#mappendants

[dojoreetsyfinds]

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[gemworld-finejewelery-blog]

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[z0ruas]

Basic Facts and Concepts

Brain is not a randomly connected mesh, there is low and high level structure

Grey matter — surface layer, cortex, mostly neurons, ~6 layers of neurons, for other animals less, varies by area

White matter — insides, connections between neurons, moslty axons

Occurent information — current representation, vector at layer in network, what is being processed now

Abeyant information — stored representation, weights at layer of network

Local conding - each neuron represents information; "Grandmother" is a single neuron

Distributed coding - multiple neurons involved in represeting information; "Grandmother" is distributed on many neurons

Connectomics - given brain is a graph what are it's properties, how it differs by animal, how it comes to being and changes with time. Fully mappend connectoms are for: C elegans, Drosophilia, Fly. Pigeon brain is good coverage, but not full. Monkey, humans not copmlete but at lot is mapped. A lot, but not all, structure of connectome and weights is encoded in genes.

Plasticity - how neurons change their connection weights: structural plasticity - changing what is connected; functional plasticity - changing strength and qualities of connections.

Translational neuroscience - how genes are encoding, guiding, controling brain structures

Hebbian rule - "what fires together, wires together". Some mechanisms in brain are anti-Hebbian and some are neither. Overall, it is important model.

Does model need to be accurate down to molecular level to understand cognition? Computational neuroscience says no, level of neurons and their connections might be sufficient

There are a lot of recurrent loops in networks

Timing is very important, as a lot of processing and wiring are in oscilating currents

There are different types of neurons

Books

The Computational Brain, Patricia S. Churchland, Terrence J. Sejnowski, MIT Press

Changing Connectomes, Marcus Kaiser, MIT Press

Neuroscience of Mathematical Cognitive Development, Rhonda Douglas Brown, Springer

Chasing Men on Fire, Stephen G Waxman, MIT Press

The Ego Tunnel, Thomas Metzinger

Strüngmann Forum Reports, yearly meeting of leading researchers regarding current status of selected topic:

The Neocortex, MIT Press, 2019

Emergent Brain Dynamics: Prebirth to Adolescence, MIT Press, 2018

Translational Neuroscience: Toward New Therapies, MIT Press, 2015

[creativedesignz]

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[assemblage333]

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[snak]

Monoid of functors

As we saw in the previous post, a function from a to a forms a monoid with respect to their composition and an identity function. Then, what we can do with a functor?

{-# LANGUAGE FlexibleInstances, RankNTypes, TypeFamilies, TypeOperators, UndecidableInstances #-} import Data.Functor.Compose (Compose(Compose)) import Data.Functor.Identity (Identity(Identity)) import Data.Kind (Type) import Data.Monoid (Endo(Endo))

First, let's revisit the monoid instance of Endo. But this time, we define our monoid class.

class Monoid' a where mu :: (a, a) -> a eta :: () -> a

This class expresses the same idea as the standard Monoid. mu is an uncurryed version of mappend, and eta is a function version of mempty. You can implement an instance for Endo easily.

instance Monoid' (Endo a) where mu (Endo f, Endo g) = Endo $ f . g eta () = Endo id

Actually, we don't necessarily use a tuple and a unit, but can use some kind of product and its unit if they satisify certain laws, such as the product is a bifunctor. You can find more details at From Monoids to Monads, and Monads Categorically. I'll skip these details in this article, but focus on get some intuitions from the code.

So, let's factor out these types using type families.

class Monoid'' a where type Product'' a type Unit'' a mu' :: Product'' a -> a eta' :: Unit'' a -> a instance Monoid'' (Endo a) where type Product'' (Endo a) = (Endo a, Endo a) type Unit'' (Endo a) = () mu' (Endo f, Endo g) = Endo $ f . g eta' () = Endo id

Then, now we're going to define a class for functors.

class MonoidF' f where type ProductF' (f :: Type -> Type) :: Type -> Type type UnitF' f :: Type -> Type muF' :: ProductF' f a -> f a etaF' :: UnitF' f a -> f a

This looks very similar to Monoid'' except that it's for functors. By introducing an operator (~>), they can look even similar.

type f ~> g = forall a. f a -> g a class MonoidF'' f where type ProductF'' (f :: Type -> Type) :: Type -> Type type UnitF'' f :: Type -> Type muF'' :: ProductF'' f ~> f etaF'' :: UnitF'' f ~> f

