290 lines
12 KiB
Racket
290 lines
12 KiB
Racket
#lang racket
|
||
|
||
;;; dds/networks
|
||
|
||
;;; This module provides some quick definitions for and analysing
|
||
;;; network models. A network is a set of variables which are updated
|
||
;;; according to their corresponding update functions. The variables
|
||
;;; to be updated at each step are given by the mode.
|
||
;;;
|
||
;;; This model can generalise Boolean networks, TBANs, multivalued
|
||
;;; networks, etc.
|
||
|
||
(require "utils.rkt" "generic.rkt" graph)
|
||
|
||
(provide
|
||
;; Structures
|
||
(struct-out dynamics)
|
||
;; Functions
|
||
(contract-out [update (-> network? state? (set/c variable? #:kind 'dont-care) state?)]
|
||
[make-state (-> (listof (cons/c symbol? any/c)) state?)]
|
||
[make-network-from-functions (-> (listof (cons/c symbol? update-function/c)) network?)]
|
||
[update-function-form->update-function (-> update-function-form? update-function/c)]
|
||
[network-form->network (-> network-form? network?)]
|
||
[make-network-from-forms (-> (listof (cons/c symbol? update-function-form?))
|
||
network?)]
|
||
[list-interactions (-> network-form? variable? (listof variable?))]
|
||
[build-interaction-graph (-> network-form? graph?)]
|
||
[build-all-states (-> domain-mapping/c (listof state?))]
|
||
[make-same-domains (-> (listof variable?) generic-set? domain-mapping/c)]
|
||
[make-boolean-domains (-> (listof variable?) (hash/c variable? (list/c #f #t)))]
|
||
[get-interaction-sign (-> network-form? domain-mapping/c variable? variable? (or/c '+ '- '0))]
|
||
[build-signed-interaction-graph (-> network-form? domain-mapping/c graph?)]
|
||
[build-boolean-signed-interaction-graph (-> network-form? graph?)]
|
||
[make-asyn (-> (listof variable?) mode?)]
|
||
[make-syn (-> (listof variable?) mode?)]
|
||
[make-dynamics-from-func (-> network? (-> (listof variable?) mode?) dynamics?)]
|
||
[make-asyn-dynamics (-> network? dynamics?)]
|
||
[make-syn-dynamics (-> network? dynamics?)]
|
||
[dds-step-one (-> dynamics? state? (set/c state?))]
|
||
[dds-step (-> dynamics? (set/c state?) (set/c state?))])
|
||
;; Predicates
|
||
(contract-out [variable? (-> any/c boolean?)]
|
||
[state? (-> any/c boolean?)]
|
||
[update-function-form? (-> any/c boolean?)]
|
||
[network-form? (-> any/c boolean?)]
|
||
[mode? (-> any/c boolean?)])
|
||
;; Contracts
|
||
(contract-out [state/c contract?]
|
||
[network/c contract?]
|
||
[update-function/c contract?]
|
||
[domain-mapping/c contract?])
|
||
;; Syntax
|
||
st nn)
|
||
|
||
|
||
;;; =================
|
||
;;; Basic definitions
|
||
;;; =================
|
||
|
||
(define variable? symbol?)
|
||
|
||
;;; A state of a network is a mapping from the variables of the
|
||
;;; network to their values.
|
||
(define state? variable-mapping?)
|
||
(define state/c (flat-named-contract 'state state?))
|
||
|
||
;;; An update function is a function computing a value from the given
|
||
;;; state.
|
||
(define update-function/c (-> state? any/c))
|
||
|
||
;;; A network is a mapping from its variables to its update functions.
|
||
(define network? variable-mapping?)
|
||
(define network/c (flat-named-contract 'network network?))
|
||
|
||
;;; Given a state s updates all the variables from xs. This
|
||
;;; corresponds to a parallel mode.
|
||
(define (update network s xs)
|
||
(for/fold ([new-s s])
|
||
([x xs])
|
||
(let ([f (hash-ref network x)])
|
||
(hash-set new-s x (f s)))))
|
||
|
||
;;; A version of make-immutable-hash restricted to creating network
|
||
;;; states (see contract).
