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A sound unification algorithm based on telescope equivalences Jesper Cockx DistriNet KU Leuven 20 April 2016 Pattern matching is awesome Agda uses unification to: check which constructors are possible specialize the result type data Vec

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A sound unification algorithm based on telescope equivalences

Jesper Cockx

DistriNet – KU Leuven

20 April 2016

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Pattern matching is awesome

Agda uses unification to: check which constructors are possible specialize the result type data Vec (A : Set) : N → Set where [] : Vec A 0 cons : (n : N) → A → Vec A n → Vec A (1 + n) f : Vec A 1 → T f (cons .0 x xs) = . . .

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Pattern matching is awesome

Agda uses unification to: check which constructors are possible specialize the result type data Vec (A : Set) : N → Set where [] : Vec A 0 cons : (n : N) → A → Vec A n → Vec A (1 + n) f : Vec A 1 → T f (cons .0 x xs) = . . .

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Pattern matching is awesome

Agda uses unification to: check which constructors are possible specialize the result type data Vec (A : Set) : N → Set where [] : Vec A 0 cons : (n : N) → A → Vec A n → Vec A (1 + n) f : Vec A 1 → T f (cons .0 x xs) = . . .

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Details of unification are important

Agda has pattern matching as a primitive, so results of unification determine Agda’s notion of equality Example: deleting reflexive equations implies K

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Details of unification are important

Agda has pattern matching as a primitive, so results of unification determine Agda’s notion of equality Example: deleting reflexive equations implies K

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Time for a quiz

Should the following code be accepted? {-# OPTIONS --without-K #-} . . . -- imports f : (Bool , true) ≡ (Bool , false) → ⊥ f ()

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Time for a quiz

Should the following code be accepted? {-# OPTIONS --without-K #-} . . . -- imports f : (Bool , true) ≡ (Bool , false) → ⊥ f () Answer: depends on the type of the equation!

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Postponing equations causes problems

If we postpone an equation, following equations can be heterogeneous Naively continuing unification is bad Equality of second projections Injectivity of type constructors . . . It’s hard to distinguish good and bad situations!

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Postponing equations causes problems

If we postpone an equation, following equations can be heterogeneous Naively continuing unification is bad Equality of second projections Injectivity of type constructors . . . It’s hard to distinguish good and bad situations!

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Postponing equations causes problems

If we postpone an equation, following equations can be heterogeneous Naively continuing unification is bad Equality of second projections Injectivity of type constructors . . . It’s hard to distinguish good and bad situations!

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We need a general way to think about unification

It’s not sufficient to “make things equal” Core idea: Unification rules are equivalences between telescopes of equations This is the basis of the new unification algorithm in Agda 2.5.1

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We need a general way to think about unification

It’s not sufficient to “make things equal” Core idea: Unification rules are equivalences between telescopes of equations This is the basis of the new unification algorithm in Agda 2.5.1

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We need a general way to think about unification

It’s not sufficient to “make things equal” Core idea: Unification rules are equivalences between telescopes of equations This is the basis of the new unification algorithm in Agda 2.5.1

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A sound unification algorithm based on telescope equivalences

1 Unifiers as equivalences 2 Unification rules 3 Higher-dimensional unification

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A sound unification algorithm based on telescope equivalences

1 Unifiers as equivalences 2 Unification rules 3 Higher-dimensional unification

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What do we want from unification?

It has to be possible to translate pattern matching to eliminators The core tool we need is specialization by unification Build a function m : Γ → ¯ u ≡∆ ¯ v → T from a function m′ : Γ′ → Tσ where σ : Γ′ → Γ is computed by unification

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What do we want from unification?

It has to be possible to translate pattern matching to eliminators The core tool we need is specialization by unification Build a function m : Γ → ¯ u ≡∆ ¯ v → T from a function m′ : Γ′ → Tσ where σ : Γ′ → Γ is computed by unification

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Intermezzo: telescopic equality

Type of an equation may depend

n solution of previous equations

Heterogeneous equality doesn’t keep enough information: Safe to consider equation homogeneous? Does equation depend on other equation? How do equations depend on each other?

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Intermezzo: telescopic equality

Type of an equation may depend

n solution of previous equations

Heterogeneous equality doesn’t keep enough information: Safe to consider equation homogeneous? Does equation depend on other equation? How do equations depend on each other?

