Minimal logic for computable functionals Helmut Schwichtenberg Mathematisches Institut der Universit¨ at M¨ unchen
Contents ◮ 1. Partial continuous functionals ◮ 2. Total and structure-total functionals ◮ 3. Terms; denotational and operational semantics ◮ 4. Adequacy ◮ 5. Implementation, computational content of proofs
1. Partial continuous functionals (Finite) types: ρ, σ ::= µ | ρ ⇒ σ . Naive model: full set theoretic hierarchy of functionals of finite types. Leads to higher cardinalities. A more appropriate semantics for typed languages has its roots in work of Kreisel (1959) (who used formal neighborhoods) and Kleene (1959). Developed in a mathematically more satisfactory way by Scott and Ershov (early 1970s). Today this theory is usually presented in the context of abstract domain theory; it is based on classical logic. Here: an attempt to develop a constructive theory of formal neighborhoods for continuous functionals, in a direct and intuitive style. Replace abstract domain theory by a more concrete and (in the case of finitary free algebras) finitary theory of representations. As a framework for this we use Scott’s information systems (1982).
References G. Kreisel. Interpretation of analysis by means of constructive functionals of finite types. In A. Heyting, editor, Constructivity in Mathematics , pages 101–128, 1959. S. C. Kleene. Countable functionals. Same volume, pages 81–100. Y. L. Ershov. Everywhere defined continuous functionals. Algebra i Logika , 11(6):656–665, 1972. D. Scott. A type theoretical alternative to ISWIM, CUCH, OWHY. 1969. Published in Theoret. Comput. Sci. 121 (1993), 411–440. D. Scott. Domains for denotational semantics. In E. Nielsen and E.M. Schmidt, editors, Automata, Languages and Programming , volume 140 of LNCS , pages 577–613, 1982.
Partial continuous functionals (continued) Information systems (Scott 1982) provide an intuitive approach to deal constructively with ideal, infinite objects in function spaces, by means of their finite approximations. One has ◮ atomic units of information, called tokens, ◮ a notion of consistency for finite sets of tokens, and ◮ an entailment relation, between consistent finite sets of tokens and single tokens. The ideals (or “objects”) of an information system are the consistent and deductively closed sets of tokens. We define the partial continuous functionals via information systems. We will only need to deal with atomic (a concept introduced by Berger) and coherent (Plotkin 1978) information systems, which allows some simplifications.
Atomic coherent information systems (acis) An acis is a triple ( A , Con , ≥ ) with A a countable set (tokens), Con a nonempty set of finite subsets of A (consistent sets), and ≥ a transitive and reflexive relation on A (entails) which satisfy (a) ∅ ∈ Con , and { a } ∈ Con for every a ∈ A , (b) X ∈ Con iff every two-element subset of X is in Con , and (c) if { a , b } ∈ Con and b ≥ c , then { a , c } ∈ Con . Write X ≥ a for ∃ b ∈ X b ≥ a , and X ≥ Y for ∀ a ∈ Y X ≥ a .
Function spaces Lemma (Acis’s are information systems) If X ≥ Y 1 , Y 2 , then Y 1 ∪ Y 2 ∈ Con . Definition (Function space) Let A = ( A , Con A , ≥ A ) and B = ( B , Con B , ≥ B ) be acis’s. Define A → B := ( C , Con , ≥ ) with C := Con A × B , { ( X 1 , b 1 ) , . . . ( X n , b n ) } ∈ Con : ↔ � � ∀ i , j X i ∪ X j ∈ Con A → { b i , b j } ∈ Con B , ( X , b ) ≥ ( Y , c ) : ↔ Y ≥ A X ∧ b ≥ B c . Lemma A → B is an acis again.
Approximable maps The ideals (or “objects”) of an information system are the consistent and deductively closed sets of tokens; write | A | for the set of ideals of A . Lemma Let A and B be acis’s. The ideals of A → B are exactly the approximable maps from A to B , that is, the relations r ⊆ Con A × B such that (a) if r ( X , b 1 ) and r ( X , b 2 ) , then { b 1 , b 2 } ∈ Con B , and (b) if r ( X , b ) , Y ≥ A X and b ≥ B c, then r ( Y , c ) . Proof. Scott 1982.
Continuity The set | A | of ideals for A carries a natural topology (the Scott topology), which has the deductive closures X of Con -sets X as basis. The continuous maps f : | A | → | B | and the ideals r ∈ | A → B | are in a bijective correspondence: ◮ With any r ∈ | A → B | we can associate a continuous | r | : | A | → | B | : | r | ( z ) := { b ∈ B | r ( X , b ) for some X ⊆ fin z } , ◮ and with any continuous f : | A | → | B | we can associate ˆ f ∈ | A → B | : ˆ f ( X , b ) : ⇐ ⇒ b ∈ f ( X ) . These assignments are inverse to each other, i.e., f = | ˆ f | and r = � | r | .
Algebras We consider free algebras (= data types), given by constructors. Definition (Inductive, of types ρ and constructor types κ ) Let � α = ( α j ) j =1 ,..., N be a list of distinct type variables. σ n ∈ T � ρ,� σ 1 , . . . ,� ( n ≥ 0) � ρ ⇒ ( � σ 1 ⇒ α j 1 ) ⇒ . . . ⇒ ( � σ n ⇒ α j n ) ⇒ α j ∈ KT ( � α ) κ 1 , . . . , κ n ∈ KT ( � α ) ρ, σ ∈ T ( n ≥ 1) ( µ� α ( κ 1 , . . . , κ n )) j ∈ T ρ ⇒ σ ∈ T The parameter types of µ are the members of all � ρ appearing in its constructor types κ 1 , . . . , κ k .
