## Computation, Dynamics, and CognitionCurrently there is growing interest in the application of dynamical methods to the study of cognition. Computation, Dynamics, and Cognition investigates this convergence from a theoretical and philosophical perspective, generating a provocative new view of the aims and methods of cognitive science. Advancing the dynamical approach as the methodological frame best equipped to guide inquiry in the field's two main research programs--the symbolic and connectionist approaches--Marco Giunti engages a host of questions crucial not only to the science of cognition, but also to computation theory, dynamical systems theory, philosophy of mind, and philosophy of science. In chapter one Giunti employs a dynamical viewpoint to explore foundational issues in computation theory. Using the concept of Turing computability, he precisely and originally defines the nature of a computational system, sharpening our understanding of computation theory and its applications. In chapter two he generalizes his definition of a computational system, arguing that the concept of Turing computability itself is relative to the kind of support on which Turing machine operate. Chapter three completes the book's conceptual foundation, discussing a form of scientific explanation for real dynamical systems that Giunti calls "Galilean explanation." The book's fourth and final chapter develops the methodological thesis that all cognitive systems are dynamical systems. On Giunti's view, a dynamical approach is likely to benefit even those scientific explanations of cognition which are based on symbolic models. Giunti concludes by proposing a new modeling practice for cognitive science, one based on "Galilean models" of cognitive systems. Innovative, lucidly-written, and broad-ranging in its analysis, Computation, Dynamics, and Cognition will interest philosophers of science and mind, as well as cognitive scientists, computer scientists, and theorists of dynamical systems. This book elaborates a comprehensive picture of the application of dynamical methods to the study of cognition. Giunti argues that both computational systems and connectionist networks are special types of dynamical systems. He shows how this dynamical approach can be applied to problems of cognition, information processing, consciousness, meaning, and the relation between body and mind. |

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### Contents

Mathematical Dynamical Systems and Computational Systems | 3 |

2 Mathematical dynamical systems | 7 |

3 Computational systems | 12 |

4 The emulation relation | 23 |

5 Reversible versus irreversible systems | 28 |

6 The realization relation | 37 |

7 Virtual systems the reversible realizability of irreversible systems and the existence of reversible universal systems | 42 |

Proofs of selected theorems | 46 |

6 Galilean frameworks of Ksystems | 120 |

7 Explicit versus implicit specification of the set of state transitions of a possible Galilean model of a Ksystem | 124 |

8 The inductive method for constructing Galilean frameworks of Ksystems | 129 |

9 The deductive method for constructing Galilean frameworks of Ksystems | 131 |

Proofs of selected theorems | 133 |

Notes | 137 |

Cognitive Systems and the Scientific Explanation of Cognition | 139 |

2 Cognitive systems as dynamical systems | 140 |

Notes | 50 |

Generalized Computational Systems | 55 |

2 Turing machines on pattern field F | 57 |

3 Is the concept of computability on pattern field F reducible to the usual concept of Turing computability? | 67 |

4 Is a nonrecursive pattern field necessary for computing nonrecursive functions? | 78 |

5 The concept of computability on pattern field F is a generalization of the usual concept of computability | 83 |

6 Computational systems on pattern field F | 88 |

Proofs of selected theorems | 92 |

Notes | 106 |

Galilean Models and Explanations | 113 |

2 Real dynamical systems versus mathematical dynamical systems | 114 |

3 Models of Ksystems | 115 |

4 Galilean explanations and the traditional practice of dynamical modeling | 116 |

5 Galilean explanations are based on Galilean models of Ksystems | 119 |

the argument | 141 |

computation theory and dynamical systems theory | 145 |

31 Dynamical systems theory and the explanation of cognition based on symbolic models | 146 |

32 Computation theory and the explanation of cognition based on neural networks or other continuous dynamical models | 148 |

4 Cognitive systems and their models | 153 |

41 Simulation models of cognitive systems | 154 |

42 Galilean models of cognitive systems | 155 |

Pexplanatory and correct Galilean frameworks of cognitive systems | 158 |

44 The Galilean approach to cognitive science | 159 |

Notes | 160 |

163 | |

173 | |

### Common terms and phrases

alphabet Ac bijection blank cellular automata chapter coding yc cognitive science cognitive system computable on F computation theory computational system concept of computability condition 2d corresponds decoding defined denoted by numeral differential or difference doubly infinite tape dynamical models dynamical systems theory effective relative emulates MDS2 emulating system emulation relation eventually periodic orbits evolution functions example explanation of cognition F relative F(AC falling body field F finite number finite strings Galilean explanation halting problem identity function instantiates internal state q0 irreversible system isomorphic to MDS K-system lemma logically irreversible magnitudes mathematical dynamical system merging orbits model of KRS nonblank symbol number denoted numeric function oracle machines orb(x orb(y pattern field possible Galilean model Proof real dynamical system real system recursive functions recursive relative regular enumeration regular relative replaces set of values strongly irreversible theorem thesis tion total recursive transition g Turing machine w e Mc Z+ satisfy

### Popular passages

Page x - M that represents all possible states through which the system can evolve; M is called the state space (or sometimes the phase space) of the system. The third element is a set of functions {#'} that tells us the state of the system at any instant t provided that we know the initial state; each function in {g'} is called a state transition (or a tadvance) of the system.