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Intermezzo: telescopic equality

Solution: use “path over” construction to keep track of dependencies For example: (e1 : m ≡N n)(e2 : u ≡e1

Vec A v)

Cubical (abuse of) notation: (e1 : m ≡N n)(e2 : u ≡Vec A e1 v)

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Intermezzo: telescopic equality

Solution: use “path over” construction to keep track of dependencies For example: (e1 : m ≡N n)(e2 : u ≡e1

Vec A v)

Cubical (abuse of) notation: (e1 : m ≡N n)(e2 : u ≡Vec A e1 v)

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Specialization by unification

The goal is to construct m : Γ → ¯ u ≡∆ ¯ v → T Input: Telescope Γ of flexible variables Telescope ¯ u ≡∆ ¯ v of equations Output: New telescope Γ′ Substitution σ : Γ′ → Γ Evidence of unification ¯ e : Γ′ → ¯ uσ ≡∆σ ¯ vσ

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Specialization by unification

The goal is to construct m : Γ → ¯ u ≡∆ ¯ v → T Input: Telescope Γ of flexible variables Telescope ¯ u ≡∆ ¯ v of equations Output: New telescope Γ′ Substitution σ : Γ′ → Γ Evidence of unification ¯ e : Γ′ → ¯ uσ ≡∆σ ¯ vσ

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Specialization by unification

The goal is to construct m : Γ → ¯ u ≡∆ ¯ v → T Input: Telescope Γ of flexible variables Telescope ¯ u ≡∆ ¯ v of equations Output: New telescope Γ′ Substitution σ : Γ′ → Γ Evidence of unification ¯ e : Γ′ → ¯ uσ ≡∆σ ¯ vσ

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Specialization by unification

The goal is to construct m : Γ → ¯ u ≡∆ ¯ v → T Input: Telescope Γ of flexible variables Telescope ¯ u ≡∆ ¯ v of equations Output: New telescope Γ′ Telescope mapping f : Γ′ → Γ(¯ u ≡∆ ¯ v)

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Two more requirements

Let f : Γ′ → Γ(¯ u ≡∆ ¯ v) be a unifier f should be most general ⇒ f needs a right inverse g1 Γ′ should be minimal ⇒ f needs a left inverse g2

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Two more requirements

Let f : Γ′ → Γ(¯ u ≡∆ ¯ v) be a unifier f should be most general ⇒ f needs a right inverse g1 Γ′ should be minimal ⇒ f needs a left inverse g2

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Two more requirements

Let f : Γ′ → Γ(¯ u ≡∆ ¯ v) be a unifier f should be most general ⇒ f needs a right inverse g1 Γ′ should be minimal ⇒ f needs a left inverse g2

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Most general unifiers as equivalences

A most general unifier of ¯ u and ¯ v is an equivalence f : Γ(¯ u ≡∆ ¯ v) ≃ Γ′ for some Γ′ Specialization by unification: m : Γ → ¯ u ≡∆ ¯ v → T m ¯ x ¯ e = subst (λ¯ x ¯

e. T) (isLinv f ¯

x ¯ e) (m′ (f ¯ x ¯ e))

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Most general unifiers as equivalences

A most general unifier of ¯ u and ¯ v is an equivalence f : Γ(¯ u ≡∆ ¯ v) ≃ Γ′ for some Γ′ Specialization by unification: m : Γ → ¯ u ≡∆ ¯ v → T m ¯ x ¯ e = subst (λ¯ x ¯

e. T) (isLinv f ¯

x ¯ e) (m′ (f ¯ x ¯ e))

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Disunifiers

A disunifier of ¯ u and ¯ v is an equivalence f : Γ(¯ u ≡∆ ¯ v) ≃ ⊥ Specialization by unification: m : Γ → ¯ u ≡∆ ¯ v → T m ¯ x ¯ e = elim⊥ T (f ¯ x ¯ e)

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Disunifiers

A disunifier of ¯ u and ¯ v is an equivalence f : Γ(¯ u ≡∆ ¯ v) ≃ ⊥ Specialization by unification: m : Γ → ¯ u ≡∆ ¯ v → T m ¯ x ¯ e = elim⊥ T (f ¯ x ¯ e)

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A sound unification algorithm based on telescope equivalences

1 Unifiers as equivalences 2 Unification rules 3 Higher-dimensional unification

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Basic unification rules

MGU is constructed by chaining together equivalences given by unification rules (k l : N)(e : suc k ≡N suc l) ≃ (k l : N)(e : k ≡N l) ≃ (k : N) f −1 : (k : N) → (k l : N)(e : suc k ≡N suc l) f −1 k = k; k; refl