|
||
(define (make-state mappings) (make-immutable-hash mappings))
|
||
|
||
;;; A shortcut for make-state.
|
||
(define-syntax-rule (st mappings) (make-state mappings))
|
||
|
||
;;; A version of make-immutable-hash restricted to creating networks.
|
||
(define (make-network-from-functions funcs) (make-immutable-hash funcs))
|
||
|
||
|
||
;;; =================================
|
||
;;; Syntactic description of networks
|
||
;;; =================================
|
||
|
||
;;; An update function form is any form which can appear as a body of
|
||
;;; a function and which can be evaluated with eval. For example,
|
||
;;; '(and x y (not z)) or '(+ 1 a (- b 10)).
|
||
(define update-function-form? any/c)
|
||
|
||
;;; A Boolean network form is a mapping from its variables to the
|
||
;;; forms of their update functions.
|
||
(define network-form? variable-mapping?)
|
||
|
||
;;; Build an update function from an update function form.
|
||
(define (update-function-form->update-function form)
|
||
(λ (s) (eval-with s form)))
|
||
|
||
;;; Build a network from a network form.
|
||
(define (network-form->network bnf)
|
||
(for/hash ([(x form) bnf])
|
||
(values x (update-function-form->update-function form))))
|
||
|
||
;;; Build a network from a list of pairs of forms of update functions.
|
||
(define (make-network-from-forms forms)
|
||
(network-form->network (make-immutable-hash forms)))
|
||
|
||
;;; A shortcut for make-network-from-forms.
|
||
(define-syntax-rule (nn forms) (make-network-from-forms forms))
|
||
|
||
|
||
;;; ============================
|
||
;;; Inferring interaction graphs
|
||
;;; ============================
|
||
|
||
;;; I allow any syntactic forms in definitions of Boolean functions.
|
||
;;; I can still find out which Boolean variables appear in those
|
||
;;; syntactic form, but I have no reliable syntactic means of finding
|
||
;;; out what kind of action do they have (inhibition or activation)
|
||
;;; since I cannot do Boolean minimisation (e.g., I cannot rely on not
|
||
;;; appearing before a variable, since (not (not a)) is equivalent
|
||
;;; to a). On the other hand, going through all Boolean states is
|
||
;;; quite resource-consuming and thus not always useful.
|
||
;;;
|
||
;;; In this section I provide inference of both unsigned and signed
|
||
;;; interaction graphs, but since the inference of signed interaction
|
||
;;; graphs is based on analysing the dynamics of the networks, it may
|
||
;;; be quite resource-consuming.
|
||
|
||
;;; Lists the variables of the network form appearing in the update
|
||
;;; function form for x.
|
||
(define (list-interactions nf x)
|
||
(set-intersect
|
||
(extract-symbols (hash-ref nf x))
|
||
(hash-keys nf)))
|
||
|
||
;;; Builds the graph in which the vertices are the variables of a
|
||
;;; given network, and which contains an arrow from a to b whenever a
|
||
;;; appears in (list-interactions a).
|
||
(define (build-interaction-graph n)
|
||
(transpose
|
||
(unweighted-graph/adj
|
||
(for/list ([(var _) n]) (cons var (list-interactions n var))))))
|
||
|
||
;;; A domain mapping is a hash set mapping variables to the lists of
|
||
;;; values in their domains.
|
||
(define domain-mapping/c (hash/c variable? list?))
|
||
|
||
;;; Given a hash-set mapping variables to generic sets of their
|
||
;;; possible values, constructs the list of all possible states.
|
||
(define (build-all-states vars-domains)
|
||
(let* ([var-dom-list (hash->list vars-domains)]
|
||
[vars (map car var-dom-list)]
|
||
[domains (map cdr var-dom-list)])
|
||
(for/list ([s (apply cartesian-product domains)])
|
||
(make-state (for/list ([var vars] [val s])
|
||
(cons var val))))))
|
||
|
||
;;; Makes a hash set mapping all variables to a single domain.
|
||
(define (make-same-domains vars domain)
|
||
(for/hash ([var vars]) (values var domain)))
|
||
|
||
;;; Makes a hash set mapping all variables to the Boolean domain.
