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Summary A crucial problem in the philosophy of computing is represented by the nature of computation. On the one hand, a computation is thought of as some representation of a formal process composed by well-defined steps, which allows to reach in a finite amount of time a given output from a given input. This is tantamount to the formulation of a mathematical or biological function or the design of an algorithm. On the other hand, a computation is inherently bound to its execution and thus to an implementation. This strongly relates to the problem of determining which physical systems can be said to implement a computation, in turn which systems can be said to be properly computational. The answer to this question can be offered by reduction to other relations (such as causation), but it triggered a widespread debate on whether it implies that almost any physical system is then by definition computational. This has been a particularly intense debate in the cognitive sciences. The duality formal-physical that affects the nature of computation is also of especially great importance in the philosophical debate on the nature of algorithms and programs, where the latter are considered physical implementations of the former.
Key works The thesis that certain human abilities cannot be considered implementation of computations is notoriously held by Dreyfus 1972 and Putnam 1987. This argument is even stronger in Searle 1980, where it is argued that even the interpretation of human abilites as implementation of computations is not enough for the mind. The thesis that a physical system implements a computation if the causal structure of the former reflects the formal structure of the latter is defended in Chalmers 1994. See also Piccinini 2007. A starting point for the  debate on the nature of algorithms is represented by Moschovakis 2001 and Gurevich 2012. Fetzer 1988 offers the very first critique of program verification in view of the formal-physical divide, with a large debate following.
Introductions See Piccinini 2010 for an overview of the notion of computation in physical systems, including an assessment of varieties of the physical Church-Turing thesis. 
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  1. Fred Boogerd, Frank Bruggeman, Catholijn Jonker, Huib Looren de Jong, Allard Tamminga, Jan Treur, Hans Westerhoff & Wouter Wijngaards (2002). Inter-Level Relations in Computer Science, Biology, and Psychology. Philosophical Psychology 15 (4):463–471.
    Investigations into inter-level relations in computer science, biology and psychology call for an *empirical* turn in the philosophy of mind. Rather than concentrate on *a priori* discussions of inter-level relations between 'completed' sciences, a case is made for the actual study of the way inter-level relations grow out of the developing sciences. Thus, philosophical inquiries will be made more relevant to the sciences, and, more importantly, philosophical accounts of inter-level relations will be testable by confronting them with what really happens (...)
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  2. Rainer P. Born (ed.) (1987). Artificial Intelligence: The Case Against. St Martin's Press.
  3. Nick Bostrom (2006). Quantity of Experience: Brain-Duplication and Degrees of Consciousness. [REVIEW] Minds and Machines 16 (2):185-200.
    If a brain is duplicated so that there are two brains in identical states, are there then two numerically distinct phenomenal experiences or only one? There are two, I argue, and given computationalism, this has implications for what it is to implement a computation. I then consider what happens when a computation is implemented in a system that either uses unreliable components or possesses varying degrees of parallelism. I show that in some of these cases there can be, in a (...)
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  4. Andrew Boucher (1997). Parallel Machines. Minds and Machines 7 (4):543-551.
    Because it is time-dependent, parallel computation is fundamentally different from sequential computation. Parallel programs are non-deterministic and are not effective procedures. Given the brain operates in parallel, this casts doubt on AI's attempt to make sequential computers intelligent.
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  5. Selmer Bringsjord (2001). In Computation, Parallel is Nothing, Physical Everything. Minds and Machines 11 (1):95-99.
    Andrew Boucher (1997) argues that ``parallel computation is fundamentally different from sequential computation'' (p. 543), and that this fact provides reason to be skeptical about whether AI can produce a genuinely intelligent machine. But parallelism, as I prove herein, is irrelevant. What Boucher has inadvertently glimpsed is one small part of a mathematical tapestry portraying the simple but undeniable fact that physical computation can be fundamentally different from ordinary, ``textbook'' computation (whether parallel or sequential). This tapestry does indeed immediately imply (...)
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  6. Selmer Bringsjord (1994). Computation, Among Other Things, is Beneath Us. Minds and Machines 4 (4):469-88.
    What''s computation? The received answer is that computation is a computer at work, and a computer at work is that which can be modelled as a Turing machine at work. Unfortunately, as John Searle has recently argued, and as others have agreed, the received answer appears to imply that AI and Cog Sci are a royal waste of time. The argument here is alarmingly simple: AI and Cog Sci (of the Strong sort, anyway) are committed to the view that cognition (...)
