Title: AUBER F16
1 CDT314 FABER Formal Languages, Automata and
Models of Computation Lecture 13 School of
Innovation, Design and Engineering Mälardalen
University 2011
1
2ContentAlan Turing and Hilbert Program
Universal Turing Machine Chomsky Hierarchy
DecidabilityReducibilityUncomputable
FunctionsRices TheoremChurch-Turing
ThesisComputation beyond Turing
ModelInteractive Computing, Persistent TMs
(Dina Goldin/Peter Wegner)
3http//www.turing.org.uk/turing
- Who was Alan Turing?
- Founder of computer science,
- mathematician,
- philosopher,
- codebreaker,
- visionary man before his time.
- http//www.cs.usfca.edu/www.AlanTuring.net/turing_
archive/index.html- Jack Copeland and Diane
Proudfoot - http//www.turing.org.uk/turing/ The Alan Turing
Home PageAndrew Hodges
4Alan Turing
- 1912 (23 June) Birth, London1926-31 Sherborne
School1930 Death of friend Christopher
Morcom1931-34 Undergraduate at King's College,
Cambridge University1932-35 Quantum mechanics,
probability, logic1935 Elected fellow of King's
College, Cambridge1936 The Turing machine,
computability, universal machine1936-38
Princeton University. Ph.D. Logic, algebra,
number theory1938-39 Return to Cambridge.
Introduced to German Enigma cipher
machine1939-40 The Bombe, machine for Enigma
decryption1939-42 Breaking of U-boat Enigma,
saving battle of the Atlantic
5Alan Turing
- 1943-45 Chief Anglo-American crypto consultant.
Electronic work.1945 National Physical
Laboratory, London1946 Computer and software
design leading the world.1947-48 Programming,
neural nets, and artificial intelligence1948
Manchester University1949 First serious
mathematical use of a computer1950 The Turing
Test for machine intelligence1951 Elected FRS.
Non-linear theory of biological growth1952
Arrested as a homosexual, loss of security
clearance1953-54 Unfinished work in biology and
physics1954 (7 June) Death (suicide) by cyanide
poisoning, Wilmslow, Cheshire.
6Hilberts Program, 1900
- Hilberts hope was that mathematics would be
reducible to finding proofs (manipulating the
strings of symbols) from a fixed system of
axioms, axioms that everyone could agree were
true. - Can all of mathematics be made algorithmic, or
will there always be new problems that outstrip
any given algorithm, and so require creative acts
of mind to solve?
7TURING MACHINES
8- Turings "Machines". These machines are humans
who calculate. - (Wittgenstein)
- A man provided with paper, pencil, and rubber,
and subject to strict discipline, is in effect a
universal machine. (Turing)
9Standard Turing Machine
Tape
......
......
Read-Write head
Control Unit
10The Tape
No boundaries -- infinite length
......
......
Read-Write head
The head moves Left or Right
11......
......
Read-Write head
The head at each time step
1. Reads a symbol 2. Writes a symbol 3. Moves
Left or Right
12Example
3. Moves Left
13The Input String
Input string
Blank symbol
......
......
head
Head starts at the leftmost position of the input
string
14States Transitions
Write
Read
Move Left
Move Right
15Time 1
......
......
Time 2
......
......
16Determinism
Turing Machines are deterministic
Not Allowed
Allowed
No lambda transitions allowed in standard TM!
17Formal Definitions for Turing Machines
18Transition Function
19Turing Machine
Input alphabet
Tape alphabet
States
Final states
Transition function
Initial state
blank
20Universal Turing Machine
21A limitation of Turing Machines
Better are reprogrammable machines.
22Solution
Universal Turing Machine
Characteristics
- Reprogrammable machine
- Simulates any other Turing Machine
23Universal Turing Machine
simulates any other Turing Machine
Input of Universal Turing Machine
- Description of transitions of
24Tape 1
Three tapes
Description of
Tape 2
Universal Turing Machine
Tape Contents of
Tape 3
State of
25Tape 1
Description of
We describe Turing machine as a string of
symbols We encode as a string of symbols
26(No Transcript)
27State Encoding
Head Move Encoding
28Transition Encoding
Transition
Encoding
separator
29Machine Encoding
Transitions
Encoding
separator
30Tape 1 contents of Universal Turing Machine
encoding of the simulated machine as
a binary string of 0s and 1s
31A Turing Machine is described with a binary
string of 0s and 1s.
