1 CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 13 Mälardalen University 2010

2 Content Alan Turing and Hilbert Program Universal Turing Machine Chomsky Hierarchy Decidability Reducibility Uncomputable Functions Rice’s Theorem Interactive Computing, Persistent TM’s (Dina Goldin)

3 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. (Wittgenstein) A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine. (Turing)

Tape Read-Write head Control Unit Standard Turing Machine

Read-Write head No boundaries -- infinite length The head moves Left or Right The Tape

Read-Write head 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right The head at each time step:

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

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

14 Read Write Move Left Move Right States & Transitions

Time Time 2

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

17 Formal Definitions for Turing Machines

18 Transition Function

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

20 Universal Turing Machine

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

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

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

24 Universal Turing Machine Description of Three tapes Tape Contents of Tape 2 State of Tape 3 Tape 1

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

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 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. The set of Turing machines forms a language: Each string of the language is the binary encoding of a Turing Machine. Therefore:

32 Language of Turing Machines L = { , , , …… } (Turing Machine 1) (Turing Machine 2) ……

33 The Chomsky Hierarchy

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

35 Unrestricted Grammars Productions String of variables and terminals String of variables and terminals

36 Example of unrestricted grammar

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

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

39 The language is context-sensitive:

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

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

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

44 Decidability

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

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

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

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 (“Acceptera”) and Decidable problems (“Avgöra”) The YES states may not be final states. For decidable problems:

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 String Question:Does halt on ?

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

55 There exists a Turing Machine that solves the halting problem YEShalts on doesn’t halt on NO

56 Input: initial tape contents Encoding of String YES NO Construction of

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

58 NO Loop forever YES

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

60 copy

61 Run machine with input itself Input: If halts on input Then loop forever Else halt (machine )

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

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

66 There exists Turing Machine that solves the halting problem. YEShalts on doesn’t halt on NO

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

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

69 Therefore is recursive. But there are recursively enumerable languages which are not recursive. Contradiction! Since is chosen arbitrarily, we have proven that every recursively enumerable language is also recursive.

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 String Question:Does accept ?

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

74 There exists a Turing Machine that solves the membership problem YES accepts NO rejects

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

76 accepts ? NO YES accept Turing Machine that accepts and halts on any input reject

77 Therefore, is recursive. But there are recursively enumerable languages which are not recursive. Contradiction! Since is chosen arbitrarily, we have proven that every recursively enumerable language is also recursive.

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

79 Reducibility

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

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

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

83 The state-entry problem Inputs: Question: Turing Machine State String Does enter state on input ?

84 Theorem The state-entry problem is undecidable. Proof 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 Algorithm for state-entry problem YES NO enters doesn’t enter Assume we have the state-entry algorithm:

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

88 Modify input machine Add new state From any halting state add transitions to halting states Single halt state

89 halts halts on state if and only if

90 Algorithm for halting problem Inputs:machine and string 2. Run algorithm for state-entry problem with inputs:,, 1. Construct machine with state

91 GenerateState-entry algorithm Halting problem algorithm YES NO YES NO

92 Since the halting problem is undecidable, it must be that the state-entry problem is also undecidable. END OF PROOF We reduced the halting problem to the state-entry problem.

93 Another example The halting problem reduced to the blank-tape halting problem.

94 The blank-tape halting problem Input:Turing Machine Question:Doeshalt when started with a blank tape?

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

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

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

98 Algorithm for halting problem YES NO halts on doesn’t halt on We want to design the halting algorithm:

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

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

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

102 Generate Blank-tape halting algorithm Halting problem algorithm YES NO YES NO

103 Since the halting problem is undecidable, the blank-tape halting problem is also undecidable. END OF PROOF We reduced the halting problem to the blank-tape halting problem.

104 Summary of Undecidable Problems Halting Problem Does machine halt on input ? Membership problem Does machine accept string ? Is a string member of a recursively enumerable language ? ) (In other words:

105 Does machine halt when starting on blank tape? Blank-tape halting problem State-entry Problem: Does machine enter state on input ?

106 Uncomputable Functions

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

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

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

110 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

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

112 Therefore, function is uncomputable. END OF PROOF

113 A language L is recursively enumerable if There is a Turing machine M accepting all Strings w in L. This is denoted L=L(M) A language L is recursive if there is a Turing Machine M, that halts on any input w and deciding weather w is in L or not. Definition

114 A language L is recursively enumerable iff it Is generated by an unrestricted grammar. Another definition

115 A language L is recursive iff L is recursively enumerable AND it’s complement L’ is recursively enumerable Implication

116 Proof sketch (if) Let L be a recursive language. Then there is A Turing machine M such that running any input w on M, M will answer yes if w is in L and no if w is not in L. Construct a machine M L accepting L by modifying M such as the yes state is a final state (all other rejecting). Construct a machine M L’ accepting L’ by modifying M such that the no state is a final state (all other rejecting).

