Formal Languages, Automata and Models of Computation

Formal Languages, Automata and Models of Computation
CDT314 FABER Formal Languages, Automataand Models of Computation Lecture 13 Mlardalen University 2012 1 ContentAlan Turing and Hilbert ProgramUniversal Turing MachineChomsky HierarchyDecidability Reducibility Uncomputable Functions Rices Theorem Church-Turing Thesis Computation beyond Turing Model Interactive Computing, Persistent TMs (Dina Goldin/Peter Wegner) TURING MACHINES Based on C Busch, RPI, Models of Computation 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) Standard Turing Machine
Tape ...... ...... Read-Write head Control Unit The Tape No boundaries -- infinite length ...... ......
Read-Write head The head moves Left or Right ...... ...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right Example Time 0 ...... Time 1 ...... 1. Reads 2. Writes 3. Moves Left The Input String Input string Blank symbol ...... ...... head
Head starts at the leftmost position of the input string States & Transitions Write Read Move Left Move Right Time 1 ...... ...... Time 2 ...... ...... Determinism Not Allowed Turing Machines are deterministic Allowed
No lambda transitions allowed in standard TM! Formal Definitions for Turing Machines Transition Function Turing Machine Input alphabet Tape alphabet States Final Transition
function Initial state blank Universal Turing Machine A limitation of Turing Machines:
Turing Machines are hardwired they execute only one program Better are reprogrammable machines. Solution: Universal Turing Machine Characteristics: Reprogrammable machine Simulates any other Turing Machine Universal Turing Machine
simulates any other Turing Machine Input ofUniversal Turing Machine Description of transitions of Initial tape contents of Three tapes Description of Universal Turing Machine Tape Contents of
States of We encode/describe Turing Machine as a string of symbols.
Tape 1 Description of We encode/describe Turing Machine as a string of symbols. Tape Contents Encoding
Alphabet Encoding Symbols: Encoding: State Encoding States: Encoding: Head Move Encoding Move: Encoding: Transition Encoding Transition: Encoding: separator Machine Encoding Transitions: Encoding: separator Tape 1 contents of Universal Turing Machine:
encoding of the simulated machine as a binary string of 0s and 1s A 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. Language of Turing Machines
, , } (Turing Machine 2) (Turing Machine 3) The Chomsky Hierarchy The Chomsky Language Hierarchy
Recursively-enumerable Recursive Context-sensitive Context-free Regular Non-recursively enumerable Recursively Enumerable Languages
Unrestricted Grammars Productions String of variables and terminals Example of unrestricted grammar Theorem A language is recursively enumerable if and only if it is generated by an unrestricted grammar. Context-Sensitive Grammars
and Productions String of variables and terminals The language is context-sensitive: A language is context sensitive if and only if
Theorem A language is context sensitive if and only if it is accepted by a Linear-Bounded automaton. 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. Left-end marker Input string Right-end Working space in tape All computation is done between end markers. Linear Bounded Automaton (LBA) Observation There is a language which is context-sensitive but not recursive. Decidability Consider problems with answer YES or NO.
Examples Does Machinehave three states ? Is string a binary number? Does DFA accept any input? A problem is decidable if some Turing machine
solves (decides) the problem. Decidable problems: Does Machinehave three states ? Is string a binary number? Does DFA accept any input? The Turing machine that solves a problem
answers YES or NO for each instance. Input problem instance YES Turing Machine NO 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. Turing Machine that decides a problem
YES NO YES and NO states are halting states Difference between Recursive Languages (Acceptera) andDecidable problems (Avgra) For decidable problems: The YES states may not be final states. Some problems are undecidable:
There is no Turing Machine that solves all instances of the problem. A famous undecidable problem:
The halting problem The Halting Problem Input: Turing Machine String Question: Does halt on ? Theorem The halting problem is undecidable. Proof Assume to the contrary that the halting problem is decidable. There exists Turing Machine
that solves the halting problem YES halts on NO doesnt halt on Construction of Input: initial tape contents YES NO Encoding of String Construct machine returns YES then loop forever. If returns NO then halt. If Loop forever YES NO Construct machine Input: (machine ) If halts on input Then loop forever Else halt copy Run machinewith input itself
If halts on input Then loop forever Else halt on input : If halts then loops forever. If doesnt halt then it halts. CONTRADICTION ! The halting problem is undecidable.
This means that The halting problem is undecidable. END OF PROOF Another proof of the same theorem:
If the halting problem was decidable then every recursively enumerable language would be recursive. Recursive vs. Recursively Enumerable Languages Theorem The halting problem is undecidable. Proof Assume to the contrary that the halting problem is decidable. There exists Turing Machine
that solves the halting problem. YES halts on NO doesnt halt on Letbe 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 acceptsand halts on any input. NO reject halts on ? YES accept Run with input reject
Turing 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 Therefore 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! Therefore, the halting problem is undecidable.
