7 June 2005
CWI
1
Interaction:
Conjectures, Results, and Myths
Dina Goldin
Univ. of Connecticut, Brown University
http://www.cse.uconn.edu/~dqg
7 June 2005
CWI
2
Fundamental Questions
Underlying Theory of Computation
What is computation?
How do we model it?
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Shared Wisdom
(from undergraduate Theory of Computation courses)
computation
: finite transformation
of input to output
input
: finite size (e.g. string or
number)
closed system
: all input available
at start, all output generated at end
behavior
: functions, transformation of
input data to output data
Church

Turing thesis
: Turing
Machines capture this (algorithmic)
notion of computation
Mathematical worldview
:
All computable problems
are function

based.
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4
The Operating System
Conundrum
Real programs… often receive an unbounded amount of input over
time, and never "finish" their task. Turing machines do not model
such ongoing computation well.
[TM entry, Wikipedia]
If a computation
does not terminate,
it’s “useless”
–
but aren’t OS’s
useful??
7 June 2005
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5
•
Rethinking the mathematical worldview
–
Historical perspective
–
Algorithmic vs. interactive computation
–
Wegner’s conjecture
–
Driving home from work
•
Persistent Turing Machines (PTMs)
•
PTM expressiveness
•
Sequential Interaction Thesis
•
The Myth of the Church

Turing Thesis
•
Future work
Outline
7 June 2005
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“The theory of computability and non

computability [is] usually
referred to as the theory of recursive functions... the notion of TM has
been made central in the development."
Martin Davis,
Computability & Unsolvability
, 1958
“Of all undergraduate CS subjects, theoretical computer science has
changed the least over the decades.”
SIGACT News
, March 2004
“A TM can do anything that a computer can do.”
Michael Sipser,
Introduction to the Theory of Computation
, 1997
The Mathematical Worldview
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Rethinking Shared Wisdom:
(what do computers do?
)
computation
: finite
transformation of input to output
input
: finite

size (string or number)
closed system
: all input available at
start, all output generated at end
behavior
: functions, algorithmic
transformation of input data to
output data
Church

Turing thesis
: Turing
Machines capture this (algorithmic)
notion of computation
computation
: ongoing process which
performs a task or delivers a service
dynamically
generated stream of input
tokens (requests, percepts, messages)
open system
: later inputs depend on
earlier outputs and vice versa (I/O
entanglement, history dependence)
behavior
: processes, components,
control devices, reactive systems,
intelligent agents
Turing Machines do not capture this
(interactive) notion of computation
7 June 2005
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•
Many interactive models
–
Reactive [MP] and embedded systems
–
Dataflow, I/O automata [Lynch], synchronous languages, finite/pushdown
automata over infinite words
–
Interaction games [Abramsky], online algorithms [Albers]
–
TM extensions: on

line Turing machines [Fischer], interactive Turing
machines [Goldreich]
•
Concurrency Theory
–
Focuses on
communication
(between concurrent agents/processes) rather
than computation [Milner]
–
Orthogonal
to the theory of computation and TMs.
•
What makes our approach unique?
–
Communication (I/O) is part of computation.
•
a paradigm change
–
Bridging the gap between concurrency theory (labeled transition systems)
and traditional TOC.
•
models borrow from both fields
–
Focus on expressiveness
•
rather than mere convenience of modeling
Modeling Interactive Computation
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Our Motivation
Wegner’s Conjecture:
Interaction is more powerful
than algorithms
[CACM’97]
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Example:
Driving home from work
Algorithmic input
: a description of the
world
(a static “map”)
Output
: a sequence of pairs of #s (time

series data)

for turning the wheel

for pressing gas/break
Similar to classic AI search/planning problems.
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Can you get the car
into my driveway
without running
over my kid?
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But
… in a real

world environment, the output depends
on every grain of sand in the road (
chaotic behavior
).
Can we possibly have a map that’s detailed enough?
Worse yet
… the domain is dynamic. The output
depends on weather conditions, and on other drivers and
pedestrians.
We can’t possibly be expected to predict that in advance!
Nevertheless
the problem is solvable!
Google “autonomous vehicle research”
Driving home from work (cont.)
?
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Driving home from work (cont.)
The problem is solvable interactively
.
Interactive input
: stream of video camera images, gathered
as we are driving
Output
: the desired time

series data, generated
as we are driving
similar to control systems, or online computation
A paradigm shift in the conceptualization of computational problem solving.
7 June 2005
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•
Rethinking the mathematical worldview
•
Persistent Turing Machines (PTMs)
–
extending N3TM computations model
–
Persistent Stream Languages
–
examples
•
PTM expressiveness
•
Sequential Interaction Thesis
•
The Myth of the Church

