Itegories J.R.B. Cockett Department of Computer Science University - - PowerPoint PPT Presentation

Itegories

Itegories

J.R.B. Cockett

Department of Computer Science University of Calgary Alberta, Canada robin@cpsc.ucalgary.ca (work with: Pieter Hofstra, Chad Nester)

Aviero, June 2015

Itegories

Introduction Restriction category Basics Extensivity Independence Disjoins Constructions Itegories Basics Traces Cartesian itegories Computability Stack objects Abstract machine

Itegories Introduction

WHAT IS THIS TALK ABOUT? Answer: a very elementary setting for describing sequential computation ... The semantics of flow diagrams ... (an old topic) (Dana Scott, Elgot, Steve Bloom, Ernie Manes, Bob Walters, ...)

Itegories Introduction

To the memory of Bob Walters: He would not have described it this way nor necessarily would he have approved but an itegory is something he certainly thought about ...

Itegories Introduction

What is an Itegory

An itegory is a disjoin restriction category .... with a Kleene wand. Idea: (a) Disjoint unions of (partial) maps: f ⊔ g (b) A Kleene wand f : A − → A g : A − → B f ⊥ g f ∗ | g : A − → B f ⊥ g means f is disjoint from g ... Kleene wand is a way of specifying a particle style trace. Begin with restriction categories ...

Itegories Restriction category Basics

A restriction category is a category X equipped with a restriction: A

f

− − → B A − − →

f

A [R.1] f f = f [R.2] f g = g f [R.3] f g = f g [R.4] f g = fgf A map f is said to be total when f = 1.

Lemma

(i) f f = f (iv) f g = f g (ii) f is monic then f = 1 (v) f = f (iii) fg = f g (vi) f , g total implies fg total.

Itegories Restriction category Basics

COMPLETENESS: Every restriction category is a full subcategory

• f a real category of partial maps!

(A convenient algebraic framework for partial maps ...)

Itegories Restriction category Basics

Partial products

A restriction category has a partial terminal object, 1, in case for each object A there is a total map !A : A − → 1 such that given any

• ther map h : A −

→ 1 we have h = h!A. A restriction category has partial products in case for each pair of

• bject A and B there is an object A × B and total maps

π0 : A × B − → A and π1 : A × B − → B such that given any pair of maps f : X − → A and g : X − → B there is a map f , g : X − → A × B with f , gπ0 = gf and f , gπ1 = f g. A X

f

• ⑧

⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧

g

• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄

f ,g

• ≥

≥

A × B

π0

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

π1

①①①①①①①①①

B

Itegories Restriction category Basics

Cartesian restriction categories

A restriction category which has partial products is a cartesian restriction category. (a.k.a. “P-category” developed by Robinson and Rossolini) Any Cartesian restriction category has a total category with real products and terminal object.

Itegories Restriction category Basics

Examples of restriction categories

• 1. The category of sets and partial maps Par(Set, monic).
• 2. Any partial map category: a favourite is CRingop with

localizations L. Then Par(CRingop, L) is a restriction category (of use in algebraic geometry).

• 3. The category of topological spaces with partial maps defined
• n an open subset and a continuous map on that subset.
• 4. The category sSLatop with stable maps (i.e. binary meet

preserving maps). The category of locales with stable maps (i.e. binary meet and join preserving maps).

• 5. Partial recursive maps on the natural numbers.
• 6. Given any partial algebraic theory T there is a classifying

cartesian restriction category C(T ) with a generic model of the partial algebraic theory. For example, there is a generic partial combinatory algebra which lives in its own environment and gives a generic version of computability.

Itegories Restriction category Basics

Partial Combinator Algebras (PCAs)

With partial product one can express arbitrary partial algebras. Here is a Partial Combinatory Algebra (PCA):

• : A × A −

→ A k, s : 1 − → A such that k = 11 and s = 11 A × A

π0

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

k×1×1

A × A × A

• 2
• A

(A × A) × A

θ×

• s×1×1×1

A × A × A × A

• 3
• (A × A) × (A × A)

(•×•)•

A

(s × 1 × 1)•2 = 1 Remark: Given an (internal) PCA one can not only express all computations but also one has the usual undecidability results associated with computability.

Itegories Restriction category Extensivity

Restriction zeros

A restriction category has a restriction zeros if for each pair of

• bjects A and B there is a map 0 : A −

→ B such that

◮ f 0g = 0 ◮ 0 = 0

If a zero map splits then the category has a zero object. If a restriction category has restriction zeros then it is enriched

• ver pointed sets.

Note that this category is also the category of sets and partial maps. Sets and partial maps have restriction zeros.