Therefore
The set of Turing machines forms a language
Each string of the language is the binary
encoding of a Turing Machine.
32Language of Turing Machines
(Turing Machine 1)
L 010100101, 00100100101111,
111010011110010101,
(Turing Machine 2)
33The Chomsky Hierarchy
34The Chomsky Language Hierarchy
Recursively-enumerable
Recursive
Context-sensitive
Context-free
Regular
Non-recursively enumerable
35(No Transcript)
36(No Transcript)
37Theorem
A language is recursively enumerable if and
only if it is generated by an unrestricted
grammar.
38(No Transcript)
39is context-sensitive
40Theorem
A language is context sensitive if and only
if it is accepted by a Linear-Bounded
automaton.
41Linear Bounded Automata (LBAs) are the same as
Turing Machines with one difference
The input string tape space is the only tape
space allowed to use.
42(No Transcript)
43Observation
There is a language which is context-sensitive but
not recursive.
44Decidability
45Consider problems with answer YES or NO.
Examples
46A problem is decidable if some Turing
machine solves (decides) the problem.
Decidable problems
47The Turing machine that solves a problem answers
YES or NO for each instance.
YES
Input problem instance
Turing Machine
NO
48The machine that decides a problem
- If the answer is YES
- then halts in a yes state
- If the answer is NO
- then halts in a no state
These states may not be the final states.
49Turing Machine that decides a problem
YES
NO
YES and NO states are halting states
50Difference between
Recursive Languages (Acceptera) and Decidable
problems (Avgöra)
For decidable problems
The YES states may not be final states.
51Some problems are undecidable
There is no Turing Machine that solves all
instances of the problem.
52A famous undecidable problem
The halting problem
53The Halting Problem
54Theorem
The halting problem is undecidable.
Proof
Assume to the contrary that the halting problem
is decidable.
55YES
halts on
doesnt halt on
NO
56Input initial tape contents
YES
NO
Encoding of
String
57(No Transcript)
58Loop forever
YES
NO
59Construct machine
Input
(machine )
halts on input
If
Then loop forever
Else halt
60copy
61Run machine with input itself
(machine )
Input
halts on input
If
Then loop forever
Else halt
62on input
If halts then loops forever.
If doesnt halt then it halts.
CONTRADICTION !
63This means that
The halting problem is undecidable.
END OF PROOF
64Another proof of the same theorem
If the halting problem was decidable then every
recursively enumerable language would be
recursive.
65Theorem
The halting problem is undecidable.
Proof
Assume to the contrary that the halting problem
is decidable.
66There exists Turing Machine that solves the
halting problem.
YES
halts on
doesnt halt on
NO
67(No Transcript)
68Turing Machine that accepts and halts on any input
NO
reject
halts on ?
YES
accept
Halts on final state
Run with input
reject
Halts on non-final state
69Therefore
is recursive.
Since is chosen arbitrarily, we have
proven that every recursively enumerable language
is also recursive.
But there are recursively enumerable languages
which are not recursive.
Contradiction!
70Therefore, the halting problem is undecidable.
END OF PROOF
71A simple undecidable problem
The Membership Problem
72The Membership Problem
Input
73Theorem
The membership problem is undecidable.
Proof
Assume to the contrary that the membership
problem is decidable.
74There exists a Turing Machine that solves the
membership problem
accepts
YES
NO
rejects
75We will prove that is also recursive
We will describe a Turing machine that accepts
and halts on any input.
76Turing Machine that accepts and halts on any input
YES
accept
accepts ?
NO
reject
77But there are recursively enumerable languages
which are not recursive.
Contradiction!
78Therefore, the membership problem is undecidable.
END OF PROOF
79Reducibility
80(No Transcript)
81(No Transcript)
82Example
the halting problem
reduced to
the state-entry problem.
83The state-entry problem
Inputs
Question
84Theorem
The state-entry problem is undecidable.
Proof
85Suppose we have an algorithm (Turing
Machine) that solves the state-entry problem.
We will construct an algorithm that solves the
halting problem.
86Assume we have the state-entry algorithm
YES
enters
Algorithm for state-entry problem
NO
doesnt enter
87We want to design the halting algorithm
YES
halts on
Algorithm for Halting problem
NO
doesnt halt on
88Modify input machine
Single halt state
halting states
89halts
if and only if
halts on state
90Algorithm for halting problem
91Halting problem algorithm
YES
YES
Generate
State-entry
NO
algorithm
NO
92We reduced the halting problem to the state-entry
problem.