Turing Thesis
•
Future work
Outline
7 June 2005
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Three

tape Turing Machines
(N3TM)
s

current
state
w
1

contents of
input
tape
w
2

contents of
work
tape
w
3

contents of
output
tape
n
1
, n
2
, n
3

tape head posns
•
N3TM configurations:
input
work
output
S
•
Computation = a sequence of transitions between
configurations, from initial to halting.
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N3TM macrosteps
w
in
, w
Notation:
w
in
S
o
w
e
w
in
S
h
w’
w
out
M

< s
0
, w
in
, w,
e
, 1, 1, 1 >
< s
h
, w
in
, w’, w
out
, 1, 1, 1 >
w’, w
out
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Extending N3TM Computations
•
Dynamic stream semantics

Inputs are
streams
of
dynamically
generated tokens (strings).

For each input token, there is an N3TM macrostep generating the
corresponding output token.
•
Persistence (memory)

The contents
w
of the work tape at the beginning of each macrostep
is the
same
as at the end of the previous one.
in
1
S
0
e
e
S
h
out
1
w
1
in
1
in
2
S
0
w
1
e
S
h
out
2
w
2
in
2
...
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•
Persistent Stream Language
of a PTM: set of streams
•
Conductive stream semantics:
...
Persistent Turing Machine (PTM)
PTM: N3TM with persistent stream

based
computational semantics
PTM
memory
Environment
Interaction Stream
stream of inputs
stream of outputs
7 June 2005
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Formal Definition
))}
'
(
(
'
,
'
,
w
M
PSL
w
w
w
w
o
i
s
(
Coinductive
definition, relative to N3TM
M
and memory
w
)
PSL(M(w))
= {
(
w
i
,
w
o
),
s
’
S

$
w
’
S
*:
PSL
=
{
PSL
(
M
) 
M
is a PTM}
7 June 2005
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PTM Example
•
Answering Machine (AM)
f
AM
(
record Y
,
X
) = (
ok
,
XY
)
f
AM
(
erase
,
X
) = (
done
,
e
)
f
AM
(
playback
,
X
) = (
X
,
X
)
–
PSL(
AM
) contains:
(
record
hello
,
ok
), (
erase
,
done
) , (
record
Farhad
,
ok
),
(
record
Arbab
,
ok
) , (
playback
,
Farhad Arbab
), …
–
Sequential objects as PTMs
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More PTM Examples
•
PowerPoint Editor
–
Input: mouse movements, keyboard clicks
–
Output: refreshed view of presentation
•
Driving Agent
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•
At each step, output first bit of
previous
step.
–
inputs
in
1
; outputs
1
–
inputs
in
2
; outputs
1
st
bit of in
1
–
inputs
in
3
; outputs
1
st
bit of in
2
–
...
•
PSL(Latch) contains:
•
PTM as a Labeled Transition System
–
Latch has 3 states, meaning “contents of worktape”
–
The labels are input/output pairs, as in the interaction stream.
PTM Example: Latch
#
1
0
(1*,1)
(0*,1)
(0*,1)
(1*,0)
(1*,1)
(0*,0)
7 June 2005
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•
Rethinking the mathematical worldview
•
Persistent Turing Machines (PTMs)
•
Interactive Transition Systems
–
Different notions of equivalence
–
Correspondence to PTMs
•
Sequential Interaction Thesis
•
The Myth of the Church