Itegories Restriction category Extensivity

Coproducts and extensivity

An extensive restriction category has coproducts, restriction zeros, and a decision operator: f : X − → Y + Z f : X − → X + X [Dec.1] ∇f = f [Dec.2] (f + f )f = (σ0 + σ1)f d : X − → X + X is a decision in case d = d. Binary decisions imply n-ary decisions d : A − → A + ... + A (these satisfy ∇d = d and (d + ... + d)d = (σ1 + ... + σn)d). Extensivity implies unique decomposition as the coproduct is a pre-biproduct: f = ((σ(−1) σ0) + (σ(−1)

1

σ1)f = (σ(−1) + σ(−1)

1

)(f + f )f = ∇(σ(−1) f + σ(−1)

1

f )f = (σ(−1) f ⊔ σ(−1)

1

f )

Itegories Restriction category Extensivity

Extensivity cont.

Why is extensivity important? Every map between coproducts an be written as a matrix .. X1 + ... + Xn − − − − − − − − − − − − − − − − →

A =   a11 ... a1n ... am1 ... amn  

Y1 + ... + Ym where the ith rows must be separated by a decision di such that aij = σjdi

Itegories Restriction category Independence

Independence

An independence space (X, X) is a set with a set of finite subsets X ⊆ Pf (X) which are down-closed and contain all singletons. A morphism f : (X, X) − → (Y , Y) between independence spaces is a partial map f : X − → Y which, when restricted to any independent subset, becomes a partial isomorphism. Each X ′ ∈ X determines a restriction idempotent eX ′ : X − → X, we require that eX ′f is a partial isomorphism and (eX ′f )(−1) determines an independent subset of Y . independent subsets cannot be “squashed” by maps ... The category of independence spaces is an extensive restriction

• category. It does not have partial products but it has various

tensors ....

Itegories Restriction category Independence

Independence spaces

Have the following (non-symmetric) tensor product: (A, A) ⊳ (B, B) = (A × B, A ⊳ B) where X ∈ A ⊳ B if and only if {a|∃b.(a, b) ∈ X} ∈ A & Xa = {b|(a, b) ∈ X} ∈ B b11 . . . bm1 . . . b1n

• · · ·

bmnm

• a1

· · · am Every object has a naturally coassociative comultiplication ...

Itegories Restriction category Independence

Enriching in Independence

Require that each X(A, B)∗ (remove zero) is an independence space and ( ) : X(A, B)∗ − → X(A, A)∗ and composition are independence space maps.

◮ Parallel arrows {f1, ..., fn} are independent if and only if

{f1, ..., fn} is.

◮ If {f1, ..., fn} are independent and fi ≤ f ′ i then {f ′ 1, ..., f ′ n} are

independent.

◮ f and g are independent then f g = 0.

Proof (of last) If {f , g} is independent then {(f , f ), (f , g), (g, f ), (g, g)} is independent in X(A, A) ⊳ X(A, A) but under composition f g = g f so these non-diagonal pairs are squashed under composition so cannot be defined. This means that these composites must be zero.

Itegories Restriction category Independence

Examples of independences

◮ Given any restriction category X with a restriction zero

declare the independent sets on X(A, B)∗ (where we remove the zero) to be the singleton sets.

◮ Given any restriction category X with a restriction zero

declare the independent sets on X(A, B)∗ to consist of the finite sets X ′ ⊆ X(A, B)∗ such that for each distinct pair f , g ∈ X ′ we have f g = 0.

◮ (Separation) Given any extensive category X declare

{f1, ..., fn} ⊆ X(A, B)∗ independent in case there is a decision d : A − → A + ... + A such that fi ≤ σ(−1)

i

d. Does separation in extensive categories cover all examples? Answer: Yes

Itegories Restriction category Disjoins

Disjoin restriction categories

A disjoin restriction category is independence enriched so that independent parallel maps have a join f ⊔ g satisfying;

◮ f ≤ f ⊔ g and g ≤ f ⊔ g ◮ Given f ≤ h and g ≤ h then f ⊔ g ≤ h. ◮ It is stable (f ⊔ g)h = (fh) ⊔ (gh)

(and universal k(f ⊔ g) = (kf ) ⊔ (kg)).

◮ {fij|i ∈ I, j ∈ Ji} indep. if and only if { j∈Ji fij|i ∈ I} is indep.

In a disjoin restriction category binary independence suffice: {f1, ..., fn} is independent iff fn−1 ⊥ fn fn−2 ⊥ fn−1 ⊔ fn ... f1 ⊥ f2 ⊔ ... ⊔ fn Extensive restriction cats are disjoin with respect to separation.

Itegories Restriction category Disjoins

Constructions ...

There are a series of constructions: Independence restriction

dj

• Disjoin restriction

Mat

• dj∞

Infinitely disjoin restriction

Extensive restriction

Itegories Restriction category Constructions

Constructions ...

