feat: proof of Myhill-Nerode theorem for DFAs

Cslib.lean

@@ -26,6 +26,7 @@ public import Cslib.Computability.Languages.Congruences.BuchiCongruence
 public import Cslib.Computability.Languages.Congruences.RightCongruence
 public import Cslib.Computability.Languages.ExampleEventuallyZero
 public import Cslib.Computability.Languages.Language
+public import Cslib.Computability.Languages.MyhillNerode
 public import Cslib.Computability.Languages.OmegaLanguage
 public import Cslib.Computability.Languages.OmegaRegularLanguage
 public import Cslib.Computability.Languages.RegularLanguage

Cslib/Computability/Languages/MyhillNerode.lean

@@ -0,0 +1,254 @@
+/-
+Copyright (c) 2026 Akhilesh Balaji. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Akhilesh Balaji
+-/
+
+module
+
+public import Cslib.Computability.Languages.RegularLanguage
+
+/-! # Myhill-Nerode Theorem
+
+Let `l` be a language on an alphabet `α`. The Nerode congruence (referred to as `c_l`
+in comments below) of a language `l` is a right congruence on strings where two strings are
+related iff all their right extensions are either both in the language or both not in it.
+
+The Myhill-Nerode theorem has three parts [WikipediaMyhillNerode2026]:
+
+(1) `l` is regular iff `c_l` has a finite number `N` of equivalence classes.
+
+(2) `N` is the number of states of the minimal DFA accepting `l`.
+
+(3) The minimal DFA is unique up to unique isomorphism. That is, for any
+    minimal DFA accepting `l`, there exists exactly an isomorphism from it to the
+    canonical DFA whose states are the equivalence classses of `c_l`, whose
+    state transitions are of the form `⟦ x ⟧ → ⟦ x ++ [a] ⟧` (where `a : α`
+    and `x : List α`), whose initial state is `⟦ [] ⟧`, and whose accepting states
+    are `{ ⟦ x ⟧ | x ∈ l }`.
+
+## References
+
+* [J. E. Hopcroft, R. Motwani, J. D. Ullman,
+  *Introduction to Automata Theory, Languages, and Computation*][Hopcroft2006]
+* [T. Malkin, *COMS W3261: Computer Science Theory, Handout 3: The Myhill-Nerode Theorem
+   and Implications*][Malkin2024]
+* [Wikipedia contributors, Myhill–Nerode theorem][WikipediaMyhillNerode2026]
+-/
+
+@[expose] public section
+
+variable {α State : Type*}
+
+namespace Language
+
+open Cslib Language Automata DA FinAcc Acceptor
+open scoped RightCongruence
+
+/-- The Nerode congruence (henceforth called `c_l`) of a language `l` is a right congruence on
+strings where two strings are related iff all their right extensions are either both in the language
+or both not in it. -/
+@[implicit_reducible]
+def NerodeCongruence (l : Language α) : RightCongruence α where
+  r x y := ∀ z, x ++ z ∈ l ↔ y ++ z ∈ l
+  iseqv.refl := fun _ _ => Iff.rfl
+  iseqv.symm := fun h z => (h z).symm
+  iseqv.trans := fun h_1 h_2 z => (h_1 z).trans (h_2 z)
+  right_cov.elim := fun a {x y} (h : ∀ z, x ++ z ∈ l ↔ y ++ z ∈ l) z =>
+    List.append_assoc x a z ▸ List.append_assoc y a z ▸ h (a ++ z)
+
+/-- The quotient type of a Nerode congruence. -/
+abbrev NerodeQuotient (l : Language α) := Quotient (l.NerodeCongruence).eq
+
+/-- The Nerode congruence of a language `l` gives rise to a DFA where each state corresponds to an
+equivalence class of the language under the Nerode congruence. Note that this is simply the DFA
