226 lines
6.1 KiB
Racket
226 lines
6.1 KiB
Racket
#lang scribble/manual
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@(require scribble/example racket/sandbox
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(for-label typed/racket/base
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graph
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(only-in typed/graph Graph)
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(submod "../networks.rkt" typed)
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"../utils.rkt"
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"../functions.rkt"))
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@(define networks-evaluator
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(parameterize ([sandbox-output 'string]
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[sandbox-error-output 'string]
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[sandbox-memory-limit 50])
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(make-evaluator 'typed/racket #:requires '((submod "networks.rkt" typed)))))
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@(define-syntax-rule (ex . args)
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(examples #:eval networks-evaluator . args))
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@(define-syntax-rule (deftypeform . args)
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(defform #:kind "type" . args))
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@(define-syntax-rule (deftype . args)
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(defidform #:kind "type" . args))
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@title[#:tag "networks"]{dds/networks: Formal Dynamical Networks}
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@defmodule[(submod dds/networks typed)]
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This module provides definitions for and analysing network models. A network
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is a set of variables which are updated according to their corresponding update
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functions. The variables to be updated at each step are given by the mode.
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This model can generalise Boolean networks, TBANs, multivalued networks, etc.
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@section[#:tag "networks-basics"]{Basic types}
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@deftypeform[(State a)]{
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An immutable mapping (a hash table) assigning elements of type @racket[a] to
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the variables. A synonym of @racket[VariableMapping].
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}
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@deftypeform[(UpdateFunction a)]{
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An update function is a function computing a value from the given
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state. This is a synonym of the type @racket[(-> (State a) a)].
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}
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@deftypeform[(Domain a)]{
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A domain which is a subset of the type @racket[a].
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@racket[(Domain a)] is a synonym of @racket[(Listof a)].
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}
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@deftypeform[(DomainMapping a)]{
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A domain mapping is a hash table mapping variables to the lists of values in
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their domains.
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}
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@section{Common examples}
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The examples in this document often use the same definitions, which are
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therefore grouped here to avoid duplicating them.
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These are two functions calculating an @italic{AND} and an @italic{OR} between
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the variables @racket[a] and @racket[b]:
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@ex[
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(: or-func (UpdateFunction Boolean))
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(define (or-func s)
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(or (hash-ref s 'a) (hash-ref s 'b)))
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(: and-func (UpdateFunction Boolean))
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(define (and-func s)
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(and (hash-ref s 'a) (hash-ref s 'b)))
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]
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These are two functions calculating an @italic{AND} and an @italic{OR} between
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two variables @racket[a] and @racket[b] whose values are in @tt{{0,1}}:
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@ex[
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(require (only-in "utils.rkt" assert-type))
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(: or-func/01 (UpdateFunction (U Zero One)))
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(define (or-func/01 s)
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(assert-type (max (hash-ref s 'a) (hash-ref s 'b))
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(U Zero One)))
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(: and-func/01 (UpdateFunction (U Zero One)))
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(define (and-func/01 s)
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(assert-type (min (hash-ref s 'a) (hash-ref s 'b))
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(U Zero One)))
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]
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@section{Utilities}
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@defproc[(01->boolean/state [s (State (U Zero One))]) (State Boolean)]{
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Converts the values 0 and 1 in a state to @racket[#f] and
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@racket[#t] respectively.
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@ex[
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(01->boolean/state (hash 'a 0 'b 1))
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]}
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@defproc[(make-same-domains [vars (Listof Variable)]
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[domain (Domain a)])
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(DomainMapping a)]{
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Makes a hash set mapping all variables to a single domain.
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@ex[
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(make-same-domains '(a b) '(1 2))
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]}
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@defproc[(make-boolean-domains [vars (Listof Variable)])
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(DomainMapping Boolean)]{
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Makes a hash set mapping all variables to the Boolean domain.
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@ex[
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(make-boolean-domains '(a b))
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]}
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@defproc[(make-01-domains [vars (Listof Variable)])
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(DomainMapping (U Zero One))]{
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Makes a hash set mapping all variables to the Boolean domain, expressed as
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@tt{{0,1}}.
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@ex[
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(make-01-domains '(a b))
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]}
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@section{Networks}
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@defstruct*[network ([functions (VariableMapping (UpdateFunction a))]
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[domains (DomainMapping a)])]{
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A network consists of a mapping from its variables to its update variables, as
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a well as of a mapping from its variables to their domains.
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Instances of @racket[network] have the type @racket[Network].
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}
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@deftype[Network]{
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The type of the instances of @racket[Network].
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@ex[
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(network (hash 'a or-func
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'b and-func)
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(hash 'a '(#f #t)
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'b '(#f #t)))
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]}
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@defproc[(make-boolean-network [funcs (VariableMapping (UpdateFunction Boolean))])
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(Network Boolean)]{
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Builds a Boolean network from a given hash table assigning functions to
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variables by attributing Boolean domains to every variable.
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@ex[
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(make-boolean-network (hash 'a or-func 'b and-func))
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]}
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@defproc[(make-01-network [funcs (VariableMapping (UpdateFunction (U Zero One)))])
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(Network (U Zero One))]{
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Build a network from a given hash table assigning functions to variables by
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attributing the domain @tt{{0,1}} to every variable.
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@ex[
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(make-01-network (hash 'a or-func/01 'b and-func/01))
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]}
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@defproc[(update [network (Network a)] [s (State a)] [xs (Listof Variable)])
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(State a)]{
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Given a state @racket[s] updates all the variables of @racket[network] from
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@racket[xs].
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@ex[
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(update (make-boolean-network (hash 'a or-func 'b and-func))
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(hash 'a #f 'b #t)
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'(a))
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]}
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@section{Syntactic description of networks}
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@section{Inferring interaction graphs}
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This section provides inference of both unsigned and signed interaction graphs.
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Since the inference of signed interaction graphs is based on analysing the
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dynamics of the networks, it may be quite resource-consuming, especially since
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I allow any syntactic forms in the definitions of the functions.
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Note the fine difference between @emph{syntactic} interaction graphs and
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interaction graphs generated from the dynamics of the network.
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Syntactic interaction graphs are based on the whether a variable appears or not
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in the form of the function for another variable. On the other hand, the
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normal, conventional interaction graph records the fact that one variable has
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an impact on the dynamics of the other variable. Depending on the model, these
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may or may not be the same.
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@section{Dynamics of networks}
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This section contains definitions for building and analysing the dynamics
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of networks.
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@section{Tabulating functions and networks}
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@section{Constructing functions and networks}
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@section{Random functions and networks}
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@section{TBF/TBN and SBF/SBN}
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This section defines threshold Boolean functions (TBF) and networks (TBN), as
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well as sign Boolean functions (SBF) and networks (SBN).
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