Algebraic theories in the presence of binding operators, substitution, etc.

Algebraic theories in the presence of binding operators, substitution, etc. Chung Kil Hur Joint work with Marcelo Fiore Computer Laboratory University of Cambridge 20th March 2006

Overview First order syntax: modelled by algebras on Set First order syntax with bindings: modelled by subst-algebras (binding-algebras) on Set F First order syntax with equations: modelled by algebraic theories (lawvere theories, or variety of algebras). First oder syntax with bindings and equations: equational theories by Kelly and Power (problematic). New equational theories.

1 Abstract Syntax with Binding First order syntax without binding First order syntax with binding Substitution Algebras 2 Equational Theories Classical approach New view on equations Examples of equations

Definition and Example Signature for First Order Syntax S = (O, a) O is a set of operators a is an arity function from O to N Terms built from Signature Term V := V V is a set of variables f (Term 1 V,..., Termn V ) f O, a(f ) = n Example: Group O = {e, i, m} a(e) = 0, a(i) = 1, a(m) = 2 V = {x, y, z,...} Term V = {x, e, m(x, m(y, z)), m(e, m(i(x), y)),...}

Categorical Construction Base Category: Set of sets and functions. Endofunctor: ΣX = f O X a(f ) Example: Group ΣX = 1 + X + X 2 State Set Depth Group S 0 S 1 V + ΣS 0 0 x, y, z,..., e S 2 V + ΣS 1 1 x, e, i(e), i(x), m(e, x),... S 3 V + ΣS 2 2 x, e, i(x), m(e, x), m(x, i(e)),....... S Term V <..., m(i(i(x)), m(x, i(e))),... Free Monad of Σ: TX = µa.x + ΣA with X + ΣTX =[η X,τ X] TX

Algebras Definition of Σ-alg objects: (X, s : ΣX X) where X, s in Set morphisms: f : (X, s) (Y, t) where f : X Y in Set satisfying ΣX s X Σf f ΣY t Y Example: Group objects: (X, 1 + X + X 2 [e,i,m] X) morphisms: f : (X, e, i, m) (Y, e, i, m ) where f : X Y 1 e X id 1 f Y e X i X f f Y Y i X 2 m X f 2 f Y 2 m Y

Free Algebras Free Algebras and Free monads (TX, ΣTX τx TX) F X Σ-alg F Set Free monad T = uf u (X, s) X u Universal property ΣTX Σ(i,r) # ΣY ΣTTX Σµ X ΣTX τ X η X TX X (i,r) # i Y r τ TX η TX τ X µ X TTX TX id TX

Definition and Example Signature for First Order Syntax with Binding S = (O, a) O is a set of operators a is an arity function from O to N, i.e.a(f ) = (a 1,..., a k ) Terms built from Signature Term({x 1,..., x n }) := {x 1,..., x n } f (x n+1,..., x n+a1.term({x 1,..., x n+a1 }),. x n+1,..., x n+ak.term({x 1,..., x n+ak })) where f O, a(f ) = (a 1,..., a k ) Example: Untyped λ-calculus O = {lam, app} a(lam) = (1), a(app) = (0, 0) Term(x 1, x 2 ) = {x 1, app(x 1, x 2 ), lam(x 3.app(x 1, app(x 2, x 3 )))...} Term({x 1,..., x n }) = {[t] α FV(t) {x 1,..., x n } in untyped λ-calculus}

Category of Presheaves Definitions F objects finite cardinals, i.e., {x 1 }, {x 1, x 2 },... let n be {x 1,..., x n } morphisms set functions from n to m Set F objects functors from F to Set morphisms natural transformations between functors Interpretations X Set F : a language. X(n): set of terms in context {x 1,..., x n }, i.e.{t x 1,..., x n t} X(n) X(ρ) X(m): x 1,..., x n t x 1,..., x m t[x ρ(1) /x 1,..., x ρ(n) /x n ] Presheaf of variables and Type constructor for context extension V Set F : V(n) = {x 1,..., x n }, V(ρ) = ρ δ : Set F Set F : (δx)(n) = X(n + 1), (δx)(ρ) = X(ρ + 1)