+given rise to by the underlying right congruence with only the accept states specified here as
+`{⟦ x ⟧ | x ∈ l}`. -/
+def NerodeCongruenceDA (l : Language α) : DA.FinAcc (l.NerodeQuotient) α :=
+  FinAcc.mk (l.NerodeCongruence).toDA ((⟦·⟧) '' l)
+
+variable {l : Language α}
+
+/-- The DFA constructed from the Nerode congruence `c_l` on `l` accepts `l`. -/
+@[simp, scoped grind =]
+theorem nerodeCongruenceDA_language_eq (l : Language α) :
+    language (l.NerodeCongruenceDA) = l := by
+  ext x
+  simp only [NerodeCongruenceDA, language, Acceptor.Accepts, congr_mtr_eq, Set.mem_image]
+  constructor
+  · rintro ⟨y, hy, heq⟩
+    have h1 := Quotient.eq.mp heq []
+    simp only [List.append_nil] at h1
+    simpa [← h1]
+  · exact fun hx => ⟨x, hx, rfl⟩
+
+/-- The statement (two strings are related by the Nerode congruence `c_l` on `l` iff all their right
+extensions are either both in the language or both not in it) is equivalent to stating that (all
+their right extensions are either both accepted or rejected by the DFA given rise to by `c_l`.) -/
+theorem da_nerodeCongruence_iff {State : Type*} (M : DA.FinAcc State α) (x y : List α) :
+    ((language M).NerodeCongruence).r x y ↔
+    ∀ z, M.mtr (M.mtr M.start x) z ∈ M.accept ↔ M.mtr (M.mtr M.start y) z ∈ M.accept := by
+  simp only [FLTS.mtr, ← List.foldl_append]
+  rfl
+
+/-- If `l` is regular then the Nerode congruence on `l` has finitely many equivalence classes. -/
+theorem IsRegular.finite_nerodeQuotient (h : l.IsRegular) :
+    Finite (l.NerodeQuotient) := by
+  rcases IsRegular.iff_dfa.mp h with ⟨State, hFin, M, hM⟩
+  rw [← hM]
+  apply Finite.of_surjective
+    (fun s : State =>
+      ⟦ Classical.epsilon (fun x => M.mtr M.start x = s) ⟧)
+  intro q
+  induction q using Quotient.inductionOn with
+  | h x =>
+    exact ⟨M.mtr M.start x, Quotient.sound (by
+      change ((language M).NerodeCongruence).r _ x
+      rw [da_nerodeCongruence_iff]
+      intro z
+      have heps : M.mtr M.start (Classical.epsilon
+          (fun y => M.mtr M.start y = M.mtr M.start x)) = M.mtr M.start x :=
+        @Classical.epsilon_spec _ (fun y => M.mtr M.start y = M.mtr M.start x) ⟨x, rfl⟩
+      rw [heps])⟩
+
+-- Myhill-Nerode (1)
+
+/-- `l` is regular iff the Nerode congruence on `l` has finitely many equivalence classes. -/
+@[scoped grind =]
+theorem IsRegular.iff_finite_nerodeQuotient {l : Language α} :
+    l.IsRegular ↔ Finite (l.NerodeQuotient) := by
+  constructor
+  · intro h
+    exact IsRegular.finite_nerodeQuotient h
+  · intro h
+    letI : Fintype (l.NerodeQuotient) := Fintype.ofFinite _
+    set e := Fintype.equivFin (l.NerodeQuotient)
+    set M := l.NerodeCongruenceDA
+    refine IsRegular.iff_dfa.mpr ⟨Fin _, inferInstance,
+      { tr := fun s x => e (M.tr (e.symm s) x)
+        start := e M.start
+        accept := e '' M.accept }, ?_⟩
+    have hfoldl : ∀ s x, List.foldl (fun s x => e (M.tr (e.symm s) x)) (e s) x =
+        e (List.foldl M.tr s x) := by
+      intro s x
+      induction x generalizing s with
+      | nil => simp
+      | cons a as ih => simp only [List.foldl, e.symm_apply_apply, ih]
+    ext x
+    change List.foldl (fun s x => e (M.tr (e.symm s) x)) (e M.start) x ∈ e '' M.accept ↔ x ∈ l
