A theory of computable functionals Helmut Schwichtenberg - PowerPoint PPT Presentation

A theory of computable functionals Helmut Schwichtenberg Mathematisches Institut, LMU, M¨ unchen University of Canterbury, Christchurch, 12 Feb 2016 1 / 29

◮ Formulas and predicates ◮ A theory of computable functionals ◮ Brouwer - Heyting - Kolmogorov and decorations ◮ The type of a formula or predicate ◮ Realizability ◮ Extracted terms 2 / 29

Simultaneously define formulas and predicates A , B ::= P � r | A → B | ∀ x A , P , Q ::= X | { � x | A } | µ X ( ∀ � x i (( A i ν ) ν< n i → X � r i )) i < k Need restriction: X at most strictly positive in A i ν . 3 / 29

T N := µ X ( X 0 , ∀ n ( Xn → X ( S n ))) , Even := µ X ( X 0 , ∀ n ( Xn → X ( S ( S n )))) , := µ X ( ∀ x Xxx ) , Eq Ex Y := µ X ( ∀ x ( Yx → X )) , Cap Y , Z := µ X ( ∀ � x ( Y � x → Z � x → X � x )) , Cup Y , Z := µ X ( ∀ � x ( Y � x → X � x ) , ∀ � x ( Z � x → X � x )) . Abbreviations ∃ x A := Ex { x | A } , P ∩ Q := Cap P , Q , P ∪ Q := Cup P , Q . 4 / 29

◮ Formulas and predicates ◮ A theory of computable functionals ◮ Brouwer - Heyting - Kolmogorov and decorations ◮ The type of a formula or predicate ◮ Realizability ◮ Extracted terms 5 / 29

Relation to type theory ◮ Main difference: partial functionals are first class citizens. ◮ “Logic enriched”: Formulas and types kept separate. ◮ Minimal logic: → , ∀ only. Eq ( x , y ) (Leibniz equality), ∃ , ∨ , ∧ inductively defined (Russell, Martin-L¨ of). ◮ F := Eq (ff , tt). Ex-falso-quodlibet: F → A provable. ◮ “Decorations” → nc , ∀ nc (i) allow abstract theory (ii) remove unused data. 6 / 29

Theory of computable functionals TCF Typed variables, ranging over the partial continuous functionals. Minimal logic, with intro and elim for → and ∀ . Axioms: ◮ I + i : ∀ � x (( A ν ( I )) ν< n → I � r ) ◮ I − : ∀ � x ( I � x i (( A i ν ( I ∩ X )) ν< n i → X � r i )) i < k → X � x → ( ∀ � x ) Induction = elimination for totality over N . T − N : ∀ n ( T N n → X 0 → ∀ n ( T N n → Xn → X ( S n )) → Xn ) . Remarks ◮ Every “competitor” X satisfying the clauses contains T N . ◮ Induction for N , which only holds for total numbers. ◮ Fits the logical elimination rules (main part comes first). ◮ “Strengthened” step formula ∀ n ( T N n → Xn → X ( S n )). 7 / 29

For nullary predicates P = { | A } and Q = { | B } we write A ∧ B for P ∩ Q and A ∨ B for P ∪ Q . Introduction axioms: ∀ x ( A → ∃ x A ) , A → B → A ∧ B , A → A ∨ B , B → A ∨ B . Elimination axioms: ∃ x A → ∀ x ( A → B ) → B ( x / ∈ FV ( B )) , A ∧ B → ( A → B → C ) → C , A ∨ B → ( A → C ) → ( B → C ) → C . 8 / 29

Equalities (i) Defined function constants D introduced by computation rules, written ℓ = r , but intended as left-to-right rewrites. (ii) Leibniz equality Eq (inductively defined). (iii) Pointwise equality between partial continuous functionals, defined inductively as well. (iv) If ℓ and r have a finitary algebra as their type, ℓ = r by (i) is a boolean term. Take Eq (( ℓ = r ) B , tt). In TCF formulas A ( r ) and A ( s ) are identified if r , s ∈ T + have a common reduct. 9 / 29

Eq + : ∀ x Eq ( x ρ , x ρ ) Eq − : ∀ x , y ( Eq ( x , y ) → ∀ x Xxx → Xxy ) . Compatibility of Eq : ∀ x , y ( Eq ( x , y ) → A ( x ) → A ( y )). (Use Eq − with { x , y | A ( x ) → A ( y ) } for X .) 10 / 29

Define falsity by F := Eq (ff , tt). Ex-falso-quodlibet: TCF ⊢ F → A where A has no strictly positive occurrences of (i) predicate variables (ii) inductive predicates without nullary clauses. Proof. 1. Show F → Eq ( x ρ , y ρ ). Eq ( R ρ B ff xy , R ρ by Eq + B ff xy ) Eq ( R ρ B tt xy , R ρ B ff xy ) by compatibility from Eq (ff , tt) Eq ( x ρ , y ρ ) by conversion . 2. Show F → A , by induction on A . Case I � s . Let K 0 be the nullary clause, with final conclusion I � t . By IH from F we can derive all parameter premises, hence I � t . From F we also have Eq ( s i , t i ) by 1. Hence I � s by compatibility. The cases A → B and ∀ x A are obvious. 11 / 29

◮ Formulas and predicates ◮ A theory of computable functionals ◮ Brouwer - Heyting - Kolmogorov and decorations ◮ The type of a formula or predicate ◮ Realizability ◮ Extracted terms 12 / 29