Categorical Construction Base Category: Set F Endofunctor: Σ(X) def = f O 1 i k a(f )=(a i) 1 i k Example: λ-calculus ΣX = δx + X 2 δ ai X State Set {x 1 } {x 1, x 2 } S 0 S 1 V + ΣS 0 x 1 x 1, x 2 S 2 V + ΣS 1 λx 1.x 1 x 1, x 1 x 1, λx 2.x 1, λx 2.x 2 x 1, x 2, x 1 x 2, λx 3.x 3,... S 3 V + ΣS 2 (λx 1.x 1 )(λx 1.x 1 ), λx 1.λx 2.x 1,... x 1,(x 1 x 1 )(λx 2.x 2 ), λx 2.λx 3.x 3,... x 2,(x 1 x 2 )(λx 3.x 3 ), λx 3.λx 4.x 3,........ S Term Term( ) Term({x 1 }) Term({x 1, x 2 }) Free Monad of Σ: TX = µa.x + ΣA with X + ΣTX =[η X,τ X] TX

Algebras Example: λ-calculus objects: (X, δx + X 2 [lam,app] X) morphisms: f : (X, lam, app) (Y, lam, app ) where f : X Y δx lam X δf f δy Y lam X 2 app X f 2 f Y 2 app Y Universality δtv + TV 2 δ(i,lam,app) # +(i,lam,app) #2 δy + Y 2 τ V η V TV V (i,lam,app) # i Y [lam,app]

Motivations for substitution as an algebraic operation We can express operations involving substitution, e.g.β equations in λ-calculus. It ensures semantics substitution lemma, i.e.compositionality of substitution. It automatically verifies some structural rules related to substitution, e.g.syntactic substitution lemma and admissibility of cut.

Substitutions on concrete terms σ k,n : (x 1,..., x k t); (x 1,..., x n u 1,..., u k ) x 1,..., x n t[u 1 /x 1,..., u k /x k ] Example: λ-calculus σ k,n (t[x 1,..., x k ]; u 1,..., u k ) = u i if t = x i (σ k,n (t 1 ; u 1,..., u k ))(σ k,n (t 2 ; u 1,..., u k )) if t = t 1 t 2 λx n+1.σ k+1,n+1 (t 1 [x 1,..., x k+1 ]; u 1,..., u k, x n+1 ) if t = λx k+1.t 1 [x 1,..., x k+1 ]

Categorical Construction Substitution: Function from Substitutable Pairs to Term Substitutable Pair: (x 1,..., x k t); (x 1,..., x n u 1,..., u k ) Type Constructor for Substitutable Pairs (X Y)({x 1,..., x n }) def = k N X({x 1,..., x k }) (Y({x 1,..., x n })) k / = {(t; u) k N, t X({x 1,..., x k }), u (Y({x 1,..., x n })) k }/ where is the equivalence relation generated by (t; u 1,..., u k ) (t ; u 1,..., u k ) iff r:k k X(r)(t) = t u i = u r(i) The tensor : Set F Set F Set F forms a closed monoidal structure with a unit V and suitable natural transformations α, λ, ρ.

Categorical Construction st X,e:V Y : ΣX Y Σ(X Y) is a natural transformation between two functors of type Set F V Set F Set F. Example: λ-calculus: st X,e:V Y : (δx + X X) Y = (δx) Y + (X X) Y δ(x Y) + (X Y) (X Y) σ : X X X is defined in the following way. ΣTV TV st TV,ηV :V TV Σ(TV TV) Σσ ΣTV τ id σ TV TV TV λ TV V TV η V id τ

Initial Algebra Semantics Category of S-subst-algebras Objects: (X, e : V X, h : ΣX X, sub : X X X) where (X, e, sub) is a monoid in Set F and (X, h) is an Σ-algebra such that the following diagram commutes. ΣX X st X,e:V X Σ(X X) Σ(sub) ΣX h id X X sub X h Morphisms: maps of Set F which are both Σ-algebra and monoid homomorphisms. Initial Object: (TV, η V : V TV, τ : ΣTV TV, σ : TV TV TV)

Abstract Theory Definition (Category of Σ-subst-algebras) Monoidal closed category C = (C,, I) Strong endofunctor Σ with a strength a st Objects: X = (X, e, h, sub), where (X, e, sub) is a monoid in C and (X, h) is a Σ-algebra such that the following diagram commutes. ΣX X st X,X Σ(X X) Σ(sub) ΣX h id X X sub X h Morphisms: maps of C which are both Σ-algebra and monoid homomorphisms. a A strength for Σ is a natural transformation of type Σ(X) Y Σ(X Y) satisfying some coherence conditions.

Abstract Theory Theorem If Σ has a free monad T then (TI, η I, σ, τ) is an initial Σ-subst-algebra, where τ : ΣTI TI is the free Σ-algebra of I and σ is given as follows: σ : TI TI st I,TI T(I TI) Tλ TI TTI µ I TI, where st : TX Y T(X Y) is the lifting of the strength st of Σ to its free monad T.

