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Lax314295.MSOFreeVariables

Formulas with free variables define regular languages of annotated strings

concepts/Lax314295/MSOFreeVariables.lean · lax-314295

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    In the paper

    Theorem

    Let φ\varphi be an mso formula over AA whose free variables are among the first-order variables x1,…,xkx_1, \ldots, x_k and the set variables X1,…,XℓX_1, \ldots, X_\ell. Then the language over A×2k+ℓA \times 2^{k+\ell} of the annotated strings w⊗{x1}⊗⋯⊗{xk}⊗X1⊗⋯⊗Xℓw \otimes \{x_1\} \otimes \cdots \otimes \{x_k\} \otimes X_1 \otimes \cdots \otimes X_\ell such that w⊨φ(x1,…,xk,X1,…,Xℓ)w \models \varphi(x_1, \ldots, x_k, X_1, \ldots, X_\ell) is regular (Lemma C.4.2 of Transducers). The proof is an induction on the formula, with a nondeterministic automaton guessing the value of a quantified variable.

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    1import Lax314295.MSOLogic
    2
    3/-!
    4---
    5title: Formulas with free variables define regular languages of annotated strings
    6type: theorem
    7---
    8Let φ\varphi be an mso formula over AA whose free variables are among the
    9first-order variables x1,…,xkx_1, \ldots, x_k and the set variables
    10X1,…,XℓX_1, \ldots, X_\ell. Then the language over A×2k+ℓA \times 2^{k+\ell} of the
    11annotated strings w⊗{x1}⊗⋯⊗{xk}⊗X1⊗⋯⊗Xℓw \otimes \{x_1\} \otimes \cdots \otimes \{x_k\} \otimes X_1 \otimes \cdots \otimes X_\ell
    12 such that
    13w⊨φ(x1,…,xk,X1,…,Xℓ)w \models \varphi(x_1, \ldots, x_k, X_1, \ldots, X_\ell) is regular
    14(Lemma C.4.2 of *Transducers*). The proof is an induction on the formula, with
    15a nondeterministic automaton guessing the value of a quantified variable.
    16
    17# Formalization notes
    18
    19The free variables are among `0, …, k-1` and `0, …, l-1`; the language is that
    20of the annotations `annotate k l w fo so` of valuations by positions and sets
    21of positions of `w` under which `w` satisfies `φ`. The alphabet is assumed
    22finite.
    23-/
    24
    25namespace Lax314295.MSOFreeVariables
    26
    27open Lax314295.MSOLogic
    28
    29/-- The annotated strings satisfying a formula form a regular language. -/
    30axiom isRegular_annotated {A : Type} [Finite A] (φ : MSO A) (k l : ℕ)
    31 (hfo : φ.freeFO ⊆ {i | i < k}) (hso : φ.freeSO ⊆ {j | j < l}) :
    32 Language.IsRegular
    33 {u : List (A × (Fin k → Bool) × (Fin l → Bool)) |
    34 ∃ (w : List A) (fo : Fin k → ℕ) (so : Fin l → Set ℕ),
    35 (∀ i, fo i < w.length) ∧ (∀ j, so j ⊆ {p | p < w.length}) ∧
    36 u = annotate k l w fo so ∧ MSO.Sat w (extFO k fo) (extSO l so) φ}
    37
    38end Lax314295.MSOFreeVariables
    39
    Show Proof

    Formalization notes

    The free variables are among 0,…,k−10, …, k-1 and 0,…,l−10, …, l-1; the language is that of the annotations annotateklwfosoannotate k l w fo so of valuations by positions and sets of positions of ww under which ww satisfies φφ. The alphabet is assumed finite.

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