You can write instances of this class for, for example, Maybe, (->) r, and (,) w.

instance MonoidF'' Maybe where type ProductF'' Maybe = Compose Maybe Maybe type UnitF'' Maybe = Identity muF'' (Compose (Just (Just x))) = Just x muF'' _ = Nothing etaF'' (Identity x) = Just x instance MonoidF'' ((->) r) where type ProductF'' ((->) r) = Compose ((->) r) ((->) r) type UnitF'' ((->) r) = Identity muF'' (Compose f) e = f e e etaF'' (Identity x) = const x instance Monoid w => MonoidF'' ((,) w) where type ProductF'' ((,) w) = Compose ((,) w) ((,) w) type UnitF'' ((,) w) = Identity muF'' (Compose (w2, (w1, x))) = (w1 <> w2, x) etaF'' (Identity x) = (mempty, x)

These functors form a monoid with respect to their composition and an identity functor just like Endo forms a monoid.

As you may have noticed, it's identical to a monad. Let's define our monad class.

class Monad' f where join' :: f (f a) -> f a pure' :: a -> f a

Instead of defining it with (>>=) and pure, I defined it with join and pure because it's easier to write. You can write an instance of this class for functors that are an instance of MonoidF''.

instance (MonoidF'' f, ProductF'' f ~ Compose f f, UnitF'' f ~ Identity) => Monad' f where join' = muF'' . Compose pure' = etaF'' . Identity

So you can think that a functor is a monad if its composition forms a monoid with respect to its composition and an identity functor.

Note that you can implement the standard Applicative and Monad like this.

instance (Functor f, MonoidF'' f, ProductF'' f ~ Compose f f, UnitF'' f ~ Identity) => Applicative f where (<*>) f m = muF'' $ Compose $ fmap (\f' -> muF'' $ Compose $ fmap (pure . f') m) f pure = etaF'' . Identity instance (Functor f, MonoidF'' f, ProductF'' f ~ Compose f f, UnitF'' f ~ Identity) => Monad f where (>>=) m f = muF'' $ Compose $ fmap f m

[gima326]

Clojure （コンパイル時の計算処理　その４−６）

Paul Graham『On Lisp』（13章「コンパイル時の計算処理」より　P187） 数値のリストを前提に、そのリストのなかで n 番目（ゼロ始まり）に大きな数値を得る処理のマクロ版をば。 Common Lisp のコードを Clojure にトレースしただけなので、たぶんもともとのコードのバグだろう。リストの並びによって、挙動が変わる。 …まぁ、マクロで書くべきないケースとして挙げられた例なので（関数版だと１行で済む）、こんなにしつこく付き合う必要もなかったのだけれど。 //==================================== ■関数 (defn nthmost-fn [n lst] (nth (sort > lst) n)) ==================================== ■マクロ (defmacro nthmost [n lst] 　(if (and (integer? n) (< n 20)) 　　(with-gensyms (syms gi) 　　　(let [syms (map0-n (fn [x] (gensym)) n)] 　　　　`(let ~(vec (apply concat (for [s syms] `(~s (ref nil))))) 　　　　　(when-not (< (count '~lst) ~(+ n 1)) 　　　　　　~@(first (rest (gen-start lst syms))) 　　　　　　(for [~gi '~(first (gen-start lst syms))] ~(nthmost-gen gi syms true)) 　　　　　　(deref ~(last syms)))))) 　　`(nth (sort > ~lst) ~n) )) (defn gen-start [glst syms] 　(loop [g glst lst (reverse syms) rslt ()] 　　(if (empty? lst) 　　　`(~g ~rslt) 　　　(recur (rest g) (rest lst) 　　　　(cons 　　　　　((fn [s] (let [v (gensym)] 　　　　　　`(let [~v (first '~g)] ~(nthmost-gen v (reverse s))))) 　　　　　lst) 　　　　rslt)))) ) (defn nthmost-gen [var vars & long?] 　(if (empty? vars) 　　nil 　　(let [else (nthmost-gen var (rest vars) long?)] 　　　(if (and (not long?) (nil? else)) 　　　　((fn [x y] `(dosync (ref-set ~x ~y))) (first vars) var) 　　　　`(if (> ~var (deref ~(first vars))) 　　　　　~(let [pairs (group (mappend list (reverse vars) (rest (reverse vars))) 2)] 　　　　　　(let [f (map (fn [lst] `(ref-set ~(first lst) (deref ~(first (rest lst))))) pairs)] 　　　　　　　`(dosync ~@f (ref-set ~(first vars) ~var)))) 　　　　~else ))))) ==================================== ■正常 user=> (nthmost 0 (4 3 2 1)) 4 user=> (nthmost 1 (4 3 2 1)) 3 user=> (nthmost 2 (4 3 2 1)) 2 user=> (nthmost 3 (4 3 2 1)) 1 user=> (nthmost 4 (4 3 2 1)) nil ■異常 user=> (nthmost 0 (1 2 3 4)) 1 user=> (nthmost 1 (1 2 3 4)) Execution error (NullPointerException) at user/eval58778 (REPL:1). null ====================================//