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Basic unification rules

MGU is constructed by chaining together equivalences given by unification rules (k l : N)(e : suc k ≡N suc l) ≃ (k l : N)(e : k ≡N l) ≃ (k : N) f −1 : (k : N) → (k l : N)(e : suc k ≡N suc l) f −1 k = k; k; refl

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Basic unification rules

MGU is constructed by chaining together equivalences given by unification rules (k l : N)(e : suc k ≡N suc l) ≃ (k l : N)(e : k ≡N l) ≃ (k : N) f −1 : (k : N) → (k l : N)(e : suc k ≡N suc l) f −1 k = k; k; refl

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Basic unification rules

MGU is constructed by chaining together equivalences given by unification rules (k l : N)(e : suc k ≡N suc l) ≃ (k l : N)(e : k ≡N l) ≃ (k : N) f −1 : (k : N) → (k l : N)(e : suc k ≡N suc l) f −1 k = k; k; refl

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Basic unification rules

MGU is constructed by chaining together equivalences given by unification rules (k l : N)(e : suc k ≡N suc l) ≃ (k l : N)(e : k ≡N l) ≃ (k : N) f −1 : (k : N) → (k l : N)(e : suc k ≡N suc l) f −1 k = k; k; refl

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Basic unification rules

Solution: (x : A)(e : x ≡A t) ≃ () Deletion: (f x ≡N f x) ≃ () Injectivity: (suc x ≡N suc y) ≃ (x ≡N y) Conflict: (inj1 x ≡A⊎B inj2 y) ≃ ⊥ Cycle: (n ≡N suc n) ≃ ⊥ + auxiliary rules for weakening and reordering

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Rules for η-equality of records

η-expansion of a flexible variable: (p : N × N)(e : fst p ≡N zero) ≃ (x : N)(y : N)(e : x ≡N zero) ≃ (y : N) η-expansion of an equation: (e : x, y ≡N×N f z) ≃ (e1 : x ≡N fst (f z)) (e2 : y ≡N snd (f z))

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Rules for η-equality of records

η-expansion of a flexible variable: (p : N × N)(e : fst p ≡N zero) ≃ (x : N)(y : N)(e : x ≡N zero) ≃ (y : N) η-expansion of an equation: (e : x, y ≡N×N f z) ≃ (e1 : x ≡N fst (f z)) (e2 : y ≡N snd (f z))

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Rules for indexed data types

Idea: rules solve equations between indices together with equations between constructors Example: (e1 : suc m ≡N suc n) (e2 : cons m x xs ≡Vec A e1 cons n y ys) ≃ (e1 : m ≡N n)(e2 : x ≡A y) (e3 : xs ≡Vec A e1 ys)

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Rules for indexed data types

Idea: rules solve equations between indices together with equations between constructors Example: (e1 : suc m ≡N suc n) (e2 : cons m x xs ≡Vec A e1 cons n y ys) ≃ (e1 : m ≡N n)(e2 : x ≡A y) (e3 : xs ≡Vec A e1 ys)

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Rules for indexed data types

This can give a real boost to power: data Im (f : A → B) : B → Set where image : (x : A) → Im f (f x) (x y : A)(e1 : f x ≡B f y) (e2 : image x ≡Im f e1 image y) ≃ (x y : A)(e : x ≡A y) ≃ (x : A)

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From this point, there be dragons

Any questions so far?

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A sound unification algorithm based on telescope equivalences

1 Unifiers as equivalences 2 Unification rules 3 Higher-dimensional unification

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Indexed rules are too restrictive

Rules for indexed datatypes require indices to be fully general This is too restrictive: (e1 : cons n x xs ≡Vec A (suc n) cons n y ys) ̸≃ (e1 : x ≡A y)(e2 : xs ≡Vec A n ys)

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Indexed rules are too restrictive

Rules for indexed datatypes require indices to be fully general This is too restrictive: (e1 : cons n x xs ≡Vec A (suc n) cons n y ys) ̸≃ (e1 : x ≡A y)(e2 : xs ≡Vec A n ys)

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Generalized rules for indexed data

The following rules can be generalized to arbitrary indices: Conflict Cycle Injectivity: only if index types satisfy K!

