1. Formal Languages, Automata and Models of Computation
CD5560
FABER
Formal Languages, Automata and Models of Computation
Lecture 13
Mälardalen University
2007

2. Content Introduction Universal Turing Machine Chomsky Hierarchy Decidability Reducibility Uncomputable Functions Rice’s Theorem Interactive Computing & Persistent TM’s (Dina Goldin)

3. http://www.turing.org.uk/turing/ Who was Alan Turing?
Founder of computer science,
mathematician,
philosopher,
codebreaker,
visionary man before his time.
Jack Copeland and Diane Proudfoot
The Alan Turing Home Page Andrew Hodges

4. Alan Turing
1912 (23 June): Birth, London : Sherborne School 1930: Death of friend Christopher Morcom : Undergraduate at King's College, Cambridge University : Quantum mechanics, probability, logic 1935: Elected fellow of King's College, Cambridge 1936: The Turing machine, computability, universal machine : Princeton University. Ph.D. Logic, algebra, number theory : Return to Cambridge. Introduced to German Enigma cipher machine : The Bombe, machine for Enigma decryption : Breaking of U-boat Enigma, saving battle of the Atlantic

5. Alan Turing
: Chief Anglo-American crypto consultant. Electronic work. 1945: National Physical Laboratory, London 1946: Computer and software design leading the world : Programming, neural nets, and artificial intelligence 1948: Manchester University 1949: First serious mathematical use of a computer 1950: The Turing Test for machine intelligence 1951: Elected FRS. Non-linear theory of biological growth 1952: Arrested as a homosexual, loss of security clearance : Unfinished work in biology and physics 1954 (7 June): Death (suicide) by cyanide poisoning, Wilmslow, Cheshire.

6. Hilbert’s Program, 1900
Hilbert’s 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?

7. TURING MACHINES

8. Turing’s "Machines". These machines are humans who calculate
Turing’s "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)

9. Standard Turing Machine
Tape
......
......
Read-Write head
Control Unit

10. The 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

12. Example
Time 0
......
Time 1
......
1. Reads
2. Writes
3. Moves Left

13. The Input String Input string Blank symbol ...... ...... head
Head starts at the leftmost position
of the input string

14. States & Transitions
Write
Read
Move Left
Move Right

15. Time 1
......
......
Time 2
......
......

16. Determinism Not Allowed Turing Machines are deterministic Allowed
No lambda transitions allowed in standard TM!

17. Formal Definitions for Turing Machines

18. Transition Function

19. Turing Machine Input alphabet Tape alphabet States Final Transition
function
Initial
state
blank

20. Universal Turing Machine

21. A limitation of Turing Machines:
Turing Machines are “hardwired”
they execute
only one program
Better are reprogrammable machines.

22. Solution:
Universal Turing Machine
Characteristics:
Reprogrammable machine
Simulates any other Turing Machine

23. Universal Turing Machine
simulates any other Turing Machine
Input of Universal Turing Machine
Description of transitions of
Initial tape contents of

24. Three tapes Description of Universal Turing Machine Tape Contents of
State of

25. We describe Turing machine as a string of symbols:
Tape 1
Description of
We describe Turing machine
as a string of symbols:
We encode as a string of symbols

26. Alphabet Encoding
Symbols:
Encoding:

27. State Encoding
States:
Encoding:
Head Move Encoding
Move:
Encoding:

28. Transition Encoding
Transition:
Encoding:
separator

29. Machine Encoding
Transitions:
Encoding:
separator

30. Tape 1 contents of Universal Turing Machine:
encoding of the simulated machine M
as a binary string of 0’s and 1’s

31. A Turing Machine is described
with a binary string of 0’s and 1’s.
Therefore:
The set of Turing machines forms a language:
Each string of the language is
the binary encoding of a Turing Machine.

