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CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 13 Mälardalen University 2010 1 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) 2 http://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 Page Andrew Hodges 3 Alan Turing 1912 (23 June): Birth, London 1926-31: Sherborne School 1930: Death of friend Christopher Morcom 1931-34: Undergraduate at King's College, Cambridge University 1932-35: Quantum mechanics, probability, logic 1935: Elected fellow of King's College, Cambridge 1936: The Turing machine, computability, universal machine 1936-38: Princeton University. Ph.D. Logic, algebra, number theory 1938-39: Return to Cambridge. Introduced to German Enigma cipher machine 1939-40: The Bombe, machine for Enigma decryption 1939-42: Breaking of U-boat Enigma, saving battle of the Atlantic 4 Alan Turing 1943-45: Chief Anglo-American crypto consultant. Electronic work. 1945: National Physical Laboratory, London 1946: Computer and software design leading the world. 1947-48: 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 1953-54: Unfinished work in biology and physics 1954 (7 June): Death (suicide) by cyanide poisoning, Wilmslow, Cheshire. 5 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? 6 TURING MACHINES 7 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) 8 Standard Turing Machine Tape ...... ...... Read-Write head Control Unit 9 The Tape No boundaries -- infinite length ...... ...... Read-Write head The head moves Left or Right 10 ...... ...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right 11 Example ...... Time 0 a b a c ...... Time 1 ...... a b k c ...... 1. Reads a 2. Writes k 3. Moves Left 12 The Input String Input string ...... Blank symbol # # a b a c # # # ...... head Head starts at the leftmost position of the input string 13 States & Transitions Read q1 Write a b, L Move Left q2 Move Right q1 a b, R q2 14 Time 1 ...... # # a b a c # # # ...... q1 Time 2 ...... # # a b b c # # # ...... q2 q1 a b, R q2 15 Determinism Turing Machines are deterministic Not Allowed Allowed a b, R q2 a b, R q2 a d, L q3 q1 q1 b d, L q3 No lambda transitions allowed in standard TM! 16 Formal Definitions for Turing Machines 17 Transition Function q1 a b, R q2 (q1, a) (q2 , b, R) 18 Turing Machine Input alphabet Tape alphabet States M (Q, , , , q0 , # , F ) Final states Transition function Initial state blank 19 Universal Turing Machine 20 A limitation of Turing Machines: Turing Machines are “hardwired” they execute only one program Better are reprogrammable machines. 21 Solution: Universal Turing Machine Characteristics: • Reprogrammable machine • Simulates any other Turing Machine 22 Universal Turing Machine simulates any other Turing Machine M Input of Universal Turing Machine • Description of transitions of M • Initial tape contents of M 23 Tape 1 Three tapes Universal Turing Machine Description of M Tape 2 Tape Contents of M Tape 3 State of M 24 Tape 1 Description of M We describe Turing machine as a string of symbols: We encode M M as a string of symbols 25 Alphabet Encoding Symbols: a b c d Encoding: 1 11 111 1111 26 State Encoding States: q1 q2 q3 q4 Encoding: 1 11 111 1111 Head Move Encoding Move: L R Encoding: 1 11 27 Transition Encoding Transition: (q1, a) (q2 , b, L) Encoding: 1 0 1 0 11 0 11 0 1 separator 28 Machine Encoding Transitions: (q1, a) (q2 , b, L) (q2 , b) (q3 , c, R) Encoding: 1 0 1 0 11 0 11 0 1 00 11 0 110 111 0 111 0 11 separator 29 Tape 1 contents of Universal Turing Machine: encoding of the simulated machine as a binary string of 0’s and 1’s M 30 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. 31 Language of Turing Machines L = { 010100101, (Turing Machine 1) 00100100101111, 111010011110010101, (Turing Machine 2) …… …… } 32 The Chomsky Hierarchy 33 The Chomsky Language Hierarchy Recursively-enumerable Recursive Context-sensitive Context-free Regular Non-recursively enumerable 34 Unrestricted Grammars Productions uv String of variables and terminals String of variables and terminals 35 Example of unrestricted grammar S aBc aB cA Ac d 36 Theorem A language L is recursively enumerable if and only if it is generated by an unrestricted grammar. 37 Context-Sensitive Grammars Productions uv String of variables and terminals and String of variables and terminals |u| |v| 38 The language n n n {a b c } is context-sensitive: S aBSc S Ba aB Bc bc Bb bb 39 Theorem A language L is context sensitive if and only if it is accepted by a Linear-Bounded automaton. 40 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. 41 Linear Bounded Automaton (LBA) Input string [ a b c d e ] Left-end marker Working space in tape Right-end marker All computation is done between end markers. 42 Observation There is a language which is context-sensitive but not recursive. 43 Decidability 44 Consider problems with answer YES or NO. Examples • Does Machine • Is string M have three states ? w a binary number? • Does DFA M accept any input? 