+    simp only [hfoldl, Set.mem_image_equiv, Equiv.symm_apply_apply,
+      ← nerodeCongruenceDA_language_eq l, language, Acceptor.Accepts, FLTS.mtr]
+    exact Iff.of_eq rfl
+
+/-- Given a set of strings all distinguishable by `l` (i.e., not related to each other by the Nerode
+congruence on `l`), the number of states in the DFA accepting `l` is at least the number of strings
+in the set. -/
+theorem dfa_num_state_ge {State : Type*} [Finite State] {l : Language α}
+    {M : DA.FinAcc State α} {ws : Set (List α)} [Finite ws]
+    (hws : ws.Pairwise (¬ (l.NerodeCongruence).r · ·)) (hM : language M = l) :
+    Nat.card State ≥ Nat.card ws := by
+  letI : Fintype State := Fintype.ofFinite _
+  letI : Fintype ws := Fintype.ofFinite _
+  rw [Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]
+  by_contra h
+  simp only [ge_iff_le, not_le] at h
+  by_cases h_card : Fintype.card ws ≤ 1
+  · exact (lt_of_lt_of_le h (le_trans h_card
+      (Nat.succ_le_of_lt (Fintype.card_pos_iff.mpr ⟨M.start⟩)))).false
+  · obtain ⟨⟨x, hx⟩, ⟨y, hy⟩, hne, heq⟩ :=
+      Fintype.exists_ne_map_eq_of_card_lt (f := fun w : ws => M.mtr M.start w) (by omega)
+    rw [← hM] at hws
+    exact hws hx hy (fun h => hne (Subtype.ext h))
+      ((da_nerodeCongruence_iff M x y).mpr (fun z => heq ▸ Iff.rfl))
+
+-- Myhill-Nerode (2)
+
+/-- All DFAs accepting `l` must have at least as many states as the number of equivalence classes
+of the Nerode congruence on `l`. -/
+theorem dfa_num_state_min {State : Type} {M : DA.FinAcc State α} [Finite State] :
+    Nat.card State ≥ Nat.card (language M).NerodeQuotient := by
+  let ws : Set (List α) := Set.range
+    (Quotient.out : Quotient ((language M).NerodeCongruence).eq → List α)
+  haveI : Finite (language M).NerodeQuotient :=
+      IsRegular.iff_finite_nerodeQuotient.mp (IsRegular.iff_dfa.mpr ⟨State, inferInstance, M, rfl⟩)
+  haveI : Finite ws := Set.finite_range _ |>.to_subtype
+  have hws : ws.Pairwise (fun x y => ¬((language M).NerodeCongruence).r x y) := by
+    rintro _ ⟨qx, rfl⟩ _ ⟨qy, rfl⟩ hne h
+    exact hne (by simpa using Quotient.sound h)
+  exact (Nat.card_congr (Equiv.ofInjective _ Quotient.out_injective).symm)
+    ▸ dfa_num_state_ge hws rfl
+
+end Language
+
+namespace Cslib.Automata.DA.FinAcc
+
+open Cslib Language Automata DA FinAcc Acceptor
+open scoped RightCongruence
+
+/-- The minimal DFA accepting `l` has the same number of states as the number of equivalence classes
+of the Nerode congruence on `l`. -/
+def IsMinimalAutomaton (M : FinAcc State α) (l : Language α) :=
+  language M = l ∧ Nat.card State = Nat.card (l.NerodeQuotient)
+
+/-- Given a DFA `M`, two strings are related iff they reach the same state under when run through
+`M`. The Nerode congruence is the state congruence with respect to the minimal DFA accepting `l`. -/
+@[implicit_reducible]
+def StateCongruence (M : FinAcc State α) : RightCongruence α where
+  r x y := ∀ z, M.mtr M.start (x ++ z) = M.mtr M.start (y ++ z)
+  iseqv.refl := by intro x z; rfl
+  iseqv.symm  := by intro x y h z; symm; exact h z
+  iseqv.trans := by intro x y z h_1 h_2 w; exact (h_1 w).trans (h_2 w)
+  right_cov.elim := by
+    intro a x y h z