Brouwer - Heyting - Kolmogorov Have → ± , ∀ ± , I ± . BHK-interpretation: ◮ p proves A → B if and only if p is a construction transforming any proof q of A into a proof p ( q ) of B . ◮ p proves ∀ x ρ A ( x ) if and only if p is a construction such that for all a ρ , p ( a ) proves A ( a ). Leaves open: ◮ What is a “construction”? ◮ What is a proof of a prime formula? Proposal: ◮ Construction: computable functional. ◮ Proof of a prime formula I � r : generation tree. Example: generation tree for Even (6) should consist of a single branch with nodes Even (0), Even (2), Even (4) and Even (6). 13 / 29

Decoration x and assumptions � Which of the variables � A are actually used in the “solution” provided by a proof of x ( � ∀ � A → I � r )? To express this we split each of → , ∀ into two variants: ◮ a “computational” one → c , ∀ c and ◮ a “non-computational” one → nc , ∀ nc (with restricted rules) and consider A → nc � B → c X � y ( � ∀ nc x ∀ c r ) . � � This will lead to a different (simplified) algebra ι I associated with the inductive predicate I . 14 / 29

Each inductive predicate is marked as computationally relevant (c.r.) or non-computational (n.c., or Harrop): µ nc X ( K 0 , . . . , K k − 1 ). Then the elimination scheme must be restricted to n.c. formulas. We usually write → , ∀ , µ for → c , ∀ c , µ c . Notice that in the clauses X � of an n.c. inductive predicate µ nc K decorations play no role. For the even numbers we now have two variants: Even := µ X ( X 0 , ∀ nc n ( Xn → X ( S ( S n )))) , Even nc := µ nc X ( X 0 , ∀ n ( Xn → X ( S ( S n )))) . Generally for every c.r. inductive predicate I (i.e., defined as µ X � K ) we have an n.c. variant I nc defined as µ nc X � K . 15 / 29

ExD Y := µ X ( ∀ x ( Yx → X )) , ExL Y := µ X ( ∀ x ( Yx → nc X )) . ExR Y := µ X ( ∀ nc x ( Yx → X )) , ExU Y := µ nc X ( ∀ x ( Yx → X )) . D for “double”, L for “left”, R for “right”, U for “uniform”. Abbreviations ∃ d x A := ExD { x | A } , ∃ l x A := ExL { x | A } , ∃ r x A := ExR { x | A } , ∃ u x A := ExU { x | A } . 16 / 29

CupD Y , Z := µ X ( Y → X , Z → X ) , := µ X ( Y → X , Z → nc X ) , CupL Y , Z := µ X ( Y → nc X , Z → X ) , CupR Y , Z := µ X ( Y → nc X , Z → nc X ) , CupU Y , Z CupNC Y , Z := µ nc X ( Y → X , Z → X ) . The final nc-variant suppresses even the information which clause has been used. Abbreviations A ∨ d B := CupD {| A } , {| B } , A ∨ l B := CupL {| A } , {| B } , A ∨ r B := CupR {| A } , {| B } , A ∨ u B := CupU {| A } , {| B } , A ∨ nc B := CupNC {| A } , {| B } . 17 / 29

◮ Formulas and predicates ◮ A theory of computable functionals ◮ Brouwer - Heyting - Kolmogorov and decorations ◮ The type of a formula or predicate ◮ Realizability ◮ Extracted terms 18 / 29

Examples Let a , b ∈ Q , x ∈ R , k ∈ Z , f ∈ R → R . ◮ ∀ a , b , x ( a < b → x ≤ b ∨ u a ≤ x ) has type Q → Q → R → B . ◮ ∀ a , b , x ( a < b → x < b ∨ d a < x ) has type Q → Q → R → Z + Z . ◮ The formula ∀ f , k ( f (0) ≤ 0 ≤ f (1) → � 1 � ∀ a , b 2 k | b − a | ≤ | f ( b ) − f ( a ) | → ∃ l x f ( x )=0) has type ( R → R ) → Z → R . 19 / 29

The type τ ( C ) of a formula or predicate C τ ( C ) type or the “nulltype symbol” ◦ . Extend use of ρ → σ to ◦ : ( ρ → ◦ ) := ◦ , ( ◦ → σ ) := σ, ( ◦ → ◦ ) := ◦ . Assume a global injective assignment of a type variable ξ to every c.r. predicate variable X . Let τ ( C ) := ◦ if C is non-computational. In case C is c.r. let τ ( P � r ) := τ ( P ) , τ ( A → nc B ) := τ ( B ) , τ ( A → B ) := ( τ ( A ) → τ ( B )) , τ ( ∀ nc τ ( ∀ x ρ A ) := ( ρ → τ ( A )) , x ρ A ) := τ ( A ) , τ ( X ) := ξ, τ ( { � x | A } ) := τ ( A ) , A i → nc � y i ( � y i ) → τ ( � τ ( µ X ( ∀ nc B i → X � ) := µ ξ ( τ ( � x i ∀ � r i )) i < k B i ) → ξ ) i < k . � � �� � � �� � ι I I ι I is the algebra associated with I . 20 / 29

◮ Formulas and predicates ◮ A theory of computable functionals ◮ Brouwer - Heyting - Kolmogorov and decorations ◮ The type of a formula or predicate ◮ Realizability ◮ Extracted terms 21 / 29

Realizability For every predicate or formula C we define an n.c. predicate C r . For n.c. C let C r := C . In case C is c.r. the arity of C r is ( τ ( C ) ,� σ ) with � σ the arity of C . For c.r. formulas define r ) r := { u | P r u � ( P � r } � { u | ∀ v ( A r v → B r ( uv )) } if A is c.r. ( A → B ) r := { u | A → B r u } if A is n.c. ( A → nc B ) r := { u | A → B r u } ( ∀ x A ) r := { u | ∀ x A r ( ux ) } x A ) r := { u | ∀ x A r u } . ( ∀ nc For c.r. predicates: given n.c. X r for all predicate variables X . x | A } r := { u ,� x | A r u } . { � 22 / 29