END OF PROOF A simple undecidable problem:
The Membership Problem The Membership Problem
Input: Turing Machine String Question: Does accept ? Theorem The membership problem is undecidable. Proof Assume to the contrary that the membership problem is decidable. There exists a Turing Machine
that solves the membership problem accepts YES NO rejects Letbe a recursively enumerable language.
Let be the Turing Machine that accepts . We will prove thatis also recursive: We will describe a Turing machine that acceptsand halts on any input. Turing Machine that accepts
and halts on any input YES accept accepts ? NO reject Therefore, is recursive. Sinceis chosen arbitrarily, we have proven that every recursively enumerable language is also recursive. But there are recursively enumerable languages which are not recursive. Contradiction! Therefore, the membership problem is undecidable.
END OF PROOF Reducibility Problemis reduced to problem
If we can solve problemthen we can solve problem . Problemis reduced to problem
Ifis decidable then is decidable. If is undecidable then is undecidable. Example the halting problem reduced to the state-entry problem. The state-entry problem
Inputs: Turing Machine State String Question: Does enter state on input ? Theorem The state-entry problem is undecidable. Proof Reduce the halting problem to the state-entry problem. Suppose we have an algorithm that solves the state-entry problem.
(Turing Machine) that solves the state-entry problem. We will construct an algorithm that solves the halting problem. Assume we have the state-entry algorithm:
YES enters Algorithm for state-entry problem NO doesnt enter We want to design the halting algorithm:
YES halts on Algorithm for Halting problem NO doesnt halt on Modify input machine Add new state From any halting state add transitions to Single halt state halting states halts if and only if halts on state Algorithm for halting problem
Inputs: machine and string 1.Construct machine with state 2. Run algorithm for state-entry problem with inputs: , Halting problem algorithm
YES YES Generate State-entry algorithm NO NO 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 Another example The halting problem reduced to the blank-tape halting problem. The blank-tape halting problem
Input: Turing Machine Question: Does halt when started with a blank tape? Theorem The blank-tape halting problem is undecidable. Proof Reduce the halting problem to the blank-tape halting problem. Suppose we have an algorithm
for the blank-tape halting problem. We will construct an algorithm for the halting problem. blank-tape halting algorithm
Assume we have the blank-tape halting algorithm halts on blank tape YES Algorithm for blank-tape halting problem NO doesnt halt on blank tape We want to design the halting algorithm:
YES halts on Algorithm for halting problem NO doesnt halt on Construct a new machine
On blank tape writes Then continues execution like step 1 step2 if blank tape execute then write with input haltson input string if and only if halts when started with blank tape. Algorithm for halting problem
Inputs: machine and string 1.Construct 2.Run algorithm for blank-tape halting problem with input Halting problem algorithm
YES Blank-tape halting algorithm YES Generate NO NO 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 Summary of Undecidable Problems
Halting Problem Does machinehalt on input? Membership problem Does machineaccept string? Is a stringmember of a recursively enumerable language?) (In other words: Blank-tape halting problem
Does machine halt when starting on blank tape? State-entry Problem: Does machineenter state on input? Uncomputable Functions Uncomputable Functions
Domain Range A function is uncomputable if it cannot be computed for all of its domain. Example An uncomputable function: maximum number of moves until any Turing machine withstates halts when started with the blank tape. Theorem Function is uncomputable. Proof Assume to the contrary that is computable. Then the blank-tape halting problem is decidable. 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 returnYES otherwise returnNO Therefore, the blank-tape halting
problem must be decidable. However, we know that the blank-tape halting problem is undecidable. Contradiction! Therefore, function is uncomputable.
END OF PROOF Rices Theorem Definition Non-trivial property of recursively enumerable languages: Any property possessed by some (not all) recursively enumerable languages. Some non-trivial properties of
recursively enumerable languages: is empty is finite contains two different strings of the same length Rices Theorem Any non-trivial property of a recursively enumerable language is undecidable. Rices Theorem If is a set of Turing-acceptable languages that contain some but not all such languages, no TM can decide for an arbitrary Turing-acceptable language L if L belongs to or not. Exempel Givet en Turingmaskin M, kan man avgra om alla strngar som accepteras av M brjar och slutar p samma tecken? Oavgrbart Problemet handlar om en icke-trivial sprkegenskap. Det finns TM:er vars accepterade strngar har egenskapen i frga, och det finns TM:er vars accepterade strngar inte har egenskapen. Formellt: = { L | TM accepterbara sprk vars strngar brjar och slutar p samma tecken. } Now we will prove some non-trivial properties
without using Rices theorem. 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. Membership problem: Does machineaccept string? Letbe the machine that accepts
Assume we have the empty language algorithm: YES empty Algorithm for empty language problem NO not empty We will design the membership algorithm:
YES accepts Algorithm for membership problem NO rejects First construct machine :
Whenenters a final state, compare original input string with. Accept if original input is the same as . if and only if is not empty Algorithm for membership problem
Inputs:machine and string 1.Construct 2.Determine if is empty YES: then NO: then Membership algorithm NO YES Check if construct YES is empty NO 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 Decidability continued 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. Letbe the machine that accepts
Assume we have the finite language algorithm: Algorithm for finite language problem YES NO finite not finite We will design the halting problem algorithm:
Algorithm for Halting problem YES NO halts on doesnt halt on First construct machine .