32. Language of Turing Machines, ,
…… }
(Turing Machine 2) ……

33. The Chomsky Hierarchy

34. The Chomsky Language Hierarchy
Recursively-enumerable
Recursive
Context-sensitive
Context-free
Regular
Non-recursively enumerable

35. Unrestricted Grammars
Productions
String of variables
and terminals

36. Example of unrestricted grammar

37. Theorem A language L is recursively enumerable
if and only if it is generated by an
unrestricted grammar.

38. Context-Sensitive Grammars
Productions
String of variables
and terminals
String of variables
and terminals
and

39. The language
is context-sensitive:

40. Theorem A language L is context sensitive
if and only if it is accepted by a Linear-Bounded automaton.

41. Linear 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. Linear Bounded Automaton (LBA)
Input string
Working space
in tape
Left-end
marker
Right-end
marker
All computation is done between end markers.

43. Observation There is a language which is recursive
but not context-sensitive.

44. Decidability

45. Consider problems with answer YES or NO.
Examples
Does Machine M have three states ?
Is string w a binary number?
Does DFA M accept any input?

46. A problem is decidable if some Turing machine
solves (decides) the problem.
Decidable problems:
Does Machine M have three states ?
Is string w a binary number?
Does DFA M accept any input?

47. The Turing machine that solves a problem
answers YES or NO for each instance.
Input
problem
instance
YES
Turing Machine NO

48. The 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.

49. YES NO Turing Machine that decides a problem
YES and NO states are halting states

50. Difference between
Recursive Languages accept (“Acceptera”) and Decidable problems decide (“Avgöra”)
For decidable problems:
The YES states may not be final states.

51. Some problems are undecidable:
There is no Turing Machine that
solves all instances of the problem.

52. A famous undecidable problem:
The halting problem

53. The Halting Problem Input: Turing Machine M String w
Question: Does M halt on w?

54. Theorem
The halting problem is undecidable.
Proof
Assume to the contrary that
the halting problem is decidable.

55. M halts on w M doesn’t halt on w There exists Turing Machine H
that solves the halting problem
YES NO
M halts on w
M doesn’t halt on w

56. YES NO Construction of H Input: initial tape contents encoding of M
string w

57. Construct machine H´
If H returns YES then loop forever.
If H returns NO then halt.

58. NO
Loop forever
YES

59. Construct machine
Input:
(machine ) If
halts on input
Then loop forever
Else halt

60. copy

61. Run machine with input itselfIf
halts on input
Then loop forever
Else halt

62. on input :
If halts then loops forever.
If doesn’t halt then it halts.
CONTRADICTION !

63. The halting problem is undecidable.
This means that
The halting problem is undecidable.
END OF PROOF

64. Another proof of the same theorem
If the halting problem was decidable then
every recursively enumerable language
would be recursive.

65. Theorem Proof The halting problem is undecidable.
Assume to the contrary that
the halting problem is decidable.

66. M halts on w M doesn’t halt on w There exists Turing Machine H
that solves the halting problem.
YES NO
M halts on w
M doesn’t halt on w

67. Let L be a recursively enumerable language.
Let M be the Turing Machine that accepts L.
We will prove that L is also recursive:
We will describe a Turing machine that accepts L and halts on any input.

68. Turing Machine that accepts L and halts on any input
reject w NO
M halts on w ?
YES
accept w
Halts on final state
Run M
with input w
reject w
Halts on non-final
state

69. Contradiction! Therefore L is recursive.
Since L 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!

70. Therefore, the halting problem is undecidable.
END OF PROOF

71. A simple undecidable problem:
The Membership Problem

72. The Membership Problem
Input:
Turing Machine M
String w
Question: Does M accept w?

73. Theorem Proof The membership problem is undecidable.
Assume to the contrary that
the membership problem is decidable.

74. M accepts w M rejects w There exists Turing Machine H
that solves the membership problem
YES NO
M accepts w
M rejects w

75. Let L be a recursively enumerable language.
Let M be the Turing Machine that accepts L.
We will prove that L is also recursive:
We will describe a Turing machine that
accepts L and halts on any input.

76. YES M accepts w? NO Turing Machine that accepts L
and halts on any input
YES
accept w
M accepts w? NO
reject w

77. Contradiction! Therefore L is recursive.
Since L 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!

78. Therefore, the membership problem
is undecidable.
END OF PROOF

79. Reducibility

80. Problem A is reduced to problem B.
If we can solve problem B then
we can solve problem A.

81. Problem A is reduced to problem B.
If B is decidable then A is decidable.
If A is undecidable then B is undecidable.

82. Example
the halting problem
reduced to
the state-entry problem.

83. The state-entry problem
Input:
Turing Machine M
State q
String w
Question:
Does M enter state q on input w?

84. Theorem Proof The state-entry problem is undecidable.
Reduce the halting problem to
the state-entry problem.

85. Suppose we have an algorithm
(Turing Machine)
that solves the state-entry problem.
We will construct an algorithm
that solves the halting problem.

