Epsilon Nondeterministic Finite Automata

This file contains the definition of an epsilon Nondeterministic Finite Automaton (εNFA), a state machine which determines whether a string (implemented as a list over an arbitrary alphabet) is in a regular set by evaluating the string over every possible path, also having access to ε-transitions, which can be followed without reading a character. Since this definition allows for automata with infinite states, a Fintype instance must be supplied for true εNFA's.

structure εNFA (α : Type u) (σ : Type v) : Type (max u v)

An εNFA is a set of states (σ), a transition function from state to state labelled by the alphabet (step), a starting state (start) and a set of acceptance states (accept). Note the transition function sends a state to a Set of states and can make ε-transitions by inputting none. Since this definition allows for Automata with infinite states, a Fintype instance must be supplied for true εNFA's.

• step : σ → Option α → Set σ

  Transition function. The automaton is rendered non-deterministic by this transition function returning Set σ (rather than σ), and ε-transitions are made possible by taking Option α (rather than α).

• start : Set σ

  Starting states.

• accept : Set σ

  Set of acceptance states.

inductive εNFA.εClosure {α : Type u} {σ : Type v} (M : εNFA α σ) (S : Set σ) : Set σ

The εClosure of a set is the set of states which can be reached by taking a finite string of ε-transitions from an element of the set.

• base: ∀ {α : Type u} {σ : Type v} {M : εNFA α σ} {S : Set σ}, ∀ s ∈ S, M.εClosure S s
• step: ∀ {α : Type u} {σ : Type v} {M : εNFA α σ} {S : Set σ} (s t : σ), t ∈ M.step s none → M.εClosure S s → M.εClosure S t

@[simp]

theorem εNFA.subset_εClosure {α : Type u} {σ : Type v} (M : εNFA α σ) (S : Set σ) : S ⊆ M.εClosure S

@[simp]

theorem εNFA.εClosure_empty {α : Type u} {σ : Type v} (M : εNFA α σ) : M.εClosure ∅ = ∅

@[simp]

theorem εNFA.εClosure_univ {α : Type u} {σ : Type v} (M : εNFA α σ) : M.εClosure Set.univ = Set.univ

def εNFA.stepSet {α : Type u} {σ : Type v} (M : εNFA α σ) (S : Set σ) (a : α) : Set σ

M.stepSet S a is the union of the ε-closure of M.step s a for all s ∈ S.

Equations

• M.stepSet S a = ⋃ s ∈ S, M.εClosure (M.step s (some a))

@[simp]

theorem εNFA.mem_stepSet_iff {α : Type u} {σ : Type v} {M : εNFA α σ} {S : Set σ} {s : σ} {a : α} : s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.εClosure (M.step t (some a))

@[simp]

theorem εNFA.stepSet_empty {α : Type u} {σ : Type v} {M : εNFA α σ} (a : α) : M.stepSet ∅ a = ∅

def εNFA.evalFrom {α : Type u} {σ : Type v} (M : εNFA α σ) (start : Set σ) : List α → Set σ

M.evalFrom S x computes all possible paths through M with input x starting at an element of S.

Equations

• M.evalFrom start = List.foldl M.stepSet (M.εClosure start)

@[simp]

theorem εNFA.evalFrom_nil {α : Type u} {σ : Type v} (M : εNFA α σ) (S : Set σ) : M.evalFrom S [] = M.εClosure S

@[simp]

theorem εNFA.evalFrom_singleton {α : Type u} {σ : Type v} (M : εNFA α σ) (S : Set σ) (a : α) : M.evalFrom S [a] = M.stepSet (M.εClosure S) a

@[simp]

theorem εNFA.evalFrom_append_singleton {α : Type u} {σ : Type v} (M : εNFA α σ) (S : Set σ) (x : List α) (a : α) : M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a

@[simp]

theorem εNFA.evalFrom_empty {α : Type u} {σ : Type v} (M : εNFA α σ) (x : List α) : M.evalFrom ∅ x = ∅

def εNFA.eval {α : Type u} {σ : Type v} (M : εNFA α σ) : List α → Set σ

M.eval x computes all possible paths through M with input x starting at an element of M.start.