Turing Thesis
•
Future work
Outline
7 June 2005
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Interactive Transition Systems
over
S
•
S
is set of
states
•
r
is
initial state (root)
•
m
is
transition relation
Required to be recursively enumerable
<
S, m, r
>
#
1
0
(1*,1)
(0*,1)
(0*,1)
(1*,0)
(1*,1)
(0*,0)
7 June 2005
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•
Infinite sequences of input/output token

pairs
emanating from a particular ITS state
•
ISL(T),
the
interactive stream language
of an
ITS
T,
is the set of all such sequences emanating from
T
’s root.
Interactive Stream
Equivalence
T
1
=
ISL
T
2
if
ISL
(
T
1
) =
ISL
(
T
2
)
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ITS Isomorphism
1.
2.
Let
T
1
,
T
2
be ITSs
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ITS Bisimulation
Let
be ITSs,
i=1,2
is a (strong)
interactive bisimulation
if:
1.
2.
3.
Clause 2. with roles of
s
and
t
reversed
T
1
=
bisim
T
2
if
$
an interactive bisim. between them
R
is defined recursively, has greatest fixpoint semantics
7 June 2005
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=
iso
=
bisim
=
ISL
Comparing Equivalence
Relations
Equivalent machines = same behavior
Equivalence classes = set of distinct behaviors
=
1
is
strictly finer
than =
2
if
(x =
1
y) implies (x =
2
y)
but not vice

versa
7 June 2005
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From PTMs to ITSs
Reachable memories of a PTM
M
:
Set of words (work

tape contents)
w
encountered
after zero or more macrosteps.
reach
(
M
),
m
,
e
ξ(M)
o
M
i
w
s
s
w
,
'
,
m
o
i
w
s
w
s
,
'
,
,
iff
where
ξ i
s a bijection between PTMs and ITSs
[I&C’04]
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Theorem
:
Proof
:
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PTMs
ITSs
=
ms
=
iso
=
bisim
=
ISL
=
PSL
PTMs vs. ITSs
Two representation of the same computational behavior
7 June 2005
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32
•
Rethinking the mathematical worldview
•
Persistent Turing Machines (PTMs)
•
Interactive Transition Systems
•
PTM expressiveness
–
Infinite equivalence hierarchy
•
Equivalence hierarchy gap
–
Unbounded nondeterminism and divergence
–
Amnesic PTMs
•
It pays to be persistent
•
Sequential Interaction Thesis
•
Future work
Outline
7 June 2005
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33
Infinite Equivalence Hierarchy
•
L
k
(
M
) = stream prefix language of PTM
M
set of prefixes of length
k for streams in
PSL
(
M
)
represents finite observations of M (testing equivalence)
•
L
∞
(
M
) =
U
k
≥
1
L
k
(
M
)
•
Corresponding
notions of equivalence
:
M
1
=
k
M
2
:
L
k
(
M
1
)
= L
k
(
M
2
)
M
1
=
∞
M
2
:
L
∞
(
M
1
)
= L
∞
(
M
2
)
•
=1
is like Turing Machine equivalence
7 June 2005
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34
Infinite Equivalence Hierarchy
(cont)
=
2
=
1
...
=
∞
=
PSL
What is your
name?
A deterministic
machine
More equivalence classes = more distinct behaviors
7 June 2005
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35
=
PSL
=
=
2
=
1
...
Equivalence Hierarchy Gap
•
Proof
: construct PTMs
M
1
and
M
2
where
L
(
M
1
) =
L
(
M
2
)
but
PSL
(
M
1
)
≠
PSL
(
M
2
)
•
Note
: M
1
exhibits
unbounded nondeterminism
•
Unbounded nondeterminism
implies
divergence.
[I&C 2004]
7 June 2005
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Proof Details
•
M1 produces output streams of the form 1*0
+
On 1st macrostep, initializes a persistent string
n
of
1
’s:
while true do
write ‘1’ on the work tape, move head to the right;
nondeterministically choose to exit loop or continue
The output at every macrostep is determined as follows:
if n > 0
then decrement n by 1 and output ‘1’;
else output ‘0’
•
M2 is the same, but also produces the stream 1*
•
M
1
, M
2
exhibit
unbounded non

