Behavioral Computation Theory: Tutorial

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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

• Intuition
• Axiomatic definition
• Behavioral equivalence
• Sequential abstract state machines
• Sequential Characterization Theorem
• Algorithms are ASMs, and vice versa, as far as
  behavior is concerned.

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Example Euclids 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.
• If a(t) 0, set d b(t) and halt.
• Set a(t1) b(t) mod a(t),set b(t1) 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
  distributed
• Positive 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 with
• a nonempty set States
• a nonempty subset Initial States
• a 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 semanticalthe
  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 whats local?
• How to measure work?

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Bounded work vs. bounded change

• Bounded work ? bounded change, butbounded
  change ? bounded work, e.g.
• if ?x?y(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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Eucs 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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Eucs associates

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

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Eucs 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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Eucs 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(t1), b(t) mod a(t), b(t1), 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 obvious
• Abstract state obvious
• Bounded 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(t1) b(t) mod a(t) b(t1)
  a(t) t t1

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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 s1
• 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/micr
  osoft/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/A
  smL/

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Part 2 Interactive Algorithms
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Collaborators

• Andreas Blass, Dean Rosenzweig, Benjamin Rossman
• Refs 166, 170, 171, 176

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Time permitting, plan would be

• Intuition
• Axiomatic definition
• Behavioral equivalence
• Interactive ASMs
• Interactive 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 MoveR, H1 undef, then import x x
  H1 H x
• Reserve
• Background

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Nondeterministic algorithms

• A contradiction in terms
• Yogi Berra When you come to a fork in the road,
  take it.
• Explanation nondeterminism is a manifestation of
  interaction
• Nondeterministic FSM
• Nondeterministic algorithms
• E.g. bipartite matching

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Nondeterministic ASMs

• q (any x x in ?(q,a))
• Alternative syntaxchoose 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 algorithms 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

• RPC
• Messages
• Single-answer queries
• Multiple-answer queries
• Etc.
• Is there one universal mechanism? Yes.

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Queries

• Getting input, printing output
• Receiving and sending messages
• Non-deterministic choices
• New objects
• Calling an external function (in ASMs)
• Implicit queries

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Whats a message(in the algorithms 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
  x1
• A query may be blocking or not depending on
  historyif (p ? q) then x1
• An 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 a ? ß then x -1
• if a ? ß then x 0
• if a ? ß 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?