Equations

• M.eval = M.evalFrom M.start

@[simp]

theorem εNFA.eval_nil {α : Type u} {σ : Type v} (M : εNFA α σ) : M.eval [] = M.εClosure M.start

@[simp]

theorem εNFA.eval_singleton {α : Type u} {σ : Type v} (M : εNFA α σ) (a : α) : M.eval [a] = M.stepSet (M.εClosure M.start) a

@[simp]

theorem εNFA.eval_append_singleton {α : Type u} {σ : Type v} (M : εNFA α σ) (x : List α) (a : α) : M.eval (x ++ [a]) = M.stepSet (M.eval x) a

def εNFA.accepts {α : Type u} {σ : Type v} (M : εNFA α σ) : Language α

M.accepts is the language of x such that there is an accept state in M.eval x.

Equations

• M.accepts = {x : List α | ∃ S ∈ M.accept, S ∈ M.eval x}

Conversions between εNFA and NFA

def εNFA.toNFA {α : Type u} {σ : Type v} (M : εNFA α σ) : NFA α σ

M.toNFA is an NFA constructed from an εNFA M.

Equations

• M.toNFA = { step := fun (S : σ) (a : α) => M.εClosure (M.step S (some a)), start := M.εClosure M.start, accept := M.accept }

@[simp]

theorem εNFA.toNFA_evalFrom_match {α : Type u} {σ : Type v} (M : εNFA α σ) (start : Set σ) : M.toNFA.evalFrom (M.εClosure start) = M.evalFrom start

@[simp]

theorem εNFA.toNFA_correct {α : Type u} {σ : Type v} (M : εNFA α σ) : M.toNFA.accepts = M.accepts

theorem εNFA.pumping_lemma {α : Type u} {σ : Type v} (M : εNFA α σ) [Fintype σ] {x : List α} (hx : x ∈ M.accepts) (hlen : Fintype.card (Set σ) ≤ x.length) : ∃ (a : List α) (b : List α) (c : List α), x = a ++ b ++ c ∧ a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * KStar.kstar {b} * {c} ≤ M.accepts

def NFA.toεNFA {α : Type u} {σ : Type v} (M : NFA α σ) : εNFA α σ

M.toεNFA is an εNFA constructed from an NFA M by using the same start and accept states and transition functions.

Equations

• M.toεNFA = { step := fun (s : σ) (a : Option α) => a.casesOn' ∅ fun (a : α) => M.step s a, start := M.start, accept := M.accept }

@[simp]

theorem NFA.toεNFA_εClosure {α : Type u} {σ : Type v} (M : NFA α σ) (S : Set σ) : M.toεNFA.εClosure S = S

@[simp]

theorem NFA.toεNFA_evalFrom_match {α : Type u} {σ : Type v} (M : NFA α σ) (start : Set σ) : M.toεNFA.evalFrom start = M.evalFrom start

@[simp]

theorem NFA.toεNFA_correct {α : Type u} {σ : Type v} (M : NFA α σ) : M.toεNFA.accepts = M.accepts

Regex-like operations

instance εNFA.instZero {α : Type u} {σ : Type v} : Zero (εNFA α σ)

Equations

• εNFA.instZero = { zero := { step := fun (x : σ) (x : Option α) => ∅, start := ∅, accept := ∅ } }

instance εNFA.instOne {α : Type u} {σ : Type v} : One (εNFA α σ)

Equations

• εNFA.instOne = { one := { step := fun (x : σ) (x : Option α) => ∅, start := Set.univ, accept := Set.univ } }

instance εNFA.instInhabited {α : Type u} {σ : Type v} : Inhabited (εNFA α σ)

Equations

• εNFA.instInhabited = { default := 0 }

@[simp]

theorem εNFA.step_zero {α : Type u} {σ : Type v} (s : σ) (a : Option α) : εNFA.step 0 s a = ∅

@[simp]

theorem εNFA.step_one {α : Type u} {σ : Type v} (s : σ) (a : Option α) : εNFA.step 1 s a = ∅

@[simp]

theorem εNFA.start_zero {α : Type u} {σ : Type v} : εNFA.start 0 = ∅

@[simp]

theorem εNFA.start_one {α : Type u} {σ : Type v} : εNFA.start 1 = Set.univ

@[simp]

theorem εNFA.accept_zero {α : Type u} {σ : Type v} : εNFA.accept 0 = ∅

@[simp]

theorem εNFA.accept_one {α : Type u} {σ : Type v} : εNFA.accept 1 = Set.univ