Algebras: examples U := µα α, Unit := µα ( α, α ) , Booleans B := µα ( α, α ⇒ α ) , N Natural numbers L ( ρ ) := µα ( α, ρ ⇒ α ⇒ α ) , Lists ρ ⊗ σ := µα ρ ⇒ σ ⇒ α, (Tensor) product ρ + σ := µα ( ρ ⇒ α, σ ⇒ α ) , Sum ( tree , tlist ) := µ ( α, β ) ( N ⇒ α, β ⇒ α, β, α ⇒ β ⇒ β ) , Bin := µα ( α, α ⇒ α ⇒ α ) , Binary trees O := µα ( α, α ⇒ α, ( N ⇒ α ) ⇒ α ) , Ordinals T 0 := N , T n +1 := µα ( α, ( T n ⇒ α ) ⇒ α ) . Trees
Algebras with approximations The acis of an algebra µ j , given by constructors C i . Tokens: ◮ a special one – written ∗ –, which carries no information; ◮ all type correct constructor expressions with an outermost C i , where at any finitary argument position we have a token, and at any other argument position we have a Con -set. Two tokens are in the entailment relation ≥ if either the right hand one is ∗ , or they start with the same constructor, and for each finitary argument position the argument tokens a , b located there satisfy a ≥ b , and at any other argument position the Con -sets X and Y located there satisfy X ≥ Y . A finite set of tokens is consistent if each two-element subset is; two tokens are consistent if one of them is ∗ , or both start with the same constructor and have consistent tokens resp. Con -sets at corresponding argument positions.
Tokens for the algebra N ... • S ( S ( S 0)) ❅ � ❅ � ❅ � • • S ( S ( S ∗ )) S ( S 0) ❅ � ❅ � ❅ � • • S ( S ∗ ) S 0 ❅ � ❅ � ❅ � • • 0 S ∗ ❅ � ❅ � ∗ ❅ � • A token a entails b iff there is a path from a (up) to b (down). In this case (and similarly for every finitary algebra) a finite set X of tokens is consistent iff it has an upper bound.
Constructors as continuous functionals Every constructor C generates r C := { ( � X , b ) | b = ∗ , or b = C � b with X i ≥ b i , b i token or Con -set } . The continuous map | r C | is defined by X ⊆ fin � z ) := { b | ( � X , b ) ∈ r C for some � | r C | ( � z } . Hence the (continuous maps corresponding to) constructors are injective and their ranges are disjoint.
Ideals for N and their inclusion relation ... • ∞ • S ( S ( S 0)) ❅ � ❅ � ❅ � S ( S 0) • • S ( S ( S ⊥ )) ❅ � ❅ � ❅ � • • S ( S ⊥ ) S 0 ❅ � ❅ � ❅ � • • S ⊥ 0 ❅ � ❅ � ❅ � • ⊥ Ideals x for µ : consistent and deductively closed sets of tokens. All non- ∗ tokens in x begin with the same constructor. For instance, { S ( S 0) , S ( S ∗ ) , S ∗ , ∗} , { S ( S ∗ ) , S ∗ , ∗} , { 0 , ∗} , {∗} are ideals for N , but also the infinite set { S n ∗ | n ≥ 0 } . ⊥ , 0, ∞ denote {∗} , { 0 , ∗} , { S n ∗ | n ≥ 0 } .
2. Total and structure-total functionals The total ideals of type ρ are defined inductively. ◮ Case µ . All | r C | ( � z ) with � z total. ◮ Case ρ ⇒ σ . An ideal r of type ρ ⇒ σ is total iff for all total z of type ρ , the result | r | ( z ) of applying r to z is total. Structure-total ideals are defined similarly; the difference is that in case µ the ideals at parameter positions of C need not be total. For N the ideals 0, S 0, S ( S 0) etc. are total, but ⊥ , S ⊥ , S ( S ⊥ ), . . . , ∞ are not. For L ( ρ ), precisely all ideals of the form Cons ( x 1 , . . . Cons ( x n , Nil ) . . . ) are structure-total. The total ones are those where in addition all list elements x 1 , . . . , x n are total. x ∈ G ρ means x is a total ideal of type ρ (Ershov’s notation).
Induction is valid for total and structure-total ideals. Examples (all variables range over total ideals): Ind p , A : A [ p := tt ] → A [ p := ff ] → ∀ p B A , Ind n , A : A [ n := 0] → ∀ n ( A → A [ n := S n ]) → ∀ n N A , Ind l , A : A [ l := Nil ] → ∀ x , l ( A → A [ l := Cons ( x , l )]) → ∀ l L ( α ) A Ind x , A : ∀ y 1 A [ x := Inl ( y 1 )] → ∀ y 2 A [ x := Inr ( y 2 )] → ∀ x ρ 1+ ρ 2 A . Induction over the structure-total ideals is defined similarly. For instance, in list induction Ind l , A we can let x range over arbitrary ideals, and l over the structure-total ones.
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