-defjnable functions and automata theory Nguyn L Thnh Dng (a.k.a. - PowerPoint PPT Presentation

LIPN, Université Paris 13 TACL summer school, Porquerolles, June 12th, 2019 1/6 λ -defjnable functions and automata theory Nguyễn Lê Thành Dũng (a.k.a. Tito) — nltd@nguyentito.eu

f n 2/6 morally, n x f f x f 2 f f f n Church encodings of natural numbers: A naive syntactic theory of functions: f x The λ -calculus ≈ f ( x ) λ x . t ≈ x �→ t ( x �→ x 2 + 1 )( 42 ) = 42 2 + 1 ( λ x . t ) u = β t { x := u } ≈

A naive syntactic theory of functions: f x Church encodings of natural numbers: 2/6 The λ -calculus ≈ f ( x ) λ x . t ≈ x �→ t ( x �→ x 2 + 1 )( 42 ) = 42 2 + 1 ( λ x . t ) u = β t { x := u } ≈ morally, n ∈ N ⇝ n : f �→ f n = f ◦ . . . ◦ f 2 = λ f . ( λ x . f ( f x ))

3/6 x f f x o is the type of natural numbers o o o Nat o o o o f 2 The simply typed λ -calculus Add a type system : specifjcations for λ -terms t : A → B ≈ “ t is a function from A to B ” Simple types : built from constant o and binary operation → f : o → o x : o f : o → o f x : o f ( f x ) : o

3/6 The simply typed λ -calculus Add a type system : specifjcations for λ -terms t : A → B ≈ “ t is a function from A to B ” Simple types : built from constant o and binary operation → f : o → o x : o f : o → o f x : o f ( f x ) : o 2 = λ f . ( λ x . f ( f x )) : ( o → o ) → ( o → o ) Nat = ( o → o ) → ( o → o ) is the type of natural numbers

1 0 for some Open question! No satisfactory characterization. Nat Nat w/o substitution: extended polynomials (Schwichtenberg 1975) Theorem ( folklore? but not very well-known) For X , X f -defjnable f ifg X is ultimately periodic . 4/6 λ -defjnable functions f : N → N λ -defjnable ifg ∃ A simple type , t : Nat { o := A } → Nat | ∀ n ∈ N , t n = β f ( n ) Question: what are the λ -defjnable functions N → N ?

1 0 for some Theorem ( folklore? but not very well-known) Open question! No satisfactory characterization. For X , X f -defjnable f ifg X is ultimately periodic . 4/6 λ -defjnable functions f : N → N λ -defjnable ifg ∃ A simple type , t : Nat { o := A } → Nat | ∀ n ∈ N , t n = β f ( n ) Question: what are the λ -defjnable functions N → N ? Nat → Nat w/o substitution: extended polynomials (Schwichtenberg 1975)

Theorem ( folklore? but not very well-known) Open question! No satisfactory characterization. ifg X is ultimately periodic . 4/6 λ -defjnable functions f : N → N λ -defjnable ifg ∃ A simple type , t : Nat { o := A } → Nat | ∀ n ∈ N , t n = β f ( n ) Question: what are the λ -defjnable functions N → N ? Nat → Nat w/o substitution: extended polynomials (Schwichtenberg 1975) For X ⊆ N , X = f − 1 ( 0 ) for some λ -defjnable f : N → N

1 0 for some Proof by semantic evaluation Proof sketch of X n determines whether n • to Nat n is a monoid morphism from • n • choose S fjnite: Nat is a fjnite monoid . -defjnable f ifg X is ultimately periodic . Canonical semantics: f , X For X Theorem 5/6 • choose set S , � o � = S , � A → B � = � B � � A � • t : A ⇝ � t � ∈ � A � , e.g. � f x � = � f � ( � x � ) • soundness: t = β u = ⇒ � t � = � u �

Proof by semantic evaluation Canonical semantics: Theorem ifg X is ultimately periodic . 5/6 • choose set S , � o � = S , � A → B � = � B � � A � • t : A ⇝ � t � ∈ � A � , e.g. � f x � = � f � ( � x � ) • soundness: t = β u = ⇒ � t � = � u � For X ⊆ N , X = f − 1 ( 0 ) for some λ -defjnable f : N → N Proof sketch of ( = ⇒ ) . • choose S fjnite: � Nat � is a fjnite monoid • n �→ � n � is a monoid morphism from ( N , +) to � Nat � • � n � determines whether n ∈ X

Finally: connections with automata Theorem (Hillebrand & Kanellakis 1995) ifg X is a regular language . Same proof (characterize reg. lang. by monoids). 6/6 Generalization to Church-encoded words over fjnite alphabet Σ : For L ⊆ Σ ∗ , L = f − 1 ( ε ) for some λ -defjnable f : Σ ∗ → Σ ∗ λ -defjnable languages are recognizable by fjnite automata. λ -defjnable functions are regularity-preserving . − → I’m looking for an automata-theoretic characterization.