Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Algebraic Datatypes are Horn-Clause Theories František Farka University of Dundee, and University of St Andrews ff32@st-andrews.ac.uk 8 December 2015 8 December 2015 1/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Introduction Logic programming Programming in Horn-clause logic Goals resolved by a search - SLD resolution Automated theorem proving (ATP) Functional programming Program specified by a term Type of a term is a proposition Interactive theorem proving (ITP) 8 December 2015 2/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Introduction Logic programming Programming in Horn-clause logic Goals resolved by a search - SLD resolution Automated theorem proving (ATP) Functional programming Program specified by a term Type of a term is a proposition Interactive theorem proving (ITP) How are the two related? 8 December 2015 2/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Introduction (cont) Propositions as Types Due to Barendregt, 1991 Relating lambda calculi and different logics λω λ C Prop ω Pred ω λ 2 λΠ 2 Prop 2 Pred 2 λω λΠω Prop ω Pred ω λ → λΠ Prop Pred 8 December 2015 3/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Propositional Logic Programming Propositional Logic Programming Infinite set of elementary propositions P , propositions denoted nat , bool , . . . A program is a set of Horn-clauses, i. e. clauses in the form H ← B 1 , . . . B n Resolution step: P ⊢ B 1 , . . . , P ⊢ B n A ← B 0 . . . B n ∈ P P ⊢ A 8 December 2015 4/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Propositional Logic Programming Propositional Logic Programming Infinite set of elementary propositions P , propositions denoted nat , bool , . . . A set of clause names α, β 1 , . . . equipped with arity ( ar ( α ) = 1, . . . ) A program is a set of Horn-clauses, i. e. clauses in the form α : H ← B 1 , . . . B n where ar ( α ) = n Resolution step: P ⊢ β 1 : B 1 , . . . , P ⊢ β n : B n α : A ← B 0 . . . B n ∈ P P ⊢ α ( β 1 , . . . , β n ) : A 8 December 2015 4/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Propositional Logic Programming (cont.) A proof in PLP Success tree - all leafs are empty goals Applicative term as a proof Example Resolution in P nat = { ζ : nat , σ : nat ← nat } ζ : nat P ⊢ ζ : nat σ : nat ← nat P ⊢ σ ( ζ ) : nat 8 December 2015 5/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Propositional Logic Programming (cont.) A proof in PLP Success tree - all leafs are empty goals Note that success tree can have infinite branches (coinductive int.) Applicative term as a proof Example Resolution in P nat = { ζ : nat , σ : nat ← nat } ζ : nat P ⊢ ζ : nat σ : nat ← nat P ⊢ σ ( ζ ) : nat . . . σ : nat ← nat P ⊢ σ ( . . . ) : nat σ : nat ← nat P ⊢ σ ( σ ( . . . )) : nat 8 December 2015 5/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Propositional Logic Programming (cont.) Theorem: Inductive soundness and completeness A proposition A is in the least model M P of a program P iff there is a finite term π s. t. P ⊢ π : A Theorem: Coinductive soundness A proposition A is in the greatest model M ω P of a program P if there is an infinite term π s. t. P ⊢ π : A 8 December 2015 6/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Propositional Logic Programming (cont.) Theorem: Inductive soundness and completeness A proposition A is in the least model M P of a program P iff there is a finite term π s. t. P ⊢ π : A Theorem: Coinductive soundness A proposition A is in the greatest model M ω P of a program P if there is an infinite term π s. t. P ⊢ π : A Models are as usual in LP: T P ( X ) = { A | α : A ← B 0 , . . . , B n − 1 ∈ P && ∀ i ∈ 0 , . . . , n − 1 B i ∈ X } And M P = µ ( T P ) and M ω P = ν ( T P ) 8 December 2015 6/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Algebraic Datatypes Simply Typed Lambda Calculus ( Λ → ) Infinite set V of variables ( x , y , . . . ), and infinite set B of type