determinism
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Divergent Computation
Theorem
: If a PTM M has unbounded
nondeterminism, then M diverges.
[I&C’04]
e
(
S
*, 1)
n = 0
n = 1
n = 2
n = 3
(
S
*, 1)
(
S
*, 1)
(
S
*, 1)
(
S
*, 1)
(
S
*, 1)
(
S
*, 1)
(
S
*, 1)
(
S
*, 1)
(
S
*, 0)
...
(
S
*,
t
)
s
div
(
S
*,
t
)
...
M
1
7 June 2005
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38
Amnesic PTM Computation:
stream

based but not persistent
))}
'
(
(
'
,
w
M
PSL
w
i
w
',
w
o
s
e
PTM
M
is
amnesic
if
PSL(M)
ASL
ASL
=
{
ASL
(
M
) 
M
is a PTM}
7 June 2005
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39
Amnesic PTMs:
“half

way” between TMs and PTMs
Example: squaring machine (
out
i
= in
i
2
)
[Prasse & Rittgen]
•
Amnesic PTMs extend TMs with stream

based semantics.
At least as expressive as TMs
•
Unlike PTMs, they lack persistence.
in
1
S
0
e
e
S
h
out
1
w
1
in
1
in
2
S
0
e
e
S
h
out
2
w
2
in
2
...
7 June 2005
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It Pays to be Persistent
ASL
PSL
Two arguments that PTMs are more expressive
than Amnesic PTMs:
1.
Collapse of the equivalence hierarchy.
2.
Smaller set of stream languages.
=
=
1
=
PSL
7 June 2005
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Summary of Results
[I&C’04]
PTMs
ITSs
=
=
=
2
=
1
=
ms
=
iso
=
bisim
=
ISL
=
PSL
...
=
TM
7 June 2005
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42
•
Rethinking the mathematical worldview
•
Persistent Turing Machines (PTMs)
•
Interactive Transition Systems
•
PTM expressiveness
•
Sequential Interaction
–
Sequential Interaction Thesis
–
Universal PTMs
–
Church

Turing Thesis revisited
•
Future work
Outline
7 June 2005
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43
Sequential Interaction
•
Sequential interactive computation:
system continuously interacts with its environment
by alternately accepting an input string
and computing a corresponding output string.
•
Examples:

method invocations of an object instance
in an OO language

a C function with static variables

queries/updates to single

user databases

recurrent neural networks

control systems

online computation

transducers

dynamic algorithms

embedded systems
7 June 2005
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44
Sequential Interaction Thesis
•
Universal PTM
: simulates any other PTM
–
Need additional input describing the PTM (only once)
•
Example
: simulating Answering Machine
(
simulate
AM
,
will

do
),
(
record
hello
,
ok
), (
erase
,
done
), (
record
Farhad
,
ok
),
(
record
Arbab
,
ok
), (
playback
,
Farhad Arbab
), …
Simulation of other sequential interactive systems is analogous.
Whenever there is an effective method for performing
sequential interactive computation, this computation
can be performed by a Persistent Turing Machine
7 June 2005
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45
Origins of the Church

Turing Thesis Myth
A TM can do anything that a computer can do.
Based on several claims:
1.
A problem is
solvable
if there exists a Turing Machine
for computing it.
2.
A problem is
solvable
if it can be specified by an algorithm.
3.
Algorithms
are what computers do.
Each claim is correct in isolation
provided we understand the underlying assumptions
Together, they induce an incorrect conclusion
TMs = solvable problems = algorithms = computation
7 June 2005
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Deconstructing the Turing Thesis Myth (1)
TMs = solvable problems
•
Assumes:
All computable problems are
function

based
.
•
Reasons
:
–
Theory of Computation started as a field of mathematics;
mathematical principles were adopted for the fundamental
notions of computation, identifying computability with the
computation of functions, as well as with Turing Machines.
–
The batch

based modus operandi of original computers did
not lend itself to other conceptualizations of computation.
7 June 2005
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47
Deconstructing the Turing Thesis Myth (2)
solvable problems = algorithms
Assumes:

Algorithmic computation is also function based;
i.e., the computational role of an algorithm
is to transform input data to output data.
•
Reasons
:
–
Original (mathematical) meaning of “algorithms”
E.g. Euclid’s greatest common divisor algorithm
–
Original (Knuthian) meaning of “algorithms”
“
An algorithm has zero or more inputs, i.e., quantities which are
given to it initially before the algorithm begins.“
[Knuth’68]
7 June 2005
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48
Deconstructing the Turing Thesis Myth (3)
algorithms = computation
•
Reasons
:
–
The ACM Curriculum
(1968): Adopted algorithms as the central
concept of CS without explicit agreement on the meaning of this term.
–
Textbooks
: When defining algorithms, the assumption of their closed
function

based nature was often left implicit, if not forgotten
“
An algorithm is a recipe, a set of instructions or the specifications
of a process for doing something. That something is usually solving
a problem of some sort
.”
[Rice&Rice’69]
“
An algorithm is a collection of simple instructions for carrying out
some task. Commonplace in everyday life, algorithms sometimes
are called procedures or recipes...”
[Sipser’97]
7 June 2005
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49
Church

Turing Thesis Revisited
•
Church

Turing Thesis:
Whenever there is an effective method
for obtaining the values of a mathematical function,
the function can be computed by a Turing Machine
•
Common Reinterpretation (Strong Church

Turing Thesis)
A TM can do (compute) anything that a computer can do
•
The equivalence of the two is a widespread myth
the function

based behavior of algorithms does
not
capture all forms of computation
•
Turing himself would have denied it
in the same paper where he introduced what we now call Turing
Machines, he also introduced
choice machines,
as a distinct
model of computation
choice machines extend Turing Machines to interaction by
allowing a human operator to make choices during the
computation.
7 June 2005
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50
•
Rethinking the mathematical worldview
•
Persistent Turing Machines (PTMs)
•
Interactive Transition Systems
•
PTM expressiveness
•
Sequential Interaction
•
Future work
Outline
7 June 2005
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51
•
Theory of Sequential Interactiove Computation
This is a robust notion of computation, admitting
notions analogous to
computational complexity,
logic, and recursive functions
•
Where are the ports?

Multi

stream interaction
multi

stream interaction is more powerful than
sequential interaction
[Wegner’97]
•
Formalizing indirect interaction
direct interaction (via message passing) does not
capture all forms of multi

agent behaviors
Future Work: 3 conjectures
7 June 2005
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52
Direct & Indirect Interaction
DIRECT INTERACTION: interaction via
message passing
.
•
Known as “communication” in concurrency theory.
•
ID of destination agent specified in the message.
INDIRECT INTERACTION: interaction via
persistent
,
observable
changes to the common environment.
•
Agents can affect each other without interacting directly, when one of them
makes changes to their environment that the other later observes.
•
Example
: multi

user document editing, stigmergy.
7 June 2005
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53
Role of the Environment
in Indirect Interaction
•
The environment is the
medium
of communication.
•
It must be
changeable
and
observable
: when one agent
changes it, another can observe the change.
•
It must be
persistent
: change to it remain, so they can be
observed later.
•
It may posess
locality
: given an agent, parts of the
environment are accessible to it while others are not (may
be different parts for actuating & sensing).
May be very simple (Dining Philosophers example)
or complex (Foraging Ants example)
7 June 2005
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54
Dining Philosophers
•
Classic problem in concurrency
and shared resources: define a
protocol for a ring

shaped
arrangement of processes
(philosophers) where no two
adjacent processes may execute
simultaneously.
•
Goal
: let the philosophers carry
on while avoiding starvation.
•
Starvation
: all philosophers
pick up one chopstick and wait
to pick up the other.
•
Mobile version
: philosophers
may leave table when sleepy, or
join at empty place when ready
to eat and think.
Indirect Interaction:
philosophers interact with neighbors
(persistent observable change to
environment = picking up and
putting down chopsticks).
Direct Interaction
:
philosophers interact with chopsticks
(modeled as simple processes).
7 June 2005
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55
Foraging Ants
•
Classic problem in artificial life
and stigmergy.
•
Ants wander
randomly
looking for
sources of food.
•
If they find food, they leave
behind a
phermone trail
(or
reinforce existing one).
•
Or, if they find a phermone trail,
they follow it in search of food.
•
Achieves coordination without
centralization.
•
An example of
swarm computing
,
and of
emergent behavior.
Indirect Interaction:
ants interact with each other
(persistent observable change to
environment = phermone trails).
Direct Interaction
:
?
7 June 2005
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56
Properties of Indirect Interaction (1)
Late binding of recipient
(anonymity)
Identity of the observer of given
state changes may be determined
by dynamic events occurring
after the change is made.
[mobile] The chopstick
X puts down will be
picked up either by X or
by who

