Behavioral Computation Theory: Tutorial

Behavioral Computation Theory: Tutorial

Yuri Gurevich (Microsoft Research)

WoLLIC 2006

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Agenda

1. Sequential algorithms 2. Interactive algorithms

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Part 1: Sequential algorithms

IntuitionAxiomatic definition Behavioral equivalence

Sequential abstract state machinesSequential Characterization Theorem Algorithms are ASMs, and vice versa,

as far as behavior is concerned.

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Example: Euclid’s algorithm

1.If a = 0, set d = b and halt.

2.Set a = b mod a, set b = a,and go to 1.

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Example: Euc

Initially t = 0; a(0), b(0) well defined.

1.If a(t) = 0, set d = b(t) and halt.

2.Set a(t+1)= b(t) mod a(t),set b(t+1) = a(t), increment t, and go to 1.

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A run of Euc

t = 0, a(0) = 6, b(0) = 9

t = 1, a(1) = 3, b(1) = 6

t = 2, a(2) = 0, b(2) = 3

t,a,b unchanged, d = 3

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Which algos are sequential?

Negative characterization: neither parallel nor distributedPositive characterization is our goal. But we cannot rely on the formal notion of algorithm, so the notion that we have to define is that of sequential algorithms.

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Seq Time Postulate

Every algorithm is associated witha nonempty set Statesa nonempty subset Initial Statesa transition function Next : States States

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Intuition on states

States are comprehensive. What are states of a Turing machine? What are states of a C program?

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Behavior Equivalence

Two algorithms are behaviorally equivalent if they have the same states, initial states and transition function. The equivalence relation is

semantical:the programs may be different indeed.

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What else can be said of seq algos

in full generality?

Constructive (tangible) inputs Not necessarily

Finite Programs Sure, but syntax is messy.

Small (local, bounded-work) step But what’s local? How to measure work?

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Bounded work vs. bounded change

Bounded work bounded change, but

bounded change ↛ bounded work, e.g.

if xy({x,y} E) then output := false

else output := true

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Abstract State Postulate

The states are structures of the same vocabulary.Base(Next(X)) = Base(X).If is an isomorphism from a state X to a structure Y, then Y is a state and is an isomorphism from Next(X) to Next(Y).

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Without loss of generality

A state comes with the equality relation = true, false and the standard

propositional connectives undef

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Euc’s states (non-logic part)

A Euclidean domain E (with mod) including the set N of natural numbers with 0 and successor +1

Unary dynamic functions a, b : N E

Nullary dynamic functions d,t

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Euc’s associates

States: as described above.A state is initial if d = undef, t = 0.Next is given by the program.

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Euc’s vocabulary (non-logic part)

Static part In principle, the vocabulary of

Euclidean domains (with mod) In fact, Euc uses only 0, +1, mod

Dynamic part Unary function symbols a,b Nullary function symbols d,t

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Actions

Locations and their contents = (f,a1,..,aj)

Content() = f(a1,..,aj)

Updates (,v)

The update set at state X is

(X) = { (,v) : v = Content() in Next(X)

Content() in X }

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Euc’s locations and actions

Dynamic locations: (a,.), (b,.), t, d.If a(0) = 6, b(0) = 9 at X then(X) = { (t,1),((a,1),3) , ((b,1),6) }

(Next(X)) = {(t,2), ((a,2),0) , ((b,2)3)}(Next(Next(X))) = { (d,3) }

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Element Accessibility

The only way to refer to an element a is via a term that evaluates to a.A finite program can refer to only boundedly many elements

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Bounded Exploration Postulate

There is a finite set T of termssuch that for all states X,Y

if ValX(t) = ValY(t) for t ∈ T

then (X) = (Y).

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A bounded exploration witness for Euc

true, false, undef

Terms a(t)=0, a(t+1), b(t) mod a(t), b(t+1), d,

and their subterms

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Definition

A sequential algorithm is any object that satisfies the postulates:sequential time,abstract state, bounded-exploration.

Is this definition too general?

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Seq ASM Rules

Syntax Semantics = ?

f(t1,..,tj):= t0 {(,a0)} where =(f,(a1,..,aj)) and each ai = Val(ti)

do in parallel R1 … Rk

(R1) … (Rk)

if t then R1 else R2

if Val(t) = true then (R1) else (R2)

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Seq Abstract State Machines

A program is just a rule (to be iterated)

An ASM of vocabulary V is given by a program of vocabulary V a non-empty set of V-structures (the states)

closed under isomorphism and the transition function defined by the program

a non-empty subset of initial statesclosed under isomorphism

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Every seq ASM is a seq algo

Sequential time: obviousAbstract state: obviousBounded exploration: take all the terms in the program all their subterms all logical constants

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Seq Characterization Theorem

For every sequential algorithm A,there exists a sequential ASM behaviorally equivalent to A. In particular, the ASM simulates A step

for step.