variables/identifiers: α, β, . . . Function types: σ → τ Lambda abstraction, for y : σ the expression ( λ x : τ. y ) is of type τ → σ Application, for x : σ → τ and y : σ is xy : τ 8 December 2015 7/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Algebraic Datatypes Simply Typed Lambda Calculus ( Λ → ) Infinite set V of variables ( x , y , . . . ), and infinite set B of type variables/identifiers: α, β, . . . , n at, b ool Function types: σ → τ Lambda abstraction, for y : σ the expression ( λ x : τ. y ) is of type τ → σ Application, for x : σ → τ and y : σ is xy : τ Algebraic Datatypes Constructors and eliminators/destructors for algebraic data types 8 December 2015 7/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Algebraic Datatypes (cont.) Algebraic Datatypes Algebraic type is a type variable α and a set C of i constructors c j Each constructor is equipped with arity n and with a n-tuple of ADTs Inference rules: Γ ⊢ t 1 : β j , 1 , . . . , Γ ⊢ t ar ( c j ) : β j , ar ( c j ) CON c j Γ ⊢ c j t 1 . . . t ar ( c j ) : α for j = 0 , . . . , i − 1, and Γ ⊢ t : α Γ, x 0 : β 0 , 1 , . . . , x ar ( c 0 ) : β 0 , ar ( c 0 ) ⊢ s 0 : γ , . . . , Γ, x i − 1 : β i − 1 , 1 , . . . , x ar ( c i − 1 ) : β i − 1 , ar ( c i − 1 ) ⊢ s i − 1 : γ CASE α Γ ⊢ case t of ( s 0 , . . . , s i − 1 ) : γ 8 December 2015 8/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Algebraic Datatypes (cont.) Example (Algebraic Datatypes) data Bool where true : () → Bool false : () → Bool data Nat where zero : Nat succ : Nat → Nat two : Nat two = succ ( succ zero ) prec : Nat → Nat prec x = case x of → zero zero succ x 0 → x 0 8 December 2015 9/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion ADTs are Horn-Clause Theories Translating ADTs to PLP, map | · | Let there be an isomorphism of type variables B and propositions P and of constructors and clause names Then for each constructor c j s. t. c j : ( β 1 , . . . , β n ) → α | c j | = γ j : A ← B 1 , . . . , B n Example ( N at and B ool) | Nat | = { ζ : nat ; σ : nat ← nat } , | Bool | = { τ : bool ; φ : bool } 8 December 2015 10/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion ADTs are Horn-Clause Theories Translating ADTs to PLP, map | · | Let there be an isomorphism of type variables B and propositions P and of constructors and clause names Then for each constructor c j s. t. c j : ( β 1 , . . . , β n ) → α | c j | = γ j : A ← B 1 , . . . , B n Example ( N at and B ool) | Nat | = { ζ : nat ; σ : nat ← nat } , | Bool | = { τ : bool ; φ : bool } Lemma: PLP resolution for ADTs For an ADT α there exist a term t such that Γ ⊢ λ → t : α iff | Γ | ⊢ PLP τ : A for some proof τ 8 December 2015 10/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Resolution with And-Or Trees And-Or Trees Due to Komendantskaya and Johann, 2015 even ζ : even ǫ : even ← odd ⊥ ǫ Example For a program P: odd ζ : even ⊥ ζ ⊥ o ǫ : odd ← even ǫ : even ← odd o : odd ← even even resolve goal even ζ : even ǫ : even ← odd ⊥ ǫ . . . 8 December 2015 11/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Resolution with And-Or Trees (cont.) Inductive success A finite subtree; all children in and-nodes any one child in or-nodes. ǫ ( o ( ζ )) : even Example ζ : even ǫ : even ← odd ⊥ ǫ For a program P: o ( ζ ) : odd ζ : even ǫ : even ← odd ⊥ ζ ⊥ o ǫ : odd ← even o : odd ← even ζ : even resolve goal even with ǫ ( o ( ζ )) ζ : even ǫ : even ← odd ⊥ ǫ . . . 8 December 2015 12/20
Introduction Calculi Resolution for ADTs Induction and Coinduction Future Work Discussion Resolution with And-Or Trees (cont.) Inductive success A finite subtree; all children in and-nodes any one child in or-nodes. ǫ ( o ( ζ )) : even Example ζ : even ǫ : even ← odd ⊥ ǫ For a program P: o ( ζ ) : odd ζ : even ǫ : even ← odd ⊥ ζ ⊥ o ǫ : odd ← even o : odd ← even ζ : even resolve goal even with ǫ ( o ( ζ )) ζ : even ǫ : even ← odd ⊥ ǫ . . . 8 December 2015 13/20
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