Given any independence restriction category there is a universal disjoin restriction category into which it embeds η : X − → dj(X) dj(X) is defined as follows: Objects: The same as X, X ∈ X; Maps: A map is an independent subset F ⊆ X(X, Y )∗; Composition: FG = {fg|f ∈ F, g ∈ G}; Identities: 1X = {1X}; Restriction: F = {f |f ∈ F}; Disjointness: {Fi|i ∈ I} is independent if and only if

i Fi is

independent. Join: F ⊔ G = F ∪ G. The embedding preserves the independence structure ...

Itegories Restriction category Constructions

Constructions ...

Given any disjoin restriction category there is a universal extensive restriction category into which it embeds η : X − → Mat(X) preserving the disjointness and independence properties: Objects: Maps A : IA − → obj(X) where I is a finite set. Maps: Matrices M : A − → B where M : IA × IB − → Map(X) with ∂0(M(i, j)) = A(i) and ∂1(M(i, j)) = B(j) and {M(i, j)|j ∈ IB} independent. Composition: MN(i, j) =

k M(i, k)N(k, j)

Identities: 1A : A − → A has 1A(i, i) = 1A(i) and 1A(i, i′) = 0

• therwise.

Restriction: M(i, i) =

j M(i, j) and M(i.i′) = 0 otherwise.

Disjointness: M ⊥ M′ if and only if M(i, j) ⊥ M′(i, j). Join: (M ⊔ N)(i, j) = M(i, j) ⊔ N(i, j). Note Mat(Mat(X)) ≡ Mat(X) ...

Itegories Restriction category Constructions

A final construction ...

An independence set is infinitely independent if all its finite subsets are independent. Given any disjoin restriction category there is a universal infinite disjoin restriction category into which it embeds η : X − → dj∞(X).

• bjects: As for X.

Maps: F ⊆ X(A, B) closed to finite disjoins and down-closed. composition: FG =⇓ {

i,j figj|f ∈ F, g ∈ G} where we must add

disjoins and down-close the set of composites. Identity: 1A = {e|e ≤ 1A}. Restriction: F = {f |f ∈ F}. Disjointness: F ⊥ G if and only if for every f ∈ F and g ∈ G we have f ⊥ g. Join: The disjoin and down-closure of the union.

Itegories Itegories Basics

Itegories

An itegory is a disjoint restriction category with a combinator, which we call the Kleene wand: A

f

− − → A A

g

− − → B f ⊥ g A − − − − →

f ∗ | g

B which satisfies (Conway equations): [W.1] When f ⊥ h then (gf ) ∗ | h = h ⊔ ((fg) ∗ | (hg))f ; [W.2] When f ⊥ g, g ⊥ h, and h ⊥ g then (f ⊔ g) ∗ | h = (f ∗ | g) ∗ | (f ∗ | h). The Kleene wand f ∗ | g means intuitively “iterate f until g can be applied.” i.e. repeat f until g do g. Note f and g must be disjoint, where f ⊥ g means {f , g} is independent, so that the decision to continue or finish in this iteration is unambiguous.

Itegories Itegories Basics

Itegories

The Kleene wand is both a colax combinator and a lax

• combinator. This means.

A

≥ a

• f

A

a

• A′

f ′

A′

A

≥ a

• g

B

b

• A′

g′

B′

f ⊥ g A

a

• ≥

f ∗

| g B

b

• A′

f ′∗

| g′

B′

where one can reverse the inequalities to get the colax form. With equalities this condition is often called uniformity. Uniformity and lax implies colax.

Itegories Itegories Basics

Itegories

A full subcategory of an itegory is an itegory. If E is any set of idempotents then SplitE(X) is an itegory whenever X is.

◮ f ∗

| h = (f ∗ | h)f ⊔ h;

◮ 0 ∗

| g = g;

◮ f ∗

| h = h(f ∗ | h);

◮ If c ⊥ d then (bac ∗

| d)ba = (acb ∗ | db)a = cba ∗ | dba;

◮ (f ∗

| h)h = h;

◮ If h∗f = f and h∗ ⊥ h then f ∗

| h = h.

◮ f ∗

| 0 = 0;

◮ f ∗

| (hg) = (f ∗ | h)g;

◮ When g ⊥ g′ then f ∗

| g ⊥ f ∗ | g′ and f ∗ | (g ⊔ g′) = (f ∗ | g) ⊔ (f ∗ | g′);

◮ If f ≤ f ′ and g ≤ g′ then f ∗

| g ≤ f ′ ∗ | g′.