45 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? 46 The Turing machine that solves a problem answers YES or NO for each instance. Input problem instance YES Turing Machine NO 47 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. 48 Turing Machine that decides a problem YES NO YES and NO states are halting states 49 Difference between Recursive Languages (“Acceptera”) and Decidable problems (“Avgöra”) For decidable problems: The YES states may not be final states. 50 Some problems are undecidable: There is no Turing Machine that solves all instances of the problem. 51 A famous undecidable problem: The halting problem 52 The Halting Problem Input: • Turing Machine • String w M Question: Does M halt on w ? 53 Theorem The halting problem is undecidable. Proof Assume to the contrary that the halting problem is decidable. 54 There exists a Turing Machine H that solves the halting problem M YES M halts on w NO doesn’t halt on H w M w 55 Construction of H Input: initial tape contents wM w q y YES q0 qn NO Encoding of M String w H 56 Construct machine H If H returns YES then loop forever. If H returns NO then halt. 57 H H Loop forever qy wM w YES qa qb q0 qn NO 58 Construct machine Input: If wM Ĥ (machine M) M halts on input wM Then loop forever Else halt 59 Ĥ wM copy wM wM wM H 60 Run machine Input: If Ĥ with input itself wHˆ (machine Ĥ ) Ĥ halts on input wHˆ Then loop forever Else halt 61 Ĥ on input wHˆ : If Ĥ halts then Ĥ loops forever. If Ĥ doesn’t halt then Ĥ halts. CONTRADICTION ! 62 This means that The halting problem is undecidable. END OF PROOF 63 Another proof of the same theorem If the halting problem was decidable then every recursively enumerable language would be recursive. 64 Theorem The halting problem is undecidable. Proof Assume to the contrary that the halting problem is decidable. 65 There exists Turing Machine H that solves the halting problem. M YES M halts on w NO doesn’t halt on H w M w 66 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. 67 Turing Machine that accepts L and halts on any input M H NO reject w M halts on w? w YES accept Run M with input w Halts on final state w reject w Halts on non-final 68 state 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! 69 Therefore, the halting problem is undecidable. END OF PROOF 70 A simple undecidable problem: The Membership Problem 71 The Membership Problem Input: • Turing Machine M • String w Question: Does M accept w ? 72 Theorem The membership problem is undecidable. Proof Assume to the contrary that the membership problem is decidable. 73 There exists a Turing Machine H that solves the membership problem M YES M accepts w NO M rejects w H w 74 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. 75 Turing Machine that accepts and halts on any input M H L YES accept w NO reject M accepts w? w w 76 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! 77 Therefore, the membership problem is undecidable. END OF PROOF 78 Reducibility 79 Problem A is reduced to problem B If we can solve problem B then we can solve problem A . B A 80 Problem If If A is reduced to problem B B is decidable then A is decidable. A is undecidable then B is undecidable. 81 Example the halting problem reduced to the state-entry problem. 82 The state-entry problem Inputs: •Turing Machine •State M q •String w Question: M enter state q on input w ? Does 83 Theorem The state-entry problem is undecidable. Proof Reduce the halting problem to the state-entry problem. 84 Suppose we have an algorithm (Turing Machine) that solves the state-entry problem. We will construct an algorithm that solves the halting problem. 85 Assume we have the state-entry algorithm: M w q Algorithm for state-entry problem YES NO M enters q M doesn’t enter q 86 We want to design the halting algorithm: M w YES M halts on w NO doesn’t halt on Algorithm for Halting problem M w 87 Modify input machine M • Add new state q • From any halting state add transitions to q M M halting states q Single halt state 88 M halts if and only if M halts on state q 89 Algorithm for halting problem Inputs: machine M and string w 1. Construct machine M with state q 2. Run algorithm for state-entry problem with inputs: M , q , w 90 Halting problem algorithm M Generate M M q State-entry w algorithm YES YES NO NO w 91 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 92 Another example The halting problem reduced to the blank-tape halting problem. 