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  7. Curtis Brown (2004). Implementation and Indeterminacy. Conferences in Research and Practice in Information Technology 37.
    David Chalmers has defended an account of what it is for a physical system to implement a computation. The account appeals to the idea of a “combinatorial-state automaton” or CSA. It is unclear whether Chalmers intends the CSA to be a computational model in the usual sense, or merely a convenient formalism into which instances of other models can be translated. I argue that the CSA is not a computational model in the usual sense because CSAs do not perspicuously represent (...)
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  8. David J. Buller (1993). Confirmation and the Computational Paradigm (Or: Why Do You Think They Call Itartificial Intelligence?). [REVIEW] Minds and Machines 3 (2):155-181.
    The idea that human cognitive capacities are explainable by computational models is often conjoined with the idea that, while the states postulated by such models are in fact realized by brain states, there are no type-type correlations between the states postulated by computational models and brain states (a corollary of token physicalism). I argue that these ideas are not jointly tenable. I discuss the kinds of empirical evidence available to cognitive scientists for (dis)confirming computational models of cognition and argue that (...)
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  9. Giacomo Cabri, Luca Ferrari & Rossella Rubino (2008). Building Computational Institutions for Agents with Rolex. Artificial Intelligence and Law 16 (1):129-145.
    While the sociality of software agents drives toward the definition of institutions for multi agent systems, their autonomy requires that such institutions are ruled by appropriate norm mechanisms. Computational institutions represent useful abstractions. In this paper we show how computational institutions can be built on top of the RoleX infrastructure, a role-based system with interesting features for our aim. We achieve a twofold goal: on the one hand, we give concreteness to the institution abstractions; on the other hand, we demonstrate (...)
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  10. G. J. Chaitin (2012). Proving Darwin: Making Biology Mathematical. Pantheon.
    Groundbreaking mathematician Gregory Chaitin gives us the first book to posit that we can prove how Darwin’s theory of evolution works on a mathematical level.
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  11. David Chalmers (2012). The Varieties of Computation: A Reply. Journal of Cognitive Science 2012 (3):211-248.
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  12. David J. Chalmers (2011). A Computational Foundation for the Study of Cognition. Journal of Cognitive Science 12 (4):323-357.
    Computation is central to the foundations of modern cognitive science, but its role is controversial. Questions about computation abound: What is it for a physical system to implement a computation? Is computation sufficient for thought? What is the role of computation in a theory of cognition? What is the relation between different sorts of computational theory, such as connectionism and symbolic computation? In this paper I develop a systematic framework that addresses all of these questions. Justifying the role of computation (...)
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  13. David J. Chalmers (1996). Does a Rock Implement Every Finite-State Automaton? Synthese 108 (3):309-33.
    Hilary Putnam has argued that computational functionalism cannot serve as a foundation for the study of the mind, as every ordinary open physical system implements every finite-state automaton. I argue that Putnam's argument fails, but that it points out the need for a better understanding of the bridge between the theory of computation and the theory of physical systems: the relation of implementation. It also raises questions about the class of automata that can serve as a basis for understanding the (...)
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  14. David J. Chalmers (1994). On Implementing a Computation. Minds and Machines 4 (4):391-402.
    To clarify the notion of computation and its role in cognitive science, we need an account of implementation, the nexus between abstract computations and physical systems. I provide such an account, based on the idea that a physical system implements a computation if the causal structure of the system mirrors the formal structure of the computation. The account is developed for the class of combinatorial-state automata, but is sufficiently general to cover all other discrete computational formalisms. The implementation relation is (...)
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  15. Ronald L. Chrisley (1994). The Ontological Status of Computational States. In European Review of Philosophy, Volume 1: Philosophy of Mind. Stanford: CSLI Publications
  16. Carol E. Cleland (2002). On Effective Procedures. Minds and Machines 12 (2):159-179.
    Since the mid-twentieth century, the concept of the Turing machine has dominated thought about effective procedures. This paper presents an alternative to Turing's analysis; it unifies, refines, and extends my earlier work on this topic. I show that Turing machines cannot live up to their billing as paragons of effective procedure; at best, they may be said to provide us with mere procedure schemas. I argue that the concept of an effective procedure crucially depends upon distinguishing procedures as definite courses (...)