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An ASM program for Euc

if a(t) = 0 then d := b(t)else [do in-parallel] a(t+1) := b(t) mod a(t) b(t+1) := a(t) t := t+1

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Euclid

if a = 0 then d := belse a := b mod a b := a

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Euclid with sessions

if a(s)=0 then

d(s) := b(s)

s := s+1

else

a(s) := b(s) mod a(s) b(s) := a(s)

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Reference

ACM Trans. on Computational Logicvol. 1, no. 1 (July 2000), p. 77-111.#141 in Annotated Articles athttp://research/microsoft/~gurevich

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Homework

Write an ASM program for your favorite sequential algorithm. If you want to execute it, go tohttp://research.microsoft.com/foundations/AsmL/

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Part 2: Interactive Algorithms

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Collaborators

Andreas Blass, Dean Rosenzweig, Benjamin RossmanRefs: #166, #170, #171, #176

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Time permitting, plan would be

IntuitionAxiomatic definition Behavioral equivalence

Interactive ASMsInteractive Characterization Theorem Algorithms are ASMs, and vice versa,

as far as behavior is concerned.

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More realistic plan

Motivation, clarification, small examples

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Interstep vs. intrastep

Sequential algorithms and ASMs are interstep interactive.The sequential characterization theorem generalizes to interstep interaction.From now on, by default, interaction is intrastep. But is there intrastep interaction?

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Import

A Turing machine with tape that is only potentially infinite How does it create new cells?

Object creation in object oriented programming Who creates the objects?

Import is a manifestation of interaction.

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Seq ASMs with import

if Move=R, H+1 = undef, … then import x x := H+1 H := x …

ReserveBackground

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Nondeterministic algorithms

A contradiction in termsYogi Berra: “When you come to a fork in the road, take it.”Explanation: nondeterminism is a manifestation of interactionNondeterministic FSMNondeterministic algorithms E.g. bipartite matching

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Nondeterministic ASMs

q := (any x: x in (q,a))Alternative syntax:choose x in (q,a) q := x …

Case of (q,a) =

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More examples

Input, remote procedure calls x := f(17)+2

In the case of parallel algorithms Receiving and sending mail Ping Print

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A more involved example

To paint a picture, an application calls an outside paint method.A paint agent is created and repeatedly calls back: which color for this detail?Consider making two such paint calls in parallel. This is viewed best as a single step.

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What is it all about?

Distributed computations from the point of view of a single agent. Setup: one algorithm interacts with the environment. From the algorithm’s point of view, the interaction is by means of messages only; there are no locations shared by the algorithm and the environment.

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What is environment?

It is everything that can affect the computation of the algorithm but is neither in the algorithm's state nor in its program.This may include other (silicon or carbon) agents, some central authority (think internet poker), OS, communication interfaces (think TCP/IP).

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Various Interaction Mechanisms

RPCMessagesSingle-answer queriesMultiple-answer queriesEtc.

Is there one universal mechanism? Yes.

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Queries

Getting input, printing outputReceiving and sending messagesNon-deterministic choicesNew objectsCalling an external function (in ASMs)Implicit queries

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What’s a message(in the algorithm’s view)?

A query or reply to a query.What if interaction is initiated from outside? The algorithm needs to pay attention

in order to notice an incoming message.

Paying attention is a (possibly implicit) query.

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Are queries blocking?

Not necessarily. It may be blocking: if p! then x:=1

A query may be blocking or not depending on history:if (p ⋎ q) then x:=1An algorithm is patient if all its queries are blocking.

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Is the environment info limited to query replies?

Almost. We argue that the only extra info is the order the replies are received.The broker example.The order of replies is quasi linear.An algorithm is time-insensitive if the order is immaterial.

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Ordinary algorithms

An algorithm is ordinary if it is patient and time-insensitive. Patient: all queries are blocking. Time-insensitive: the order of replies

is immaterial.

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An impatient, time-sensitive algorithm

do in parallel

if α ≺ β then x := -1

if α β then x := 0 if α ≻ β then x := 1

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Characterization Theorem

For every interactive algorithm A,there exists an interactive ASM behaviorally equivalent to A. In particular, the ASM simulates A

step for step.

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Questions?