ever happens to
be sitting next to X at
the time they get
hungry.
X’s pheromone trail
will be detected by
whoever happens to
wander past it.
Time decoupling (asynchrony)
There may be a delay between
the change and its observation,
whose duration may be
determined by dynamic events.
There may be a delay
after X puts down a
chopstick before it’s
picked up again; cannot
know when the neighbor
will get hungry.
There may be a delay
after X puts down
pheromone before it’s
detected; cannot know
when someone will
wander by to detect it.
Space Decoupling
Indirect interaction need not
imply co

location (for mobile
agents); first agent may leave
after making changes, second
arriving later.
[mobile] X may go
away after putting down
chopstick; when Y picks
it up, he interacts with X
without ever being in
the same location.
Ants who lay down
and pick up a
pheromone trail
usually do not actually
meet each other.
Dining Philosophers Foraging Ants
7 June 2005
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57
Properties of Indirect Interaction (2)
Localization
Agents may be restricted to
make/ observe changes in only
a part of the shared space (that
is
local
to them).
A philosopher can
pick up / put down
only the chopsticks
next to him.
An ant can leave or
sense pheromones
only in its local
neighborhood.
Non

intentionality
Intent to communicate is not
required, nor awareness that
communication is occurring; agents
may simply be doing their own task.
X puts down
chopsticks because
he is done, NOT
because she wants
others to know that
they can eat.
X leaves pheromones
because he is
programmed to do
this when carrying
food, not out of
intent to notify
others.
Hybrid nature
The real world may serve as the
medium of indirect interaction; the
system will involve real

world
processes, and will be hybrid rather
than fully digital.
[analog] X will not
eat with dirty
chopsticks (where
the notion of
dirtiness is fuzzy).
Pheromones
evaporate over time.
Dining Philosophers Foraging Ants
7 June 2005
CWI
58
Ubiquity of Indirect Interaction
•
Social Biology
: Social insects living in colonies interact
indirectly by making changes to common structures (termite
piles) or through pheromones.
•
Operating Systems
: Processes exchange information via
semaphores
in shared memory
•
Programming Languages
:
Tuple spaces
in Linda enable
coordination by indirect interaction.
•
Anatomy
: Cells exchange information via
hormones
in the
blood stream.
•
Economics
: the
value
of stocks, bonds, and currency acts as
medium of interaction between buyers and sellers as they
negotiate prices.
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Formalizing Indirect Interaction
•
There is a need for a formalization of indirect
interaction that
–
models its properties explicitly, without the intermediate
protocol layer
–
directly reflects problem
semantics
(i.e. chopsticks are passive
objects, not active agents)
–
is domain

independent
–
allows for real world as medium of interaction
–
extends existing models of interaction (Persistent Turing
Machines)
•
Conjecture
–
decentralized coordination in multiagent systems of simple
agents
requires
indirect interaction (to provide asynchrony and
anonymity)
7 June 2005
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60
References
http://www.cse.uconn.edu/~dqg/papers/
[Wegner’97]
Peter Wegner
Why Interaction is more Powerful than Algorithms
Communications of the ACM
, May 1997
[EGW’04]
Eugene Eberbach, Dina Goldin, Peter Wegner
Turing's Ideas and Models of Computation
book chapter, in
Alan Turing: Life and Legacy of a Great Thinker
, Springer
2004
[I&C’04]
Dina Goldin, Scott Smolka, Paul Attie, Elaine Sonderegger
Turing Machines, Transition Systems, and Interaction
Information & Computation Journal
, 2004
[GW’04]
Dina Goldin, Peter Wegner
The Church

Turing Thesis: Breaking the Myth
presented at CiE 2005, Amsterdam, June 2005
to be published in LNCS
7 June 2005
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61
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