Groupoid Free Energy--论文代写范文精选

它允许混合对称,在这个模型中,系统可以进行间断广群对称性。降低代表基本的相变。动态行为远离临界点。我们下一个应用这种形式主义纯粹内部化。下面的paper代写范文进行阐述。

Abstract
Equivalence classes define groupoids, by the mechanisms described in the Mathematical Appendix. The basic equivalence classes of a cognitive or biological structure will define the basic transitive groupoids, and higher order systems can be constructed by the union of these transitive groupoids, having larger alphabets that allow more complicated statements in the sense of Ash above. We associate information sources 2 with transitive groupoids, and with the larger groupoids constructed from them. The more complicated the groupoid, the greater the information source uncertainty, following Ash’s reasoning.

In terms of its relation to a fixed, appropriately normalized, inverse system temperature. This gives a statistical thermodynamic means of defining a ‘higher’ free energy construct – FG[K] – to which we can now apply Landau’s fundamental heuristic phase transition argument (Landau and Lifshitz 2007; Skierski et al. 1989; Pettini 2007). Absent a high value of the temperature-equivalent, in this model, only the simplest transitive groupoid structures can be manifest. A full treatment from this perspective requires invocation of groupoid representations, no small matter (e.g., Bos, 2007; Buneci, 2003). 

Somewhat more rigorously, the elaborate renormalization schemes of Wallace (2005) may now be imposed on FG[K] itself, leading to a spectrum of highly punctuated transitions in the overall system of information sources. The essential point is that FG[K] is unlikely to scale with a renormalization transform as simply as does physical free energy, and this leads to very complicated ‘biological’ renormalization strategies. See Wallace (2005), Wallace and Fullilove, (2008) or Wallace et al., (2007) for details. 

Most deeply, however, an extended version of Pettini’s (2007) Morse-Theory-based topological hypothesis can now be invoked, i.e., that changes in underlying groupoid structure are a necessary (but not sufficient) consequence of phase changes in FG[K]. Necessity, but not sufficiency, is important, as it allows for mixed symmetries. An outline of the theory is presented in the Appendix. For details see, e.g., Matsumoto (2002) or Pettini (2007). As the temperature-analog declines, in this model, the system can undergo punctuated groupoid symmetry reductions representing fundamental phase transitions. Dynamical behavior away from critical points will be determined, in this model, by Generalized Onsager Relations, also explored more fully in the Appendix. We next apply this formalism to examples of of purely internal, and of linked internal-external, cognitive function.

High Order Cognition 
According to Atlan and Cohen (1998), the essence of cognition is comparison of a perceived external signal with an internal, learned picture of the world, and then, upon that comparison, the choice of one response from a much larger repertoire of possible responses. Such reduction in uncertainty inherently carries information, and, following Wallace (2000, 2005) it is possible to make a very general model of this process as an information source. Our focus is on those composite paths x that trigger pattern recognition-and-response. That is, given a fixed initial state a0, such that h(a0) ∈ B0, we examine all possible subsequent paths x beginning with a0 and leading to the event h(x) ∈ B1. Thus h(a0, ..., aj ) ∈ B0 for all 0 ≤ j < m, but h(a0, ..., am) ∈ B1. For each positive integer n, let N(n) be the number of grammatical and syntactic high probability paths of length n which begin with some particular a0 having h(a0) ∈ B0 and lead to the condition h(x) ∈ B1. 

We shall call such paths meaningful and assume N(n) to be considerably less than the number of all possible paths of length n – pattern recognition-andresponse is comparatively rare. We again assume that the longitudinal finite limit H ≡ limn→∞ log[N(n)]/n both exists and is independent of the path x. We will – not surprisingly – call such a cognitive process ergodic. Note that disjoint partition of state space may be possible according to sets of states which can be connected by meaningful paths from a particular base point, leading to a natural coset algebra of the system, a groupoid. This is a matter of some importance. 

It is thus possible to define an ergodic information source X associated with stochastic variates Xj having joint and conditional probabilities P(a0, ..., an) and P(an|a0, ..., an−1) such that appropriate joint and conditional Shannon uncertainties may be defined which satisfy the relations of equation (1) above. This information source is taken as dual to the ergodic cognitive process. Dividing the full set of possible responses into the sets B0 and B1 may itself require higher order cognitive decisions by another module or modules, suggesting the necessity of choice within a more or less broad set of possible quasi-languages. This would directly reflect the need to shift gears according to the different challenges faced by the organism, machine, or social group. ‘Meaningful’ paths – creating an inherent grammar and syntax – have been defined entirely in terms of system response, as Atlan and Cohen (1998) propose. 

This formalism can easily be applied to the stochastic neuron in a neural network, as done in Wallace (2005). A formal equivalence class algebra can be constructed for a cognitive process characterized by a dual information source by choosing different origin points a0, in the sense above, and defining equivalence of two states by the existence of a high-probability meaningful path connecting them with the same origin. Disjoint partition by equivalence class, analogous to orbit equivalence classes for dynamical systems, defines the vertices of a network of cognitive dual languages. Each vertex then represents a different information source dual to a cognitive process. This is not a direct representation as in a neural network, or of some circuit in silicon. It is, rather, an abstract set of ‘languages’ dual to the cognitive processes instantiated by biological structures, machines, social process, or their hybrids. 

Our particular interest, however, is in an interacting network of cognitive processes. This structure generates a groupoid, in the sense of the Appendix. Recall that states aj , ak in a set A are related by the groupoid morphism if and only if there exists a highprobability grammatical path connecting them to the same base point, and tuning across the various possible ways in which that can happen – the different cognitive languages – parametizes the set of equivalence relations and creates the groupoid. We now envision an average mean field mutual information linking different information sources associated with the transitive groupoids defined by this network. Call that mean field I. Another possible interpretation is of an average probability of nondisjuctive ‘weak’ ties P (sensu Granovetter, 1973) linking the different ergodic dual information sources. 

Then, for the Groupoid Free Energy calculation above, take K ∝ 1/I, 1/P. Increasing I or P then, increases the linkage across the transitive groupoids of the cognitive system, leading, in a highly punctuated way, to larger and larger processes of collective cognition using progressively larger ‘alphabets’ and having, in the sense of Ash above, progressively larger values of the associated dual information source. A second model arises in a natural manner by taking 1/K as the mean number, N , of linkages between dual information sources in the abstract network. This leads to generalizations of the Erdos/Renyi random network formalism, and its inherent phase transitions. Both approaches can be extended to second order as an analog to hierarchical regression. The first generalization is via a kind of universality class tuning, and the second by means of a renormalization in which couplings at or above a tunable limit are set to 1 and those below to 0. A Morse Theory topological tuning results directly from the latter approach. Evolutionary process, or engineering design, are not necessarily restricted, however, to these two exactly solvable models.

Wallace (2005) and Wallace and Fullilove (2008) use simpli- fied forms of this argument to characterize consciousness and distributed institutional cognition, respectively. Our particular interest, however, is in the ways such cognitive structures respond to challenges in real time: individual and institutional distributed cognition do not occur in a vacuum, but in the context of demands for prompt action from an embedding ecological structure. We will attempt to characterize pathologies of such real time response. To do this we must iterate the argument.(paper代写)

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