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  17. Carol E. Cleland (2002). 'Turing Limit'. Some of Them (Steinhart, Copeland) Represent Extensions of Tur-Ing's Account, Whereas Others Defend Alternatives Notions of Effective Computability (Bringsjord and Zenzen, Wells). Minds and Machines 12:157-158.
  18. Carol E. Cleland (2001). Recipes, Algorithms, and Programs. Minds and Machines 11 (2):219-237.
    In the technical literature of computer science, the concept of an effective procedure is closely associated with the notion of an instruction that precisely specifies an action. Turing machine instructions are held up as providing paragons of instructions that "precisely describe" or "well define" the actions they prescribe. Numerical algorithms and computer programs are judged effective just insofar as they are thought to be translatable into Turing machine programs. Nontechnical procedures (e.g., recipes, methods) are summarily dismissed as ineffective on the (...)
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  19. Carol E. Cleland (1995). Effective Procedures and Computable Functions. Minds and Machines 5 (1):9-23.
    Horsten and Roelants have raised a number of important questions about my analysis of effective procedures and my evaluation of the Church-Turing thesis. They suggest that, on my account, effective procedures cannot enter the mathematical world because they have a built-in component of causality, and, hence, that my arguments against the Church-Turing thesis miss the mark. Unfortunately, however, their reasoning is based upon a number of misunderstandings. Effective mundane procedures do not, on my view, provide an analysis of ourgeneral concept (...)
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  20. John Collier, Information, Causation and Computation.
    Causation can be understood as a computational process once we understand causation in informational terms. I argue that if we see processes as information channels, then causal processes are most readily interpreted as the transfer of information from one state to another. This directly implies that the later state is a computation from the earlier state, given causal laws, which can also be interpreted computationally. This approach unifies the ideas of causation and computation.
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  21. B. Jack Copeland (2002). Accelerating Turing Machines. Minds and Machines 12 (2):281-300.
    Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary (...)
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  22. B. Jack Copeland (2002). Hypercomputation. Minds and Machines 12 (4):461-502.
    A survey of the field of hypercomputation, including discussion of a variety of objections.
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  23. B. Jack Copeland (1996). What is Computation? Synthese 108 (3):335-59.
    To compute is to execute an algorithm. More precisely, to say that a device or organ computes is to say that there exists a modelling relationship of a certain kind between it and a formal specification of an algorithm and supporting architecture. The key issue is to delimit the phrase of a certain kind. I call this the problem of distinguishing between standard and nonstandard models of computation. The successful drawing of this distinction guards Turing's 1936 analysis of computation against (...)
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  24. B. Jack Copeland & Oron Shagrir (2007). Physical Computation: How General Are Gandy's Principles for Mechanisms? [REVIEW] Minds and Machines 17 (2):217-231.
    What are the limits of physical computation? In his ‘Church’s Thesis and Principles for Mechanisms’, Turing’s student Robin Gandy proved that any machine satisfying four idealised physical ‘principles’ is equivalent to some Turing machine. Gandy’s four principles in effect define a class of computing machines (‘Gandy machines’). Our question is: What is the relationship of this class to the class of all (ideal) physical computing machines? Gandy himself suggests that the relationship is identity. We do not share this view. We (...)
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  25. Jack Copeland, Even Turing Machines Can Compute Uncomputable Functions.
    Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability.
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  26. Eric Dietrich (2000). A Counterexample T o All Future Dynamic Systems Theories of Cognition. J. Of Experimental and Theoretical AI 12 (2):377-382.
    Years ago, when I was an undergraduate math major at the University of Wyoming, I came across an interesting book in our library. It was a book of counterexamples t o propositions in real analysis (the mathematics of the real numbers). Mathematicians work more or less like the rest of us. They consider propositions. If one seems to them to be plausibly true, then they set about to prove it, to establish the proposition as a theorem. Instead o f setting (...)
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  27. Ronald P. Endicott (1996). Searle, Syntax, and Observer-Relativity. Canadian Journal of Philosophy 26 (1):101-22.