Itegories Itegories Basics

Inductive itegories

A uniform wand combinator on a disjoint restriction category is an inductive Kleene wand in case it satisfies: [iW.1] (f ∗ | g)f ⊔ g ≤ f ∗ | g; [iW.2] xf ≤ x and hg ≤ x implies (f ∗ | g)h ≤ x; Proposition: A disjoin restriction category with an inductive Kleene wand is an itegory. Corollary: Every disjoin restriction category can be embedded into an itegory. Proof Infinitely complete the disjoin restriction category. Define f ∗ | g :=

∞

• i=0

gf i We must show {f ig|i ≤ n} is independent: note that g ⊥ fg and g ⊥ f (g ⊔ fg) = fg ⊔ f 2g, but then g ⊥ f (g ⊔ fg ⊔ f 2g ⊔ f 3g ⊔ ....

Itegories Itegories Traces

Traces and itegories

We shall denote the trace of a map f = f00 f01 f10 f11

• : A + X −

→ B + X Tr(f ) = f00 f01 f10 f11

• : A −

→ B We may (inter)define: f ∗ | g := 1 g f

• a

b c d

• := a ⊔ (d ∗

| c)b

Theorem

A disjoint restriction category is an itegory if and only if Mat(X) is a (lax and colaxly) traced extensive category: the trace and Kleene wand are inter-related as above.

Itegories Itegories Cartesian itegories

Distributive restriction catgeories

Proposition: A Cartesian restriction category with coproducts is extensive if and

• nly it it has a restriction zero and the product distributes over the

coproduct (i.e. it is a distributive restriction category). A cartesian disjoin restriction category is a disjoint restriction category with partial products in which the product functors A × preserve disjunction and independence. Proposition: The extensive (matrix) completion of a cartesian disjoin restriction category has partial products and, thus, is a distributive restriction category.

Itegories Itegories Cartesian itegories

Cartesian itegories

In a cartesian itegory is a cartesian disjoin restriction category in which one requires that the product functors preserve iteration: that (A × f ) ∗ | (A × g) = A × (f ∗ | g) This ensures that iteration in context behaves correctly. Claim that this now provides a basic semantics of flow diagrams ...

Itegories Computability

Computabilty in cartesian itegories

We can work without loss in the extensive completion ... SO assume we are working in a distributive itegory ...

Itegories Computability Stack objects

Stack objects

Our aim is to build a partial combinator algebra for this we need: A stack object is an object with maps: put = nl|cns : 1 + A × A − → A get : A − → 1 + A × A such that put get = 11+A×A. This does force put to be a total map but there is, importantly, no requirement that get need be total.

Lemma

The following are equivalent conditions for an object: (i) 1 ⊳ A, A + A ⊳ A and A × A ⊳ A; (ii) 1 + A ⊳ A and A × A ⊳ A; (iii) 1 + A × A ⊳ A (it is a stack object).

Itegories Computability Abstract machine

Creating a PCA from an abstract machine

Code Value Stack Code Value Stack end x [] exit with x k x s end k0(x) s k0(x) y s end x s s x s end s0(x) s s0(x) y s end s1(x, y) s s1(x, y) z s x z cns(c0(y, z), s) end v cns(c0(y, z), s) y z cns(c1(v), s) end v′ cns(c1(v), s) v v′ s Combinator reduction A × A × A

(step exit)

− − − − − − − → A × A × A + A A × A × A − − − − − − − →

step ∗ | exit

A

• = (step ∗

| exit)(1 × 1 × [])

Itegories Computability Abstract machine

The identity for s is the only difficulty: ((s • x) • y) • z = s1(x, y) • z = step ∗ | exit(s1(x, y), z, []) = σ0(c, v, d) → step ∗ | exit(c, v, d) σ1(x) → x

• step(s1(x, y), z, [])

= step ∗ | exit(x, z, cons(c1(y, z), [])) = step ∗ | exit(x • z, end, cons(c1(y, z), [])) = step ∗ | exit(y, z, cons(c0(x • z), [])) = step ∗ | exit(y • z, end, cons(c0(x • z), [])) = step ∗ | exit(y • z, x • z, []) = (y • z) • (x • z) Where we use repeatedly the identity: step ∗ | exit(x, y, S) = step ∗ | exit(x • y, end, S) which, unfortunately, for a general trace and stack object may not hold!

Itegories Computability Abstract machine

For this identity we require that our stack object is appending that is has a map append : A × A − → A which satisfies A × A × A

cons×1

• 1×append A × A

cons

• A × A

append

A

1 × A

nil×1 π1

• A × A

append

qqqqqqqqqqqq

A A × A

snoc×1

• append

A

snoc

• (1 + A × A) × A

≃

• A + A × A × A

snoc+1

• 1 + A × A + A × A × A

σ0|(1×nil×1)σ1|σ1

• 1 + A × A × A

1+1×append

1 + A × A

Itegories Computability Abstract machine

EXIT!