Since the halting problem is undecidable, it must
be that the state-entry problem is also
undecidable.
END OF PROOF
93Another example
The halting problem
reduced to
the blank-tape halting problem.
94The blank-tape halting problem
Input
Turing Machine
Question
Does
halt when started with
a blank tape?
95(No Transcript)
96Suppose we have an algorithm for the blank-tape
halting problem.
We will construct an algorithm for the halting
problem.
97Assume we have the blank-tape halting algorithm
halts on blank tape
YES
Algorithm for blank-tape halting problem
NO
doesnt halt on blank tape
98We want to design the halting algorithm
YES
halts on
Algorithm for halting problem
NO
doesnt halt on
99step 1
step2
if blank tape
execute
then write
with input
100halts on input string
if and only if
halts when started with blank tape.
101Algorithm for halting problem
102Halting problem algorithm
YES
Blank-tape halting algorithm
YES
Generate
NO
NO
103We reduced the halting problem to the blank-tape
halting problem.
Since the halting problem is undecidable, the
blank-tape halting problem is also undecidable.
END OF PROOF
104Summary of Undecidable Problems
105Blank-tape halting problem
Does machine halt when starting on blank
tape?
State-entry Problem
106Uncomputable Functions
107Uncomputable Functions
Domain
Range
A function is uncomputable if it cannot be
computed for all of its domain.
108Example
An uncomputable function
109Then the blank-tape halting problem is
decidable.
110Algorithm for blank-tape halting problem
111Therefore, the blank-tape halting problem must be
decidable.
However, we know that the blank-tape halting
problem is undecidable.
Contradiction!
112END OF PROOF
113Rices Theorem
114Definition
Non-trivial property of recursively enumerable
languages
Any property possessed by some (not
all) recursively enumerable languages.
115Some non-trivial properties of recursively
enumerable languages
116Rices Theorem
Any non-trivial property of a recursively
enumerable language is undecidable.
117Summary in Swedish Rices sats
Om ? är en mängd av Turing-accepterbara språk som
innehåller något men inte alla sådana språk, så
kan ingen TM avgöra för ett godtyckligt
Turing-accepterbart språk L om L tillhör ? eller
ej.
118Exempel
Givet en Turingmaskin M, kan man avgöra om alla
strängar som accepteras av M börjar och slutar på
samma tecken?
119Oavgörbart
Problemet handlar om en icke-trivial
språkegenskap. Det finns TMer vars accepterade
strängar har egenskapen i fråga, och det finns
TMer vars accepterade strängar inte har
egenskapen.
120Formellt
- ? L TM accepterbara språk vars strängar
börjar och slutar på samma tecken.
121Now we will prove some non-trivial
properties without using Rices theorem.
122Theorem
For any recursively enumerable language
it is undecidable whether it is empty.
Proof
We will reduce the membership problem to the
problem of deciding whether is empty.
123Membership problem Does machine accept
string ?
124Assume we have the empty language algorithm
YES
empty
Algorithm for empty language problem
NO
not empty
125We will design the membership algorithm
YES
accepts
Algorithm for membership problem
NO
rejects
126First construct machine
When enters a final state, compare original
input string with .
127if and only if
is not empty
128Algorithm for membership problem
1. Construct
2. Determine if is empty
YES then
NO then
129Membership algorithm
NO
YES
Check if
construct
YES
is empty
NO
130We reduced the empty language problem to the
membership problem.
Since the membership problem is undecidable, the
empty language problem is also undecidable.
END OF PROOF
131Decidabilitycontinued
132Theorem
Proof
We will reduce the halting problem to the finite
language problem.
133Assume we have the finite language algorithm
134We will design the halting problem algorithm
135Otherwise accept nothing (finite language).
136(No Transcript)
137(No Transcript)
138Machine for halting problem
139We reduced the finite language problem to the
halting problem.
Since the halting problem is undecidable, the
finite language problem is also undecidable.
END OF PROOF
140Theorem
Proof
We will reduce the halting problem to the two
strings of equal length- problem.
141Assume we have the two-strings algorithm
142We will design the halting problem algorithm
143Initially, simulates on input .
When enters a halt state, accept symbols
or .
