he has formalised McNaughton & Yamada
  • . . . but he invented his own variation:
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Marked regular expressions II

Asperti [ITP 2012]:

  • Verified ≡-checker via marked rexp in Matita
  • Says he has formalised McNaughton & Yamada
  • . . . but he invented his own variation:

Puts the mark before the atom

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Example: (a · a + a · b)∗

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Example: (a · a + a · b)∗

(a · a + a · b)∗

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Example: (a · a + a · b)∗

(a · a + a · b)∗ (a · a + a · b)∗ a

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Example: (a · a + a · b)∗

(a · a + a · b)∗ (a · a + a · b)∗ a a, b

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Instantiation of framework

Similar but a bit more complicated

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Before vs After

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Before vs After

Transitions can be decomposed into two steps:

  • Before: read; follow
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Before vs After

Transitions can be decomposed into two steps:

  • Before: read; follow
  • After: follow; read
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Before vs After

Transitions can be decomposed into two steps:

  • Before: read; follow
  • After: follow; read

Theorem

The before-automaton is a homorphic image of the after-automaton.

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Before vs After

Transitions can be decomposed into two steps:

  • Before: read; follow
  • After: follow; read

Theorem

The before-automaton is a homorphic image of the after-automaton. Proof idea due to Helmut Seidl.

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1 The Unified Framework 2 Derivatives of Regular Expressions 3 Partial Derivatives of Regular Expressions 4 Marked regular expressions 5 Empirical Comparison

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200 400 600 2 4 6 8 n Time (s)

(a0 + · · · + an−1) · (an)∗ ≡ a∗

Deriv. Marked Part.Deriv.

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200 400 600 2 4 6 8 n Time (s)

(a0 + · · · + an−1) · (an)∗ ≡ a∗

Deriv. Marked Part.Deriv.

For randomly generated examples:

  • Deriv. ≫ Part.Deriv. ≫ Fischer, Asperti
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Extended regular expressions

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Extended regular expressions

Complement and intersection:

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Extended regular expressions

Complement and intersection:

  • Trivial for derivatives (Brzozowski)
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Extended regular expressions

Complement and intersection:

  • Trivial for derivatives (Brzozowski)
  • Harder for partial derivatives

(Champarnaud and Mignot)

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Extended regular expressions

Complement and intersection:

  • Trivial for derivatives (Brzozowski)
  • Harder for partial derivatives

(Champarnaud and Mignot)

  • Unclear for marked regular expressions
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Extended regular expressions

Complement and intersection:

  • Trivial for derivatives (Brzozowski)
  • Harder for partial derivatives

(Champarnaud and Mignot)

  • Unclear for marked regular expressions

. . . and projection:

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Extended regular expressions

Complement and intersection:

  • Trivial for derivatives (Brzozowski)
  • Harder for partial derivatives

(Champarnaud and Mignot)

  • Unclear for marked regular expressions

. . . and projection:

  • Traytel & N. [ICFP 13] extend derivatives
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Extended regular expressions

Complement and intersection:

  • Trivial for derivatives (Brzozowski)
  • Harder for partial derivatives

(Champarnaud and Mignot)

  • Unclear for marked regular expressions

. . . and projection:

  • Traytel & N. [ICFP 13] extend derivatives

decision procedure for MSO on finite strings

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Summary

Equivalence-checkers for regular expressions can be defined purely functionally

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Summary

Equivalence-checkers for regular expressions can be defined purely functionally via (partial) derivatives

  • r marked regular expressions
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Summary

Equivalence-checkers for regular expressions can be defined purely functionally via (partial) derivatives

  • r marked regular expressions

Perfect proof assistant fodder