117 Proof sketch (only if) Let M be a Turing machine accepting L and let M’ be a Turing machine accepting L’ Let M L be the Turing machine deciding L. Run M and M’ in parallel on input w. One of the machines will accept w. Let M L answer yes if M accepts, and no if M’ Accepts. Since M L clearly can be constructed. L is a recursive language.

118 Let ∑ = {a}. Let H be the set of all Turing Machines with this alphabet. Note that H is countably infinite! Thus, H = {M 0,M 1,M 2,…} Let L(M i ) denote the recursively enumerable Language of M i (that is, the set of strings that Drives M i into a final state.) Not all recursive languages are recursively enumerable

119 Let L is recursively enumerable. Pseudo-code: 1.Find i by counting a’s in w 2.Find M i by the known encoding 3.Run M i with w. Accept if M i accepts Not all recursive languages are recursively enumerable

120 Let L’ is not recursively enumerable. Assume for contradiction it is: there is a TM M’ that Accepts w if w is in L’. Note: M’ has {a} as alphabet, so M’ = M k for some integer k. Not all recursive languages are recursively enumerable

121 Let M k be the machine accepting L’. That is L’ = L(M k ). Let Is w in L’? Not all recursive languages are recursively enumerable

122 Let M k be the machine accepting L’. That is L’ = L(M k ). Contradiction!! L’ is not RE Not all recursive languages are recursively enumerable

123 L’ is not RE, but L is. According to previous Result a language is recursive iff L and L’ is RE. Conclusion: L is recursively enumerable but not recursive. Not all recursive languages are recursively enumerable

124 Rice’s Theorem

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

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

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

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

129 Theorem 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.

130 Membership problem: Does machine accept string ?

131 Algorithm for empty language problem YES NO Assume we have the empty language algorithm: Let be the machine that accepts empty not empty

132 Algorithm for membership problem YES NO accepts rejects We will design the membership algorithm:

133 First construct machine : When enters a final state, compare original input string with. Accept if original input is the same as. Construct

134 is not empty if and only if

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

136 construct Check if is empty YES NO YES Membership algorithm

137 Since the membership problem is undecidable, the empty language problem is also undecidable. END OF PROOF We reduced the empty language problem to the membership problem.

138 Decidability …continued…

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

140 Assume we have the finite language algorithm: Algorithm for finite language problem YES NO finite not finite Let be the machine that accepts

141 We will design the halting problem algorithm: Algorithm for Halting problem YES NO halts on doesn’t halt on

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

143 halts on is not finite. if and only if

144 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

145 construct Check if is finite YES NO YES Machine for halting problem

146 Since the halting problem is undecidable, the finite language problem is also undecidable. END OF PROOF We reduced the finite language problem to the halting problem.

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

148 Assume we have the two-strings algorithm: Let be the machine that accepts Algorithm for two-strings problem YES NO contains doesn’t contain two equal length strings

149 We will design the halting problem algorithm: Algorithm for Halting problem YES NO halts on doesn’t halt on

150 First construct machine. When enters a halt state, accept symbols or. Initially, simulates on input. (two equal length strings)

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

152 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

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

154 Since the halting problem is undecidable, the two strings of equal length problem is also undecidable. END OF PROOF We reduced the two strings of equal length - problem to the halting problem.

155 Rices theorem If  is a set of recursively enumerable languages containing some but not all such languages, then no TM can decide for an arbitrary Recursively enumerable language L, if L belongs to  or not.

156 Example Given a Turing Machine M, is it possible to decide weather all strings acceted by M begins and ends with the same symbol?

157 Undecidable The problem is about a non-trivial language property. There are recursively enumerable languages with this property and there are recursively enumerable langages without this property.

158 Formally:  = { L | L is a recursively enumerable language where all strings begin and end with the same symbol }

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

160 Fundamental Questions Underlying Theory of Computation What is computation? How do we model it?

161 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.

162 “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

163 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??

164 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

165 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.

166 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.) ?

167 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.

168 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 Outline

169 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

170 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 John, ok), (record Hopkins, ok), (playback, John Hopkins), … 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

171 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 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.

172 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

173 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.

174 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]

175 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]

176 Rethinking the mathematical worldview Persistent Turing Machines (PTMs) PTM expressiveness Sequential Interaction The Myth of the Church-Turing Thesis Conclusions and future work Outline

177 The Shift to Interaction in CS Computation = transforming input to output Computation = carrying out a task over time Logic and search in AIIntelligent agents, partially observable environments, learning Procedure-oriented programming Object-oriented programming Closed systemsOpen systems Compositional behaviorEmergent behavior Rule-based reasoningSimulation, control, semi-Markov processes Algorithmic Interactive

178 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]

179 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. Modeling Interactive Computation: PTMs in Perspective

180 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 Future Work: 3 conjectures

181 References [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