86. Assume we have the state-entry algorithm:
YES
M enters q
Algorithm for
state-entry
problem NO
M doesn’t enter q

87. We want to design the halting algorithm:
YES
M halts on q
Algorithm for
Halting problem
M doesn’t halt on q NO

88. From any halting state add transitions to q
Modify input machine M
Add new state q
From any halting state add transitions to q
single
halt state
halting states

89. halts
if and only if
halts on state

90. Algorithm for halting problem
Input: machine M and string w
1. Construct machine M’ with state q
2. Run algorithm for state-entry problem
with inputs: M’, q, w

91. Halting problem algorithm
YES
YES
State-entry
algorithm
Generate NO NO

92. We 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

93. Another example
The halting problem reduced to the blank-tape halting problem.
Input: Turing Machine M
Question: Does M halt when started with a blank tape?

94. Theorem Proof The blank-tape halting problem is undecidable.
Reduce the halting problem to the blank-tape halting problem.

95. Suppose we have an algorithm
for the blank-tape halting problem.
We will construct an algorithm
for the halting problem.

96. M doesn’t halt on blank tape
Assume we have the
blank-tape halting algorithm
YES
M halts on
blank tape
Algorithm for
blank-tape
halting problem NO
M doesn’t halt on blank tape

97. YES NO M halts on w M doesn’t We want to design the halting algorithm:
Algorithm for
halting problem
YES NO
M halts on w
M doesn’t
halt on w

98. Construct a new machine
On blank tape writes
Then continues execution like
step 1
step2
if blank tape
execute
then write
with input

99. halts on input string
if and only if
halts when started with blank tape.

100. Algorithm for halting problem
Inputs:
machine and string
1. Construct
2. Run algorithm for
blank-tape halting problem
with input

101. YES NO Halting problem algorithm YES Blank-tape Generate halting

102. We 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

103. Summary of Undecidable Problems
Halting Problem
Does machine M halt on input w?
Membership problem
Does machine M accept string w?
In other words: Is a string w a member of a
recursively enumerable language L?)

104. Blank-tape halting problem
Does machine M halt when starting
on blank tape?
State-entry Problem:
Does machine M enter state q
on input w?

105. Uncomputable Functions

106. Uncomputable Functions
Domain
Range
A function is uncomputable if it cannot
be computed for all of its domain.

107. Example An uncomputable function: maximum number of moves until
any Turing machine with n states halts when started with the blank tape.

108. Theorem Proof is computable. Function is uncomputable.
Assume to the contrary that
is computable.
Then the blank-tape halting problem
is decidable.

109. Algorithm for blank-tape halting problem
Input: machine
1. Count states of :
2. Compute
3. Simulate for steps
starting with empty tape
If halts then return YES
otherwise return NO

110. Contradiction! Therefore, the blank-tape halting
problem must be decidable.
However, we know that the blank-tape
halting problem is undecidable.
Contradiction!

111. Therefore, function is uncomputable.
END OF PROOF

112. Rice’s Theorem

113. Definition Non-trivial properties of recursively enumerable languages:
any property possessed by some (not all)
recursively enumerable languages.

114. Some non-trivial properties of
recursively enumerable languages:
L is empty
L is finite
L contains two different strings of the same length

115. Rice’s Theorem Any non-trivial property of
a recursively enumerable language
is undecidable.

116. We will prove some non-trivial properties
without using Rice’s theorem.

117. Theorem Proof For any recursively enumerable language L
it is undecidable whether it is empty.
Proof
We will reduce the membership problem
to the problem of deciding whether L
is empty.

118. Membership problem:
Does machine M accept string w?

119. Let M be the machine that accepts L
Assume we have the empty language algorithm:
YES
L(M) empty
Algorithm for
empty language
problem NO
L(M) non-empty

120. YES M accepts w NO M rejects w
We will design the membership algorithm:
YES
M accepts w
Algorithm for
membership
problem NO
M rejects w

121. First construct machine
When M enters a final state,
compare original input string with w.
Accept if original input is
the same as w.