[kjunichi]

SemigroupがMonoidに恋するとき - あどけない話 [はてなブックマーク]

SemigroupがMonoidに恋するとき - あどけない話

��習 Semigroup 結合則 (a・b)・c = a・(b・c) class Semigroup a where (<>) :: a -> a -> a Monoid 結合則 (a・b)・c = a・(b・c) 単位元 e・a = a・e = a class Semigroup a => Monoid a where mempty :: a mappend :: a -> a -> a mappend = (<>) 本題 Haskellerの中には、「設定はMonoidであるべき」宗派が存在する...

from kjw_junichiのはてなブックマーク http://bit.ly/2EPw1Xs

[spiritualmomentum-blog]

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[di--es---can-ic-ul-ar--es]

An author on Haskell from first principles was describing the practice of shoehorning languages into other languages (like writing mappend and mempty macros in clojure, attempting to be like Haskell) as "exporting freedom" like bringing democracy to the unwashed masses" it was great

[nolajaneshop]

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[papertownsvintage]

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[creativedesignz]

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[assemblage333]

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[wildoatsandbillygoats]

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[gima326]

Clojure （コンパイル時の計算処理　その４−５）

Paul Graham『On Lisp』（13章「コンパイル時の計算処理」より　P187） …動かぬ（記録として）。 //==================================== (defmacro nthmost [n lst] 　(if (and (integer? n) (< n 20)) 　　(with-gensyms (glst syms gi) 　　　(let [glst (ref lst)] 　　　　(let [syms (map0-n (fn [x] (gensym)) n)] 　　　　　`(let ~(vec (apply concat (for [s syms] `(~s (ref nil))))) 　　　　　　(when-not (< (count '~(deref glst)) ~(+ n 1)) 　　　　　　　~@(gen-start glst syms) 　　　　　　　(for [~gi (deref ~glst)] ~(nthmost-gen gi syms true)) 　　　　　　　~(last syms)))))) 　　`(nth (sort > ~lst) ~n) )) (defn gen-start [glst syms] 　(loop [g @glst lst (reverse syms) rslt ()] 　　(if (empty? lst) 　　　(do 　　　　(dosync (ref-set glst g)) 　　　　rslt) 　　　(recur (rest g) (rest lst) 　　　　(cons 　　　　　((fn [s] (let [v (gensym)] 　　　　　　`(let [~v (first '~g)] ~(nthmost-gen v (reverse s))))) 　　　　　lst) 　　　　rslt)))) ) (defn nthmost-gen [var vars & long?] 　(if (empty? vars) 　　nil 　　(let [else (nthmost-gen var (rest vars) long?)] 　　　(if (and (not long?) (nil? else)) 　　　　((fn [x y] `(dosync (ref-set ~x ~y))) (first vars) var) 　　　　`(if (> ~var ~(first vars)) 　　　　　~(let [pairs (group (mappend list (reverse vars) (rest (reverse vars))) 2)] 　　　　　　(let [foo (map (fn [lst] `(ref-set ~(first lst) (deref ~(first (rest lst))))) pairs)] 　　　　　　　`(dosync ~@foo (ref-set ~(first vars) ~var)))) 　　　　~else ))))) ==================================== user=> (nthmost 3 (1 2 3 4 5)) Syntax error compiling fn* at (REPL:1:1). Can't embed object in code, maybe print-dup not defined: clojure.lang.Ref@5de3ad34 ====================================//