Semantics Inductive/coinductive definitions Computational content Simultaneous inductive/coinductive definition of continuous functions Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Mathematisches Institut, LMU, M¨ unchen Schloß Dagstuhl, 10. Oktober 2011 Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Inductive/coinductive definitions Computational content Main points ◮ Semantics for constructive analysis based on the Scott/Ershov partial continuous functionals. ◮ Simultaneous inductive/coinductive definition of (uniformly) continuous functions. ◮ Extraction of computational content from proofs in an abstract theory. Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Corecursion Inductive/coinductive definitions Free algebras Computational content Partial continuous functionals Corecursion (1/2) Example: transformation of an “abstract” real in I := [ − 1 , 1] into a stream representation using “signed digits” − 1 , 0 , 1. ◮ Assume an abstract (axiomatic) theory of reals, having an unspecified type ρ , and a type σ for rationals. ◮ Assume that the abstract theory provides us with a function g : ρ → σ → σ → B comparing a real x with a proper rational interval p < q : g ( x , p , q ) = tt → x ≤ q , g ( x , p , q ) = ff → p ≤ x . Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Corecursion Inductive/coinductive definitions Free algebras Computational content Partial continuous functionals Corecursion (2/2) ◮ From g define a function h : ρ → U + ρ + ρ + ρ by if g ( x , − 1 inl ( inl ( inr (2 x + 1))) 2 , 0) = tt if g ( x , − 1 2 , 0) = ff, g ( x , 0 , 1 h ( x ) := inl ( inr (2 x )) 2 ) = tt if g ( x , 0 , 1 inr (2 x − 1) 2 ) = ff ◮ h is definable by a closed term M h in G¨ odel’s T . Then the desired function f transforming an abstract real x into a stream can be defined by f ( x ) := co R ρ I xM h . Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Corecursion Inductive/coinductive definitions Free algebras Computational content Partial continuous functionals Free algebras ◮ N with constructors 0, S . ◮ I with constructors I (for [ − 1 , 1]) and C − 1 , C 0 , C 1 (for the left, middle, right part of the interval, of half its length). For example, C − 1 I , C 0 I and C 1 I denote [ − 1 , 0], [ − 1 2 , 1 2 ] and [0 , 1]. ◮ R ( α ) with constructors R d : α → R ( α ) , R : R ( α ) → R ( α ) → R ( α ) → R ( α ) . Using R ( α ) define W with constructors W 0 : W , W : R ( W ) → W . Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Corecursion Inductive/coinductive definitions Free algebras Computational content Partial continuous functionals Information systems C ρ for partial continuous functionals ◮ Types ρ : from algebras ι by ρ → σ . ◮ C ρ := ( C ρ , Con ρ , ⊢ ρ ). ◮ Tokens a (= atomic pieces of information): constructor trees C a ∗ 1 , . . . a ∗ n with a ∗ i a token or ∗ . Example: S ( S ∗ ). ◮ Formal neighborhoods U : { a 1 , . . . , a n } , consistent. ◮ Entailment U ⊢ a . Ideals x ∈ | C ρ | (“points”, here: partial continuous functionals): consistent deductively closed sets of tokens. Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Corecursion Inductive/coinductive definitions Free algebras Computational content Partial continuous functionals Tokens and entailment for N ... S ( S ( S 0)) • ❅ � ❅ � ❅ � S ( S 0) • • S ( S ( S ∗ )) ❅ � ❅ � ❅ � • • S ( S ∗ ) S 0 ❅ � ❅ � ❅ � • • 0 S ∗ Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Corecursion Inductive/coinductive definitions Free algebras Computational content Partial continuous functionals Constructors as continuous functions ◮ Every constructor C generates an ideal in the function space: r C := { ( U , C a ∗ ) | U ⊢ a ∗ } . Associated continuous map: | r C | ( x ) = { C a ∗ | ∃ U ⊆ x ( U ⊢ a ∗ ) } . ◮ Constructors are injective and have disjoint ranges: | r C | ( � x ) ⊆ | r C | ( � y ) ↔ � x ⊆ � y , | r C 1 | ( � x ) ∩ | r C 2 | ( � y ) = ∅ . ◮ Both properties are false for flat information systems (for them, by monotonicity, constructors need to be strict). | r C | ( ∅ , y ) = ∅ = | r C | ( x , ∅ ) , | r C 1 | ( ∅ ) = ∅ = | r C 2 | ( ∅ ) . Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Corecursion Inductive/coinductive definitions Free algebras Computational content Partial continuous functionals Total and cototal ideals of base type ◮ Total ideals of I : 2 k , k := [ i 2 k − 1 2 k , i 2 k + 1 for − 2 k < i < 2 k . 2 k ] I i ◮ Cototal ideals of I : reals in [ − 1 , 1], in (non-unique) stream representation using signed digits − 1 , 0 , 1. Examples: { C n − 1 I | n ≥ 0 } (representing the real − 1), { I } ∪ { C n 1 C − 1 I | n ≥ 0 } (representing 0), { I } ∪ { C n − 1 C 1 I | n ≥ 0 } (representing 0). ◮ Cototal ideals x : every constructor tree P ( ∗ ) ∈ x has a “ ≻ 1 -successor” P ( C � ∗ ) ∈ x . ◮ Total ideals: the cototal ones with ≻ 1 well-founded. Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics Corecursion Inductive/coinductive definitions Free algebras Computational content Partial continuous functionals W and continuous real functions ◮ Consider a well-founded “read tree”, i.e., a constructor tree built from R (ternary) with R d at its leaves. The digit d at a leaf means that, after reading all input digits on the path leading to the leaf, the output d is written. ◮ Let R d 1 , . . . , R d n be all leaves. At a leaf R d i continue with W (i.e., write d i ), and continue reading. ◮ Result: a “ W -cototal R ( W )-total” ideal, representing a uniformly continuous real function f : I → I . Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics A formal theory of computable functionals Inductive/coinductive definitions Cototality Computational content Real functions Formalization ◮ TCF : theory of computable functionals. ◮ Minimal logic ( → , ∀ ), plus inductive & coinductive definitions. ◮ Variables range over partial continuous functionals. ◮ Constants denote computable functionals. ◮ Terms: from T + , a common extension of G¨ odel’s T and Plotkin’s PCF . Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics A formal theory of computable functionals Inductive/coinductive definitions Cototality Computational content Real functions Cototality for N ◮ In the algebra N totality is inductively defined by the clauses T N 0 , T N m → T N ( S m ) . ◮ As for every inductively defined predicate we have a corresponding coinductively defined one, written co T : co T N n → n = 0 ∨ ∃ m ( n = S m ∧ co T N m ) . ◮ Its greatest-fixed-point axiom (coinduction) is Pn → ∀ n ( Pn → n = 0 ∨∃ m ( n = S m ∧ ( co T N ∨ P ) m )) → co T N n . It expresses that every “competitor” P satisfying the same clause is a subset of co T N . Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics A formal theory of computable functionals Inductive/coinductive definitions Cototality Computational content Real functions Cototality for I ◮ In the algebra I of standard rational intervals cototality is coinductively defined by co T I x → x = I ∨ ∃ y ( x = C − 1 y ∧ co T I y ) ∨ ∃ y ( x = C 0 y ∧ co T I y ) ∨ ∃ y ( x = C 1 y ∧ co T I y ) . ◮ A model is provided by the set of all finite or infinite streams of signed digits − 1 , 0 , 1, i.e., the (non-unique) stream representation of real numbers. Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
Semantics A formal theory of computable functionals Inductive/coinductive definitions Cototality Computational content Real functions A simultaneous inductive/coinductive definition (1/4) ◮ Example: formalization of an abstract theory of (uniformly) continuous real functions f : I → I where I := [ − 1 , 1]. ◮ Let Cf express that f is a continuous real function, and I p , k := [ p − 2 − k , p + 2 − k ]. Assume the abstract theory proves Cf → ∀ k ∃ l B l , k f , with B l , k f := ∀ p ∃ q ( f [ I p , l ] ⊆ I q , k ). ◮ Let I − 1 := [ − 1 , 0], I 0 := [ − 1 2 , 1 2 ] and I 1 := [0 , 1]. Define in d , out d such that in d [ I ] = I d and out d [ I d ] = I by in d ( x ) := d + x , out d ( x ) := 2 x − d . 2 Both functions are inverse to each other. Helmut Schwichtenberg (j.w.w. Kenji Miyamoto) Sim. inductive/coinductive definition of continuous functions
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