122. if and only if
is not empty

123. Algorithm for membership problem
Inputs: machine and string
1. Construct
2. Determine if is empty
YES: then
NO: then

124. Membership algorithm NO
YES
Check if
construct
YES
is empty NO

125. We 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

126. Decidability …continued…

127. Theorem
For a recursively enumerable language L it is undecidable whether L is finite.
Proof
We will reduce the halting problem
to the finite language problem.

128. YES L(M) finite NO L(M) not finite Let M be the machine that accepts L
Assume we have the finite language algorithm:
YES
L(M) finite
Algorithm for
finite language
problem NO
L(M) not finite

129. YES M halts on w NO M doesn’t
We will design the halting problem algorithm:
Algorithm for
halting problem
YES NO
M halts on w
M doesn’t
halt on w

130. First construct machine .
Initially simulates M on input w.
When M enters a halt state,
accept any input (infinite language).
Otherwise accept nothing (finite language).

131. halts on
is not finite.
if and only if

132. Algorithm for halting problem:
Inputs: machine and string
1. Construct
2. Determine if is finite
YES: then doesn’t halt on
NO: then halts on

133. Machine for halting problem
construct
Check if
is finite
YES NO

134. We 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

135. Theorem
For a recursively enumerable language L it is undecidable whether L contains
two different strings of same length.
Proof
We will reduce the halting problem
to the two strings of equal length- problem.

136. L(M) does not contain two equal length strings NO
Let M be the machine that accepts L:
Assume we have the equal-strings algorithm:
YES
L(M) contains
Algorithm for
two-strings
problem
L(M) does not contain two equal length strings NO

137. YES NO M halts on w M doesn’t
We will design the halting problem algorithm:
Algorithm for
halting problem
YES NO
M halts on w
M doesn’t
halt on w

138. First construct machine
Initially simulates M on input w.
When M enters a halt state,
accept symbols a and b.
(two equal length strings)

139. halts on
if and only if
accepts and
(two equal length strings)

140. Algorithm for halting problem
Inputs: machine and string
1. Construct
2. Determine if accepts
two strings of equal length
YES: then halts on
NO: then doesn’t halt on

141. YES NO Machine for halting problem YES Check if construct NO has two
equal length
strings

142. We 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

143. Rice's theorem: (Henry Gordon Rice ) If S is a non-trivial property of Turing-acceptable languages, then the problem ‘Does L(M) have the property S? is undecidable.
Any nontrivial property of the language recognized by a Turing machine is undecidable. (Only trivial properties of programs are algorithmically decidable. Undecidability is not the exception when it comes to recursively enumerable sets, it is the rule. )
If  is a set of Turing-acceptable languages, containing some but not all such languages, no TM can decide for an arbitrary Turing-acceptable language L if L belongs to  or not.

144. Example
Given a Turing machine M, is it possible to decide if all strings accepted by M start and end with the same symbol?

145. Undecidable
Problem is about non-trivial property of a language. There exist TM’s with given property, and TM’s without.

146. Formally
 = { L | TM acceptable languages With strings that start and end with the same symbol. }

147. Interaction: Conjectures, Results, and Myths
Dina Goldin Univ. of Connecticut, Brown University

148. Fundamental Questions Underlying Theory of Computation
What is computation?
How is computation modeled?

149. Shared 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.

150. The 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

151. The 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, it’s “useless” – but aren’t OS’s useful??
* Enigma

152. 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
Wegner’s conjecture: Interaction is more powerful than algorithms

153. 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.

154. Driving 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 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”

155. 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.

156. 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

157. Sequential Interaction Thesis
Whenever there is an effective method for performing sequential interactive computation, this computation can be performed by a Persistent Turing Machine
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.

158. Church-Turing Thesis Revisited
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 any computer can do!

159. Church-Turing Thesis Revisited
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.

160. Origins 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

161. 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.

162. 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]

163. 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]

164. The 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

165. The 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]

166. Modeling 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.

167. Future 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 [Wegner’97]
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

168. 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 to be published in LNCS