+    simpa [List.append_assoc, FLTS.mtr_concat_eq] using h (a ++ z)
+
+variable {M : FinAcc State α}
+
+/-- The Nerode congruence is the most coarse state congruence given a language. -/
+theorem stateCongruence_le_nerodeCongruence :
+    ∀ x y, (StateCongruence M).r x y → ((language M).NerodeCongruence).r x y := by
+  intro x y h z
+  constructor
+  · intro hx
+    have := h z
+    simpa [language, Acceptor.Accepts, FLTS.mtr_concat_eq] using
+      congrArg (fun s => s ∈ M.accept) this ▸ hx
+  · intro hy
+    have := h z
+    simpa [language, Acceptor.Accepts, FLTS.mtr_concat_eq] using
+      congrArg (fun s => s ∈ M.accept) this ▸ hy
+
+-- Myhill-Nerode (3)
+
+/-- The minimal DFA `M` accepting the language `l` is unique up to unique isomorphism. -/
+theorem unique_minimal [Finite State]
+    (l : Language α) (hReg : l.IsRegular) (hMin : M.IsMinimalAutomaton l) :
+    ∃! φ : State ≃ l.NerodeQuotient, ∀ x, φ (M.mtr M.start x) = ⟦ x ⟧ := by
+  obtain ⟨hML, hCard⟩ := hMin
+  haveI : Finite (l.NerodeQuotient) :=
+    Language.IsRegular.iff_finite_nerodeQuotient.mp hReg
+  letI : Fintype State := Fintype.ofFinite _
+  letI : Fintype (l.NerodeQuotient) := Fintype.ofFinite _
+  subst hML
+  let φ : State → Quotient ((language M).NerodeCongruence).eq :=
+    fun s => ⟦ Classical.epsilon (fun x : List α => M.mtr M.start x = s) ⟧
+  have hφ : ∀ x, φ (M.mtr M.start x) = ⟦ x ⟧ := fun x => by
+    apply Quotient.sound
+    apply stateCongruence_le_nerodeCongruence
+    intro z
+    have := @Classical.epsilon_spec _ (fun y : List α => M.mtr M.start y = M.mtr M.start x) ⟨x, rfl⟩
+    simp only [FLTS.mtr, List.foldl_append] at this ⊢; rw [this]
+  have hφ_surj : Function.Surjective φ := fun q =>
+    q.inductionOn (fun x => ⟨M.mtr M.start x, hφ x⟩)
+  have hφ_inj : Function.Injective φ :=
+    hφ_surj.injective_of_finite (Fintype.equivOfCardEq (by
+      rwa [← Nat.card_eq_fintype_card, ← Nat.card_eq_fintype_card]))
+  let φ_equiv := Equiv.ofBijective φ ⟨hφ_inj, hφ_surj⟩
+  refine ⟨φ_equiv, hφ, fun ψ hψ => ?_⟩
+  ext s
+  obtain ⟨x, rfl⟩ : ∃ x, M.mtr M.start x = s := by
+    induction h : φ s using Quotient.inductionOn with
+    | h x => exact ⟨x, hφ_inj ((hφ x).trans h.symm)⟩
+  simp [φ_equiv, Equiv.ofBijective, hφ, hψ]
+
+end Cslib.Automata.DA.FinAcc

references.bib

@@ -94,6 +94,24 @@ @book{            Hopcroft2006
 address = {USA}
 }
 
+@misc{         Malkin2024,
+  author       = {Tal Malkin},
+  title        = {{COMS W3261: Computer Science Theory, Handout 3: The Myhill-Nerode Theorem and Implications}},
+  howpublished = {Columbia University Course Handout},
+  year         = {2024},
+  month        = {September},
+  url          = {https://www.cs.columbia.edu/~tal/3261/fall24/Handouts/3_Myhill_Nerode.pdf}
+}
+
+@misc{         WikipediaMyhillNerode2026,
+  author       = {{Wikipedia contributors}},
+  title        = {{Myhill--Nerode theorem}},
+  howpublished = {Wikipedia, The Free Encyclopedia},
+  year         = {2026},
+  url          = {https://en.wikipedia.org/wiki/Myhill%E2%80%93Nerode_theorem},
+  note         = {[Online; accessed 9-April-2026]}
+}
+
 @article{         Chargueraud2012,
 	title = {The {Locally} {Nameless} {Representation}},
 	volume = {49},