93 The blank-tape halting problem Input: Turing Machine Question: Does M M halt when started with a blank tape? 94 Theorem The blank-tape halting problem is undecidable. Proof 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 Assume we have the blank-tape halting algorithm YES M Algorithm for blank-tape halting problem M halts on blank tape NO M doesn’t halt on blank tape 97 We want to design the halting algorithm: YES M M halts on w Algorithm for halting problem w NO M doesn’t halt on w 98 Construct a new machine Mw • On blank tape writes w • Then continues execution like M Mw step 1 if blank tape then write w step2 M with input w execute 99 M halts on input string w if and only if M w halts when started with blank tape. 100 Algorithm for halting problem Inputs: machine M and string w 1. Construct M w 2. Run algorithm for blank-tape halting problem with input M w 101 Halting problem algorithm M w Generate Mw Mw Blank-tape halting algorithm YES YES NO NO 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 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 on input w ? M enter state q 105 Uncomputable Functions 106 Uncomputable Functions f Domain Range A function is uncomputable if it cannot be computed for all of its domain. 107 Example An uncomputable function: f (n) maximum number of moves until any Turing machine with n states halts when started with the blank tape. 108 Theorem Function f (n) is uncomputable. Proof Assume to the contrary that f (n) is computable. Then the blank-tape halting problem is decidable. 109 Algorithm for blank-tape halting problem Input: machine 1. Count states of M M: m f (m) 3. Simulate M for f (m) steps 2. Compute starting with empty tape If M halts then return YES otherwise return NO 110 Therefore, the blank-tape halting problem must be decidable. However, we know that the blank-tape halting problem is undecidable. Contradiction! 111 Therefore, function f (n) is uncomputable. END OF PROOF 112 Definition 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. 113 Another definition A language L is recursively enumerable iff it Is generated by an unrestricted grammar. 114 Implication A language L is recursive iff L is recursively enumerable AND it’s complement L’ is recursively enumerable 115 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 ML accepting L by modifying M such as the yes state is a final state (all other rejecting). Construct a machine ML’ accepting L’ by modifying M such that the no state is a final state (all other rejecting). 116 Proof sketch (only if) Let M be a Turing machine accepting L and let M’ be a Turing machine accepting L’ Let ML be the Turing machine deciding L. Run M and M’ in parallel on input w. One of the machines will accept w. Let ML answer yes if M accepts, and no if M’ Accepts. Since ML clearly can be constructed. L is a recursive language. 117 Not all recursive languages are recursively enumerable Let ∑ = {a}. Let H be the set of all Turing Machines with this alphabet. Note that H is countably infinite! Thus, H = {M0,M1,M2,…} Let L(Mi) denote the recursively enumerable Language of Mi (that is, the set of strings that Drives Mi into a final state.) 118 Not all recursive languages are recursively enumerable Let L {a | a L( M i )} L is recursively enumerable. Pseudo-code: 1. Find i by counting a’s in w 2. Find Mi by the known encoding 3. Run Mi with w. Accept if Mi accepts i i 119 Not all recursive languages are recursively enumerable Let L {a | a L( M i )} 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’ = Mk for some integer k. i i 120 Not all recursive languages are recursively enumerable i i L {a | a L( M i )} Let Mk be the machine accepting L’. That is L’ = L(Mk). Let w a Is w in L’? k 121 Not all recursive languages are recursively enumerable i i L {a | a L( M i )} Let Mk be the machine accepting L’. That is L’ = L(Mk). w L w L( M k ) L w L w L( M k ) L Contradiction!! L’ is not RE 122 Not all recursive languages are recursively enumerable 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. 123 Rice’s Theorem 124 Definition Non-trivial properties of recursively enumerable languages: any property possessed by some (not all) recursively enumerable languages. 125 Some non-trivial properties of recursively enumerable languages: • L is empty • L is finite • L contains two different strings of the same length 126 Rice’s Theorem Any non-trivial property of a recursively enumerable language is undecidable. 127 We will prove some non-trivial properties without using Rice’s theorem. 128 Theorem 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. 129 Membership problem: Does machine M accept string w? 130 Let M be the machine that accepts L L( M ) L Assume we have the empty language algorithm: M Algorithm for empty language problem YES L(M ) empty NO L(M ) not empty 131 We will design the membership algorithm: M w YES Algorithm for membership problem NO M accepts w M rejects w 132 First construct machine Mw : When M enters a final state, compare original input string with w. Accept if original input is the same as w. M Construct w Mw 133 w L if and only if L( M w ) is not empty L( M w ) {w} 134 Algorithm for membership problem Inputs: machine 1. Construct M and string w Mw 2. Determine if L( M w ) is empty YES: then w L(M ) NO: then w L(M ) 135 Membership algorithm M w construct Mw Mw Check if YES NO NO YES L( M w ) is empty 136 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 137 Decidability …continued… 138 Theorem For a recursively enumerable language it is undecidable to determine whether L is finite. L Proof We will reduce the halting problem to the finite language problem. 