Abstract Theory Definition (Category of Σ-ptd-subst-algebras) Monoidal closed category C = (C,, I) u-strong endofunctor Σ with a u-strength a st for the forgetful functor u from I C Objects: X = (X, e, h, sub), where (X, e, sub) is a monoid in C and (X, h) is a Σ-algebra such that the following diagram commutes. ΣX X st X,e:I X Σ(X X) Σ(sub) ΣX h id X X sub X h Morphisms: maps of C which are both Σ-algebra and monoid homomorphisms. a A u-strength for Σ is a natural transformation of type Σ(X) uy Σ(X uy) satisfying some coherence conditions.

Abstract Theory Theorem (TI, η I, σ, µ I ) is an initial Σ-ptd-subst-algebra, where σ is given by the following composite: TI TI = TI u(η I : I TI) st I,ηI T(I u(η I )) = T(I TI) Tλ TI TTI µ I TI.

Motivation for equations Group Axioms Given a group algebra (X, e, i, m), (G1) x, y, z X m(m(x, y), z) = m(x, m(y, z)) (G2) x X m(e, x) = x (G3) x X m(i(x), x) = e β, η equations for λ-calculus Given a λ-calculus algebra (X, e, lam, app, sub), (β-eqn) t δx(n), u X(n) (η-eqn) t X(n) app(lam(x n+1.t), u) = sub(t; (x 1,..., x n, u)) lam(x n+1.app(t, x n+1 )) = t

Kelly & Power equational theories KP Theory Groups Sig S : C fp C Sig S : N Set (0, 1, 2) (1, 1, 1) ΣX = a C fp C(a, X) S(a) ΣX = a N Xa S(a) = 1 + X + X 2 TX = µa.x + ΣA TX = µa.x + ΣA TX = all terms freely generated from X TX = all group terms freely generated from a variable set X (Equation) (Equation) D : C fp C together with D : N Set : (1, 3) (2, 1) a C fp D(a) EL(a) T(a) E R(a) D(1) = 2 EL(1) T(1) E R(1) E L (1)(1) = m(e, x 1 ), E R (1)(1) = x 1 E L (1)(2) = m(i(x 1 ), x 1 ), E R (1)(2) = e D(3) = 1 EL(3) E R(3) T(3) E L (3)(1) = m(m(x 1, x 2 ), x 3 ), E R (3)(1) = m(x 1, m(x 2, x 3 ))

Free Constructions KP Theory There exists a finitary monad T such that T -Alg = Σ-alg/ EL=E R F C where Σ-alg/ EL=E R is a full subcategory of Σ-alg whose objects are Σ-algebras satisfying the equations. u Groups There exists a finitary monad T such that T -Alg = Σ-alg/ EL=E R F Set where Σ-alg/ EL=E R is a full subcategory of Σ-alg whose objects are group-algebras, i.e.(x, e, i, m) satisfying the group axioms G1, G2, and G3 u

Problems with applying K.P. theory to subst-algebras Recall: subst-algebras for λ-calculus Objects: (X, e : V X, lam : δx X, app : X 2 X, sub : X X X) satisfying V X e id X X λ X X sub X V id e X X ρ X X sub (X X) X sub id X X sub α X,X,X X (X X) sub X X X id sub (δx + X 2 ) X st X,e:V X δ(x X) + (X X) 2 δsub+sub 2 δx + X 2 [lam,app] id X X sub X [lam,app]

A new view on equational theories: Observation Example: Group Given a group algebra (X, 1 + X + X 2 [e,i,m] X), (G1) x, y, z X m(m(x, y), z) = m(x, m(y, z)) (G2) x X m(e, x) = x (G3) x X m(i(x), x) = e X X X m id X X X i,id X X id m X X G1 m X m! 1 e G3 X m X 1 X X X!,id e id id G2 X m

A new view on equational theories Observation Σ-alg 1 + X + X 2 [e,i,m] X D-alg X 3 + X + X E L X E R Definition (Equations for algebras) a category C and an endofunctor Σ Equation: a domain endofunctor D with functors E L and E R from Σ-alg to D-alg preserving base-objects of algebras, i.e.u E L = u and u E R = u. Σ-alg u C E L E R u D-alg

Free constructions Theorem If either (Cond1) C is cocomplete and Σ, D preserve colimits of ω-chains. (Cond2) C is cocomplete and well-copowered, and Σ preserves epimorphisms. then L E 1 Σ-Alg/ E1=E 2 Σ-Alg D-Alg K E 2 u u =uk u C