    I critically examine some provocative arguments that John Searle presents in his book The Rediscovery of Mind to support the claim that the syntactic states of a classical computational system are "observer relative" or "mind dependent" or otherwise less than fully and objectively real. I begin by explaining how this claim differs from Searle's earlier and more well-known claim that the physical states of a machine, including the syntactic states, are insufficient to determine its semantics. In contrast, his more recent (...)
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  28. Steven Ericsson-Zenith (forthcoming). Explaining Experience In Nature: The Foundations Of Logic And Apprehension. Institute for Advanced Science & Engineering.
    At its core this book is concerned with logic and computation with respect to the mathematical characterization of sentient biophysical structure and its behavior. -/- Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. -/- These theories are based upon a new explanation of experience in (...)
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  29. R. P. Feynman (1982). Simulating Physics with Computers. International Journal of Theoretical Physics 21 (6):467-488.
  30. Nicolas Fillion & Robert M. Corless (2014). On the Epistemological Analysis of Modeling and Computational Error in the Mathematical Sciences. Synthese 191 (7):1451-1467.
    Interest in the computational aspects of modeling has been steadily growing in philosophy of science. This paper aims to advance the discussion by articulating the way in which modeling and computational errors are related and by explaining the significance of error management strategies for the rational reconstruction of scientific practice. To this end, we first characterize the role and nature of modeling error in relation to a recipe for model construction known as Euler’s recipe. We then describe a general model (...)
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  31. C. Foster (1990). Algorithms, Abstraction and Implementation. Academic Press.
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  32. Nir Fresco (2010). Explaining Computation Without Semantics: Keeping It Simple. [REVIEW] Minds and Machines 20 (2):165-181.
    This paper deals with the question: how is computation best individuated? -/- 1. The semantic view of computation: computation is best individuated by its semantic properties. 2. The causal view of computation: computation is best individuated by its causal properties. 3. The functional view of computation: computation is best individuated by its functional properties. -/- Some scientific theories explain the capacities of brains by appealing to computations that they supposedly perform. The reason for that is usually that computation is individuated (...)
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  33. Nir Fresco (2008). An Analysis of the Criteria for Evaluating Adequate Theories of Computation. Minds and Machines 18 (3):379-401.
  34. John Haugeland (2003). Syntax, Semantics, Physics. In John M. Preston & Michael A. Bishop (eds.), Views Into the Chinese Room: New Essays on Searle and Artificial Intelligence. Oxford University Press
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  35. Leon Horsten (1995). The Church-Turing Thesis and Effective Mundane Procedures. Minds and Machines 5 (1):1-8.
    We critically discuss Cleland''s analysis of effective procedures as mundane effective procedures. She argues that Turing machines cannot carry out mundane procedures, since Turing machines are abstract entities and therefore cannot generate the causal processes that are generated by mundane procedures. We argue that if Turing machines cannot enter the physical world, then it is hard to see how Cleland''s mundane procedures can enter the world of numbers. Hence her arguments against versions of the Church-Turing thesis for number theoretic functions (...)
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  36. B. Jack Copeland & Oron Shagrir (2011). Do Accelerating Turing Machines Compute the Uncomputable? Minds and Machines 21 (2):221-239.
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  37. J. R. Kazez (1994). Computationalism and the Causal Role of Content. Philosophical Studies 75 (3):231-60.
  38. Robert W. Kentridge (1995). Symbols, Neurons, Soap-Bubbles and the Neural Computation Underlying Cognition. Minds and Machines 4 (4):439-449.
    A wide range of systems appear to perform computation: what common features do they share? I consider three examples, a digital computer, a neural network and an analogue route finding system based on soap-bubbles. The common feature of these systems is that they have autonomous dynamics — their states will change over time without additional external influence. We can take advantage of these dynamics if we understand them well enough to map a problem we want to solve onto them. Programming (...)
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  39. Colin Klein (2008). Dispositional Implementation Solves the Superfluous Structure Problem. Synthese 165 (1):141 - 153.
    Consciousness supervenes on activity; computation supervenes on structure. Because of this, some argue, conscious states cannot supervene on computational ones. If true, this would present serious difficulties for computationalist analyses of consciousness (or, indeed, of any domain with properties that supervene on actual activity). I argue that the computationalist can avoid the Superfluous Structure Problem (SSP) by moving to a dispositional theory of implementation. On a dispositional theory, the activity of computation depends entirely on changes in the intrinsic properties of (...)