139 Let M be the machine that accepts L L( M ) L Assume we have the finite language algorithm: M Algorithm for finite language problem YES L(M ) finite NO L(M ) not finite 140 We will design the halting problem algorithm: YES M Algorithm for Halting problem w NO M halts on w M doesn’t w halt on 141 First construct machine Mw . Initially, simulates M on input w. When M enters a halt state, accept any input (infinite language). Otherwise accept nothing (finite language). 142 M halts on w if and only if L( M w ) is not finite. 143 Algorithm for halting problem: Inputs: machine 1. Construct M and string w Mw 2. Determine if L( M w ) is finite YES: then M doesn’t halt on w NO: then M halts on w 144 Machine for halting problem M w construct Mw Check if YES L( M w ) is finite NO NO YES 145 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 146 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. 147 Let M be the machine that accepts L L( M ) L Assume we have the two-strings algorithm: YES M Algorithm for two-strings problem NO L(M ) contains L(M ) doesn’t contain two equal length strings 148 We will design the halting problem algorithm: M Algorithm for Halting problem w YES M halts on w NO doesn’t halt on M w 149 First construct machine Initially, simulates M w. M on input w. When M enters a halt state, accept symbols a or b . (two equal length strings) 150 M halts on w if and only if M w accepts a and b (two equal length strings) 151 Algorithm for halting problem Inputs: machine 1. Construct M and string w Mw 2. Determine if M w accepts two strings of equal length M halts on w NO: then M doesn’t halt on w YES: then 152 Machine for halting problem M w construct Mw Check if YES YES NO NO L( M w ) has two equal length strings 153 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 154 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. 155 Example Given a Turing Machine M, is it possible to decide weather all strings acceted by M begins and ends with the same symbol? 156 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. 157 Formally: = { L | L is a recursively enumerable language where all strings begin and end with the same symbol } 158 Interaction: Conjectures, Results, and Myths Dina Goldin Univ. of Connecticut, Brown University http://www.cse.uconn.edu/~dqg 159 Fundamental Questions Underlying Theory of Computation What is computation? How do we model it? 160 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 Mathematical worldview: All computable problems are function-based. behavior: functions, transformation of input data to output data Church-Turing thesis: Turing Machines capture this (algorithmic) notion of computation 161 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 162 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?? 163 Rethinking Shared Wisdom: (what do computers do?) computation: finite transformation of input to output computation: ongoing process which performs a task or delivers a service input: finite-size (string or number) dynamically generated stream of input tokens (requests, percepts, messages) open system: later inputs depend on closed system: all input available at earlier outputs and vice versa (I/O start, all output generated at end entanglement, history dependence) behavior: functions, algorithmic transformation of input data to output data behavior: processes, components, control devices, reactive systems, intelligent agents Church-Turing thesis: Turing Wegner’s conjecture: Interaction is Machines capture this (algorithmic) more powerful than algorithms notion of computation 164 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. 165 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” 166 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. 167 Outline • 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 168 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 169 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. 170 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. 171 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 172 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. 173 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] 174 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] 175 Outline • Rethinking the mathematical worldview • Persistent Turing Machines (PTMs) • PTM expressiveness • Sequential Interaction • The Myth of the Church-Turing Thesis • Conclusions and future work 176 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 177 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] 178 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. 179 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 180 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 181