Free constructions Theorem If either (Cond1) C has coequalizers, pushouts and colimits of chains whose cardinals are less than or equal to a regular cardinal λ, and Σ, D preserve colimits of chains whose cardinals are equal to λ. (Cond2) C has coequalizers, pushouts and colimits of chains (whose cardinals are less than a regular cardinal λ) and is (λ-)well-copowered, and Σ preserves epimorphisms. then L E 1 Σ-Alg/ E1=E 2 Σ-Alg D-Alg K E 2 u u =uk u C

Sketch of the Proof ΣX X h

Sketch of the Proof ΣX 0 h 0 X 0

Sketch of the Proof ΣX 0 E Lh 0 E Rh 0 DX 0 X 0 h 0

Sketch of the Proof ΣX 0 X 0 X0 1 E Lh 0 E Rh 0 DX 0 h 0

Sketch of the Proof ΣX 0 ΣX 1 0 X 0 X0 1 E Lh 0 E Rh 0 DX 0 h 0

Sketch of the Proof ΣX 0 ΣX0 1 X 0 X0 1 E Lh 0 E Rh 0 DX 0 h 0 X 2 0

Sketch of the Proof ΣX 0 ΣX0 1 ΣX 0 2 X 0 X0 1 E Lh 0 E Rh 0 DX 0 h 0 X 2 0

Sketch of the Proof ΣX 0 ΣX0 1 ΣX 0 2 X 0 X0 1 E Lh 0 E Rh 0 DX 0 h 0 X 2 0 X 3 0

Sketch of the Proof ΣX 0 ΣX0 1 ΣX 2 0 ΣX 0 3 ΣX 1 X 0 X0 1 E Lh 0 E Rh 0 DX 0 h 0 X 2 0 X 3 0 X 1

Sketch of the Proof ΣX 0 ΣX0 1 ΣX 2 0 ΣX 0 3 ΣX 1 X 0 X0 1 E Lh 0 E Rh 0 DX 0 h 0 X 2 0 X 3 0 X 1 h 1

Sketch of the Proof ΣX 0 ΣX 1 X 0 X 1 E Lh 0 E Rh 0 DX 0 h 0 h 1

Sketch of the Proof ΣX 0 ΣX 1 X 0 X 1 E Lh 0 E Lh 1 E Rh 0 E DX 0 Rh 1 DX 1 h 0 h 1

Sketch of the Proof ΣX 0 ΣX 1 ΣX 2 h 0 X 0 X 1 X 2 E Lh 0 E Lh 1 E Lh 2 E Rh 0 E Rh 1 DX 0 E DX 1 Rh 2 DX 2 h 1 h 2

Sketch of the Proof ΣX 0 ΣX 1 ΣX 2 ΣX ω h 0 X 0 X 1 X 2 X ω E Lh 0 E Lh 1 E Lh 2 E Lh ω E Rh 0 E Rh 1 DX 0 E Rh 2 E DX 1 Rh ω DX 2 DXω h 1 h 2 h ω

Examples of Equations subst-algebras for λ-calculus Objects: (X, e : V X, lam : δx X, app : X 2 X, sub : X X X) satisfying V X e id X X λ X X sub X V id e X X ρ X X sub (X X) X sub id X X sub α X,X,X X (X X) sub X X X id sub (δx + X 2 ) X st X,e:V X δ(x X) + (X X) 2 δsub+sub 2 δx + X 2 [lam,app] id X X sub X [lam,app]

Examples of Equations Example: λ-calculus (β-eqn) t δx(n), u X(n) app(lam(x n+1.t), u) = sub(t; (x 1,..., x n, u)) δx X emb X X lam id sub X X X app (η-eqn) t X(n) lam(x n+1.app(t, x n+1 )) = t X id,! X 1 id up λe δx δx δ(x X) δapp δx X lam

Examples of Equations π-calculus Base category Set I ν : δx X, where δx = (V X) νx.νy.t = νy.νx.t t δ 2 X(n) νx n+1.νx n+2.t = νx n+1.νx n+2.t[x n+2 /x n+1, x n+1 /x n+2 ] δδx swap δδx δν δν δx δx ν ν X

Future Work Modelling DILL (Dual Intuitionistic Linear Logic) Modelling Dependent type languages. Develop the equational theories in pre-ordered setting, and apply them to Term Rewriting Systems. Find more applications of the equational theories. Example: the free-algebra models for the π-calculus by Ian Stark.