Initially, simulates on input. When enters a halt state, accept any input (infinite language). Otherwise accept nothing (finite language). halts on is not finite. if and only if Algorithm for halting problem:
Inputs:machine and string 1.Construct 2.Determine if is finite YES: then doesnt halt on NO: then halts on Machine for halting problem
construct Check if is finite YES NO 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 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. Letbe the machine that accepts
Assume we have the two-strings algorithm: Algorithm for two-strings problem YES NO contains doesnt contain two equal length strings We will design the halting problem algorithm:
Algorithm for Halting problem YES NO halts on doesnt halt on First construct machine .
Initially, simulates on input . When enters a halt state, accept symbols or . (two equal length strings) halts on if and only if acceptsand (two equal length strings) Algorithm for halting problem
Inputs:machine and string 1.Construct 2.Determine if accepts two strings of equal length YES: then halts on NO: thendoesnt halt on Machine for halting problem
YES YES Check if construct NO NO has two equal length strings 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 Church-Turing Thesis*
*Source: Stanford Encyclopaedia of Philosophy A Turing machine is an abstract representation of a computing device.
It is more like a computer program (software) than a computer (hardware). were first proposed by Alan Turing, in an attempt to give
LCMs [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". The Church-Turing thesis
concerns an effective or mechanical method in logic and mathematics. 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. 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. 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. 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. Churchs thesis: A function of positive integers is effectively calculable only if recursive. Misunderstandings 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". 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. 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. Among a machines repertoire of atomic operations there may be those that no human being unaided by machinery can perform. 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. Turings "Machines". These machines are humans who calculate
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) Computation 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. Computation 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. Computation 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). Interaction: Conjectures, Results, and Myths
Additional reading Interaction:Conjectures, Results, and Myths Dina Goldin Univ. of Connecticut, Brown University Fundamental Questions Underlying Theory of Computation
What is computation? How do we model it? 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 availableat 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 problemsare function-based. 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 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,its useless but arent OSs useful?? 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 Wegners conjecture: Interaction is more powerful than algorithms 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. 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 thats detailed enough? ? Worse yetthe 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 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, generatedas we are driving similar to control systems, or online computation A paradigm shift in the conceptualization of computational problem solving. Rethinking the mathematical worldview
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 Sequential Interaction
Sequential interactive computation: System continuously interacts with its environmentby alternately accepting an input stringand computing a corresponding output string. Examples: method invocations of an object instancein 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, etc. Sequential Interaction Thesis
Whenever there is an effective method for performing sequential interactive computation, this computationcan 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. 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 a computer can do. The equivalence of the two is a myth! This myth has been dogmatically accepted by the CS community. The function-based behavior of algorithms does not capture all forms of computation. Turing himself would have denied it. In the same paper where he introduced what we now call Turing Machines, he also introduced choice machines, as a distinct model of computation. Choice machines extend Turing Machines to interaction by allowing a human operator to make choices during the computation. 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 Machinefor 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 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 fundamentalnotions of computation, identifying computability with the computation of functions, as well as with Turing Machines. The batch-based operation of original computers didsuggest other conceptualizations of computation. Deconstructing the Turing Thesis Myth (2)
solvable problems = algorithms Assumes: Algorithmic computation is also function based;i.e., the computational role of an algorithmis 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 aregiven to it initially before the algorithm begins. [Knuth68] 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. Deconstructing the Turing Thesis Myth (3)
algorithms = computation Reasons: 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&Rice69] 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] Outline Rethinking the mathematical worldview Persistent Turing Machines (PTMs) PTM expressiveness Sequential Interaction The Myth of the Church-Turing Thesis Conclusions and future work 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 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] Modeling Interactive Computation: PTMs in Perspective
Many other interactive models Reactive 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. 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 [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 (GDC: except in case when environment is represented as an agent in an multi-agentmodel) References http://www.cse.uconn.edu/~dqg/papers/
[Wegner97] Peter Wegner Why Interaction is more Powerful than Algorithms Communications of the ACM, May 1997 [EGW04] Eugene Eberbach, Dina Goldin, Peter WegnerTuring's Ideas and Models of Computation book chapter, in Alan Turing: Life and Legacy of a Great Thinker, Springer 2004 [I&C04] Dina Goldin, Scott Smolka, Paul Attie, Elaine Sonderegger Turing Machines, Transition Systems, and Interaction Information & Computation Journal, 2004 [GW04] Dina Goldin, Peter Wegner The Church-Turing Thesis: Breaking the Myth presented at CiE 2005, Amsterdam, June to be published in LNCS