(two equal length strings)
144(No Transcript)
145Algorithm for halting problem
146Machine for halting problem
YES
YES
Check if
construct
NO
NO
has two equal length strings
147We reduced the two strings of equal length -
problem to the halting problem.
Since the halting problem is undecidable, the two
strings of equal length problem is also
undecidable.
END OF PROOF
148Church-Turing Thesis
Source Stanford Encyclopaedia of Philosophy
149A Turing machine is an abstract representation of
a computing device.
It is more like a computer program (software)
than a computer (hardware).
150LCMs Logical Computing Machines Turings
expression for Turing machines were first
proposed by Alan Turing, in an attempt to give
a mathematically precise definition of
"algorithm" or "mechanical procedure".
151- The Church-Turing thesis
- concerns an
- effective or mechanical method
- in logic and mathematics.
152- A method, M, is called effective or
mechanical just in case - M is set out in terms of a finite number of exact
instructions (each instruction being expressed by
means of a finite number of symbols) - M will, if carried out without error, always
produce the desired result in a finite number of
steps - M can (in practice or in principle) be carried
out by a human being unaided by any machinery
except for paper and pencil - M demands no insight or ingenuity on the part of
the human being carrying it out.
153- Turings thesis LCMs logical computing
machines TMs can do anything that could be
described as "rule of thumb" or "purely
mechanical". (Turing 1948) - He adds This is sufficiently well
established that it is now agreed amongst
logicians that "calculable by means of an LCM" is
the correct accurate rendering of such phrases.
154- Turing introduced this thesis in the course of
arguing that the Entscheidungsproblem, or
decision problem, for the predicate calculus -
posed by Hilbert (1928) - is unsolvable.
155- Churchs account of the Entscheidungsproblem
- By the Entscheidungsproblem of a system of
symbolic logic is here understood the problem to
find an effective method by which, given any
expression Q in the notation of the system, it
can be determined whether or not Q is provable in
the system. -
156- Churchs thesis A function of positive integers
is effectively calculable only if recursive.
157Misunderstandings of the Turing Thesis
- Turing did not show that his machines can solve
any problem that can be solved "by instructions,
explicitly stated rules, or procedures" and nor
did he prove that a universal Turing machine "can
compute any function that any computer, with any
architecture, can compute".
158- Turing proved that his universal machine can
compute any function that any Turing machine can
compute and he put forward, and advanced
philosophical arguments in support of, the thesis
here called Turings thesis.
159- A thesis concerning the extent of effective
methods - procedures that a human being unaided
by machinery is capable of carrying out - has no
implication concerning the extent of the
procedures that machines are capable of carrying
out, even machines acting in accordance with
explicitly stated rules.
160- Among a machines repertoire of atomic operations
there may be those that no human being unaided by
machinery can perform.
161- Turing introduces his machines as an idealised
description of a certain human activity, the
tedious one of numerical computation, which until
the advent of automatic computing machines was
the occupation of many thousands of people in
commerce, government, and research
establishments. -
162- Turings "Machines". These machines are humans
who calculate. (Wittgenstein) - A man provided with paper, pencil, and rubber,
and subject to strict discipline, is in effect a
universal machine. (Turing)
163Computation Beyond Turing Model
- Traditionally, the dynamics of computing
systems, their unfolding behavior in space and
time has been a mere means to the end of
computing the function which specifies the
algorithmic problem which the system is solving.
In much of contemporary computing, the situation
is reversed the purpose of the computing system
is to exhibit certain behaviour. () We need a
theory of the dynamics of informatic processes,
of interaction, and information flow, as a basis
for answering such fundamental questions as What
is computed? What is a process? What are the
analogues to Turing completeness and universality
when we are concerned with processes and their
behaviours, rather than the functions which they
compute? (Abramsky, 2008) - Abramsky, S. (2008) Information, processes and
games. In Philosophy of Information van Benthem,
J., Adriaans, P., Eds. North Holland Amsterdam,
The Netherlands pp. 483549.
164Computation Beyond Turing Model
- According to Abramsky, there is the need for
second generation models of computation, and in
particular there is the need for process models
such as Petri nets, Process Algebra, and similar.
- The first generation models of computation were
originating from problems of formalization of
mathematics and logic, while processes or agents,
interaction, and information flow are genuine
product of the modern development of computing. - In the second generation models of computation,
previous isolated systems with limited
interactions with the environment are replaced by
processes or agents for which the interactions
with each other and with the environment are
fundamental.
