A Monad For Randomized Algorithms

A Monad For Randomized Algorithms

Tyler BarkerTulane University1

Domains XIIAugust 26, 2015

1Supported by AFOSRTyler Barker Tulane University A Monad For Randomized Algorithms Domains XII August 26, 2015 1 / 30

Introduction

Randomized Algorithms

A randomized algorithm can be represented by a probabilistic Turingmachine.

A probabilistic Turing machine is a Turing machine that can use randomcoin flips to choose transitions.

The goal is to provide a domain-theoretic monad to use in denotationalsemantics for randomized algorithms.

Tyler Barker Tulane University A Monad For Randomized Algorithms Domains XII August 26, 2015 2 / 30

Introduction

Related Work: Probabilistic Powerdomain

The probabilistic powerdomain of Jones and Plotkin gives a monad overdomains.

However, there is no Cartesian closed category of domains known to beinvariant under the probabilistic powerdomain.

Also, the probabilistic powerdomain does not share a distributive law withany of the nondeterministic powerdomains.


The Functor

Motivation for the Functor

One of most well known randomized algorithms is the Miller-Rabinprimality test.

The algorithm is in the complexity class randomized polynomial time(RP).

This means that it always runs in polynomial time and has a one-sidederror probability.


The Functor


To test whether a given number n is prime, a random number is chosenbetween 2 and n− 2.

Tests using modular arithmetic are performed with this number beforereturning false or probably true.

A test on a prime number will always return probably true.

A test on a composite number will sometimes return probably true.


The Functor


For a composite number, at most 14 of the possible random choices

between 2 and n− 2 will result in the test returning probably true.

To minimize the error probability, we can repeat the test (choosing a newrandom number) only when the test returns probably true.

Running the test m times results in an error probability of at most 14m .


The Functor

Simplified View of Miller-Rabin - One Iteration

F F F ⊥

Here F represents “composite” while ⊥ represents “probably prime”, or“not sure”.


The Functor

Simplified View of Miller-Rabin - Two Iterations

F F F

F F F ⊥


The Functor

Simplified View of Miller-Rabin - Three Iterations

F F F

F F F

F F F ⊥


The Functor

The Functor Defined

Definition

An antichain of {0, 1}∞ is a subset of words such that no two distinctwords are comparable (no word is a prefix of another word).

Definition

An antichain M is full if {0, 1}ω ⊆↑M , or equivalently, M vEM {0, 1}ω.Denote the full antichains by AC({0, 1}∞).


The Functor

The Functor Defined

For a category of domains, D, the functor of random choice, RC, isdefined on objects by

RC(D) ={(M,f) | M ∈ AC({0,1}∞), f :M →D

}where f is Scott continuous (giving M the subspace topology from theScott topology of {0, 1}∞).

For a : D → D′ and (M,f) ∈ RC(D), we define

RC(a)(M,f) = (M,a ◦ f)


The Functor

The Functor Defined

(M,f) v (N, g) iff

M vEM N

w ≤ z⇒ f(w) v g(z),∀w ∈M,z ∈N

Since the antichains are required to be full, M vEM N is equivalent toM ⊆↓N , or dually, N ⊆↑M .

Another characterization for the order on functions is∀z ∈ N, f ◦ πM (z) v g(z), where πM (z) sends z to the unique element ofM below z.


The Functor

The Functor Defined

Proposition

If D is a dcpo, then so is RC(D).

Proposition

If D is a bounded complete domain, then so is RC(D).


The Monad

The Monad

A monad can be formed exhibiting a unit, η : D → RC(D), and for anymorphism, h : D → RC(E), a Kleisli extension, h† : RC(D)→ RC(E).

The unit is defined byη(d) = (ε,χd)

where ε is the antichain only containing the empty word, with the constantfunction whose value is d.


The Monad

Motivation for the Kleisli Extension

One function of the Kleisli extension is to lift binary operations.

If we have a binary operation ∗ : D×E → F , the Kleisli extension lifts thisoperation to ∗† : RC(D)×RC(E)→ RC(F ). This is achieved by setting

∗† = (λa.RC(λb.a ∗ b))†


The Monad


Suppose these are Miller-Rabin tests for two different composite numbers.

F F F ⊥ ⊥ F F F

How should the or operation be lifted?


The Monad


It may seem natural to perform the two tests one after the other. Thiswould result in:

⊥ F F F ⊥ F F F ⊥ F F F ⊥ ⊥ ⊥ ⊥

The probability of error in this case is 716 .


The Monad


However, this approach has two main problems.

What do we do in the infinite case?

The Kleisli extension that results in this behavior is not monotone inour order.


The Monad


Instead of performing the tests one after the other, we will perform themconcurrently, feeding the result of each coin flip to both tests. This resultsin:

⊥ F F ⊥

This results in an error probability of 12 . However, this only uses two coin

flips.


The Monad


Using four coin flips gives:

F F

⊥ F F F F F F ⊥

which only has an error probability of 18 .


The Monad

The Kleisli Extension Defined

Consider h : D → RC(E). To define the extension h† : RC(D)→ RC(E)on (M,f), h ◦ f(w) must be considered for each w ∈M .

π1 ◦ h ◦ f(w) gives an antichain, but the entire antichain is not used.

Instead, the Kleisli extension only considers the part of π1 ◦ h ◦ f(w) thatis on the same “branch” of the tree as w, namely ↑w ∪ ↓w.


The Monad


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e0 e1

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h◦f..


The Monad


e0 •

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e2

e0 e1

e3

e4 e5 e6 e7

h◦f//

h◦f..


The Monad


e0 e3

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e2

e0 e1

e3

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The Monad


e0 e3 e6 e7

e2

e0 e1

e3

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The Monad


h† is defined as

π1 ◦ h†(M,f) =⋃⋃⋃

w∈M

Min(↑w∩ ↑π1 ◦ h ◦ f(w))

((π2 ◦ h†)(M,f))(z) = g(πN(z)) where (N,g) = h ◦ f ◦ πM(z)


The Monad

The Monad

Theorem

RC forms a monad in the category of bounded complete domains.

Proposition

RC shares a distributive law with the Hoare powerdomain.


The Monad

Relation to Scott’s Stochastic Lambda Calculus

Scott has added random variables to his P(N) model to form a stochasticlambda calculus.

Definition

A random variable is a function X : [0, 1]→ P(N) where{t ∈ [0, 1] | n ∈ X(t)} is Lebesgue measurable for all n in P(N).

Compare this to infinite elements in our monad, which are of the form{0, 1}ω → D.


The Monad

Relation to Scott’s Stochastic Lambda Calculus

The lambda calculus has an application operation that must be lifted tothese random variables.

Definition

Given two random variables X,Y : [0, 1]→ P(N), the application operationis defined by

X(Y)(t) = X(t)(Y(t))

This coincides with how our Kleisli extension would lift the applicationoperation.


Conclusion

Concluding Remarks

Alterations can be made to the monad to obtain distributive laws with theSmyth and Plotkin powerdomains.

We have used the monad to give a semantics for a randomized PCF.

An interpreter for rPCF has been implemented in SML along with aprogram to perform the Miller-Rabin test.

For an operational version of our monad, the monad laws andmonotonicity of the Kleisli extension have been formally proven in Isabelle.


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A Monad For Randomized Algorithms