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  40. Colin Klein, Maudlin on Computation.
    I argue that computationalism is compatible with a plausible supervenience thesis about conscious states. The most plausible way of making it compatible, however, involves abandoning counterfactual conditions on implementation.
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  41. Bruce J. MacLennan, (Position Paper for Symposium, \What is Computing?").
    The central claim of computationalism is generally taken to be that the brain is a computer, and that any computer implementing the appropriate program would ipso facto have a mind. In this paper I argue for the following propositions: (1) The central claim of computationalism is not about computers, a concept too imprecise for a scienti c claim of this sort, but is about physical calculi (instantiated discrete formal systems). (2) In matters of formality, interpretability, and so forth, analog computation (...)
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  42. Jacques Mallah, The Many Computations Interpretation (MCI) of Quantum Mechanics.
    Computationalism provides a framework for understanding how a mathematically describable physical world could give rise to conscious observations without the need for dualism. A criterion is proposed for the implementation of computations by physical systems, which has been a problem for computationalism. Together with an independence criterion for implementations this would allow, in principle, prediction of probabilities for various observations based on counting implementations. Applied to quantum mechanics, this results in a Many Computations Interpretation (MCI), which is an explicit form (...)
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  43. Jacques Mallah (forthcoming). Structure and Dynamics in Implementation of Computations. In Yasemin J. Erden (ed.), Proceedings of the 7th AISB Symposium on Computing and Philosophy:. AISB
    Without a proper restriction on mappings, virtually any system could be seen as implementing any computation. That would not allow characterization of systems in terms of implemented computations and is not compatible with a computationalist philosophy of mind. Information-based criteria for independence of substates within structured states are proposed as a solution. Objections to the use of requirements for transitions in counterfactual states are addressed, in part using the partial-brain argument as a general counterargument to neural replacement arguments.
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  44. Jacques Mallah, The Partial Brain Thought Experiment: Partial Consciousness and its Implications.
    The ‘Fading Qualia’ thought experiment of Chalmers purports to show that computationalism is very probably true even if dualism is true by considering a series of brains, with biological parts increasingly substituted for by artificial but functionally analagous parts in small steps, and arguing that consciousness would not plausibly vanish in either a gradual or sudden way. This defense of computationalism inspired an attack on computationalism by Bishop, who argued that a similar series of substitutions by parts that have the (...)
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  45. Pavel Materna (2009). Concepts and Recipes. Acta Analytica 24 (1):69-90.
    If concepts are explicated as abstract procedures, then we can easily show that each empirical concept is a not an effective procedure. Some, but not all empirical concepts are shown to be of a special kind: they cannot in principle guarantee that the object they identify satisfies the intended conditions.
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  46. Marcin Miłkowski (2014). Computational Mechanisms and Models of Computation. Philosophia Scientiæ 18:215-228.
    In most accounts of realization of computational processes by physical mechanisms, it is presupposed that there is one-to-one correspondence between the causally active states of the physical process and the states of the computation. Yet such proposals either stipulate that only one model of computation is implemented, or they do not reflect upon the variety of models that could be implemented physically. -/- In this paper, I claim that mechanistic accounts of computation should allow for a broad variation of models (...)
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  47. Marcin Miłkowski (2012). Is Computation Based on Interpretation? Semiotica 188 (1):219-228.
    I argue that influential purely syntactic views of computation, shared by such philosophers as John Searle and Hilary Putnam, are mistaken. First, I discuss common objections, and during the discussion I mention additional necessary conditions of implementation of computations in physical processes that are neglected in classical philosophical accounts of computation. Then I try to show why realism in regards of physical computations is more plausible, and more coherent with any realistic attitude towards natural science than the received view, and (...)
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  48. Ivan Moura (2006). A Model of Agent Consciousness and its Implementation. Neurocomputing 69 (16-18):1984-1995.
  49. Sang-Chul Park (2001). Globalisation and Local Innovation System: The Implementation of Government Policies to the Formation of Science Parks in Japan. [REVIEW] AI and Society 15 (3):263-279.
  50. Anne Pascasio (1968). Implementation: Closing the Gap. In Peter Koestenbaum (ed.), Proceedings. [San Jose? Calif. 13--97.
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