165Computation Beyond Turing Model
- As a result of interactions among agents and with
the environment, complex behavior emerges. - The basic building block of this interactive
approach is the agent, and the fundamental
operation is interaction. - The ideal features sought for are computational
qualities of computational agents such as
biological organisms such as resilience and
self- capabilities. - This approach works at both macro-scale (such as
processes in operating systems, software agents
on the Internet, transactions, etc.) and on
micro-scale (program implementation down to
hardware).
166 Interaction Conjectures, Results, and Myths
Additional reading
- Dina GoldinUniv. of Connecticut, Brown
Universityhttp//www.cse.uconn.edu/dqg
167Fundamental Questions Underlying Theory of
Computation
- What is computation?
- How do we model it?
168Shared Wisdom(from our 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.
169The Mathematical Worldview
- 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
170The Operating System Conundrum
Real programs, such as operating systems and word
processors, 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, its
useless but arent OSs useful??
171Rethinking Shared Wisdom(what do computers do?)
- 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 - Wegners conjecture Interaction is more powerful
than algorithms
- 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
172Example 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.
173Driving home from work (cont.)
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
thats detailed enough?
Worse yet the domain is dynamic. The output
depends on weather conditions, and on other
drivers and pedestrians. We cant possibly be
expected to predict that in advance!
Nevertheless the problem is solvable! Google
autonomous vehicle research
174Driving home from work (cont.)
The problem is solvable interactively. Interactiv
e 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.
175Outline
- Rethinking the mathematical worldview
- Persistent Turing Machines (PTMs)
- PTM expressiveness
- Sequential Interaction
- Sequential Interaction Thesis
- The Myth of the Church-Turing Thesis
- the origins of the myth
- Conclusions and future work
176Sequential 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
177Sequential 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 John,
ok), - (record Hopkins, ok), (playback, John Hopkins),
- Simulation of other sequential interactive
systems is analogous.
178Church-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 myth
- the function-based behavior of algorithms does
not capture all forms of computation - this myth has been dogmatically accepted by the
CS community - 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.
179Origins of the Church-Turing Thesis Myth
- A TM can do anything that a computer can do.
- Based on several claims
- A problem is solvable if there exists a Turing
Machine for computing it. - A problem is solvable if it can be specified by
an algorithm. - 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
180Deconstructing 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.
181Deconstructing 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. Euclids 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. Knuth68
182Deconstructing 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. RiceRice69 - An algorithm is a collection of simple
instructions for carrying out some task.
Commonplace in everyday life, algorithms
sometimes are called procedures or recipes...
Sipser97
183Outline
- Rethinking the mathematical worldview
- Persistent Turing Machines (PTMs)
- PTM expressiveness
- Sequential Interaction
- The Myth of the Church-Turing Thesis
- Conclusions and future work
184The Shift to Interaction in CS
Algorithmic
Interactive
Computation transforming input to output Computation carrying out a task over time
Logic and search in AI Intelligent agents, partially observable environments, learning
Procedure-oriented programming Object-oriented programming
Closed systems Open systems
Compositional behavior Emergent behavior
Rule-based reasoning Simulation, control, semi-Markov processes
185The Interactive Turing Test
- From answering questions to holding discussions.
- Learning from -- and adapting to -- the
questioner. - Book intelligence vs. street smarts.
- It is hard to draw the line at what is
intelligence and what is environmental
interaction. In a sense, it does not really
matter which is which, as all intelligent systems
must be situated in some world or other if they
are to be useful entities. Brooks
186Modeling Interactive Computation PTMs in
Perspective
- Many other 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 PTMs unique?
- Provably more expressive than TMs.
- Bridging the gap between concurrency theory
(labeled transition systems) and traditional TOC.
187Future Work 3 conjectures
- Theory of Sequential Interaction
- conjecture notions analogous to computational
complexity, logic, and recursive functions can be
developed for sequential interaction computation - Multi-stream interaction
- From hidden variables to hidden interfaces
- conjecture multi-stream interaction is more
powerful than sequential interaction Wegner97 - Formalizing indirect interaction
- Interaction via persistent, observable changes to
the common environment - In contrast to direct interaction (via message
passing) - conjecture direct interaction does not
capture all forms of multi-agent behaviors
188Referenceshttp//www.cse.uconn.edu/dqg/papers/
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