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Reverse unification rules

Idea: we can generalize the indices by applying unification rules in reverse

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Reverse unification rules: example

(n : N)(x y : A)(xs ys : Vec A n) (e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n) (e2 : cons m x xs ≡Vec A (suc e1) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : suc m ≡N suc n) (e2 : cons m x xs ≡Vec A e1 cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) ≃ (n : N)(x : A)(xs : Vec A n)

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Reverse unification rules: example

(n : N)(x y : A)(xs ys : Vec A n) (e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n) (e2 : cons m x xs ≡Vec A (suc e1) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : suc m ≡N suc n) (e2 : cons m x xs ≡Vec A e1 cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) ≃ (n : N)(x : A)(xs : Vec A n)

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Reverse unification rules: example

(n : N)(x y : A)(xs ys : Vec A n) (e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n) (e2 : cons m x xs ≡Vec A (suc e1) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : suc m ≡N suc n) (e2 : cons m x xs ≡Vec A e1 cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) ≃ (n : N)(x : A)(xs : Vec A n)

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Reverse unification rules: example

(n : N)(x y : A)(xs ys : Vec A n) (e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n) (e2 : cons m x xs ≡Vec A (suc e1) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : suc m ≡N suc n) (e2 : cons m x xs ≡Vec A e1 cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) ≃ (n : N)(x : A)(xs : Vec A n)

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Reverse unification rules: example

(n : N)(x y : A)(xs ys : Vec A n) (e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n) (e2 : cons m x xs ≡Vec A (suc e1) cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : suc m ≡N suc n) (e2 : cons m x xs ≡Vec A e1 cons n y ys) ≃ (m n : N)(x y : A)(xs : Vec A m)(ys : Vec A n) (e1 : m ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) ≃ (n : N)(x : A)(xs : Vec A n)

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Reverse unification rules: problems

Applicability is limited: indices need to be linear patterns Hard to implement Not clear how to apply injectivity for indexed data in reverse

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Going beyond the first level

Realization: same problem as for case splitting,

nly for equations instead of variables

We can solve it in the same way as well: by specialization by unification

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Going beyond the first level

Realization: same problem as for case splitting,

nly for equations instead of variables

We can solve it in the same way as well: by specialization by unification

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Going beyond the first level

Realization: same problem as for case splitting,

nly for equations instead of variables

We can solve it in the same way as well: by specialization by unification

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Higher-dimensional unification: example

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (f : e1 ≡suc n≡Nsuc n refl) ≃ (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (f : cong suc e1 ≡suc n≡Nsuc n refl) ≃ (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (f : e1 ≡n≡Nn refl) ≃ (e2 : x ≡A y)(e3 : xs ≡Vec A n ys)

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Higher-dimensional unification: example

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (f : e1 ≡suc n≡Nsuc n refl) ≃ (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (f : cong suc e1 ≡suc n≡Nsuc n refl) ≃ (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (f : e1 ≡n≡Nn refl) ≃ (e2 : x ≡A y)(e3 : xs ≡Vec A n ys)

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Higher-dimensional unification: example

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (f : e1 ≡suc n≡Nsuc n refl) ≃ (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (f : cong suc e1 ≡suc n≡Nsuc n refl) ≃ (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (f : e1 ≡n≡Nn refl) ≃ (e2 : x ≡A y)(e3 : xs ≡Vec A n ys)

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Higher-dimensional unification: example

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (f : e1 ≡suc n≡Nsuc n refl) ≃ (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (f : cong suc e1 ≡suc n≡Nsuc n refl) ≃ (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (f : e1 ≡n≡Nn refl) ≃ (e2 : x ≡A y)(e3 : xs ≡Vec A n ys)

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Representing higher-order problems using first-order syntax

An n-dimensional unification problem consists of a telescope Γ of flexible variables equation telescopes ∆1, . . . , ∆n such that ⊢ Γ∆1 . . . ∆n left- and right-hand sides ¯ u1, ¯ v1, . . . ¯ un, ¯ vn such that Γ∆1 . . . ∆i−1 ⊢ ¯ ui, ¯ v1 : ∆i

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Discussion

Higher-dimensional unification seems easier to implement than reverse rules But maybe it goes too far? Alternative: use reflection to implement a case splitting tactic based on unification

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Discussion

Higher-dimensional unification seems easier to implement than reverse rules But maybe it goes too far? Alternative: use reflection to implement a case splitting tactic based on unification

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