|
||
(define (make-boolean-domains vars)
|
||
(make-same-domains vars '(#f #t)))
|
||
|
||
;;; Given two interacting variables of a network form and the domains
|
||
;;; of the variables, returns '+ if the interaction is monotonously
|
||
;;; increasing, '- if it is monotonously decreasing, and '0 otherwise.
|
||
;;;
|
||
;;; This function does not check whether the two variables indeed
|
||
;;; interact. Its behaviour is undefined if the variables do not
|
||
;;; interact.
|
||
;;;
|
||
;;; /!\ This function iterates through almost all of the states of the
|
||
;;; network, so its performance decreases very quickly with network
|
||
;;; size.
|
||
(define (get-interaction-sign network-form doms x y)
|
||
(let* ([dom-x (hash-ref doms x)]
|
||
[dom-y (hash-ref doms y)]
|
||
[network (network-form->network network-form)]
|
||
;; Replace the domain of x by a dummy singleton.
|
||
[doms-no-x (hash-set doms x '(#f))]
|
||
;; Build all the states, but as if x were not there: since I
|
||
;; replace its domain by a singleton, all states will contain
|
||
;; the same value for x.
|
||
[states-no-x (build-all-states doms-no-x)]
|
||
;; Go through all states, then through all ordered pairs of
|
||
;; values of x, generate pairs of states (s1, s2) such that x
|
||
;; has a smaller value in s1, and check that updating y in s1
|
||
;; yields a smaller value than updating y in s2. I rely on
|
||
;; the fact that the domains are ordered.
|
||
[x-y-interactions (for*/list ([s states-no-x]
|
||
[x1 dom-x] ; ordered pairs of values of x
|
||
[x2 (cdr (member x1 dom-x))])
|
||
(let* ([s1 (hash-set s x x1)] ; s1(x) < s2(x)
|
||
[s2 (hash-set s x x2)]
|
||
[y1 ((hash-ref network y) s1)]
|
||
[y2 ((hash-ref network y) s2)])
|
||
;; y1 <= y2?
|
||
(<= (index-of dom-y y1) (index-of dom-y y2))))])
|
||
(cond
|
||
;; If, in all interactions, y1 <= y2, then we have an
|
||
;; increasing/promoting interaction between x and y.
|
||
[(andmap (λ (x) (eq? x #t)) x-y-interactions) '+]
|
||
;; If, in all interactions, y1 > y2, then we have an
|
||
;; decreasing/inhibiting interaction between x and y.
|
||
[(andmap (λ (x) (eq? x #f)) x-y-interactions) '-]
|
||
;; Otherwise the interaction is neither increasing nor
|
||
;; decreasing.
|
||
[else '0])))
|
||
|
||
;;; Constructs a signed interaction graph of a given network form,
|
||
;;; given the ordered domains of its variables. The order on the
|
||
;;; domains determines the signs which will appear on the interaction
|
||
;;; graph.
|
||
;;;
|
||
;;; /!\ This function iterates through almost all states of the
|
||
;;; network for every arrow in the unsigned interaction graph, so its
|
||
;;; performance decreases very quickly with the size of the network.
|
||
(define (build-signed-interaction-graph network-form doms)
|
||
(let ([ig (build-interaction-graph network-form)])
|
||
(weighted-graph/directed
|
||
(for/list ([e (in-edges ig)])
|
||
(match-let ([(list x y) e])
|
||
(list (match (get-interaction-sign network-form doms x y)
|
||
['+ 1] ['- -1] ['0 0])
|
||
x y))))))
|
||
|
||
;;; Calls build-signed-interaction-graph with the Boolean domain for
|
||
;;; all variable.
|
||
;;;
|
||
;;; /!\ The same performance warning applies as for
|
||
;;; build-signed-interaction-graph.
|
||
(define (build-boolean-signed-interaction-graph network-form)
|
||
(build-signed-interaction-graph network-form
|
||
(make-boolean-domains (hash-keys network-form))))
|
||
|
||
|
||
;;; ====================
|
||
;;; Dynamics of networks
|
||
;;; ====================
|
||
|
||
;;; This section contains definitions for building and analysing the
|
||
;;; dynamics of networks.
|
||
|
||
;;; A mode is a set of sets of variables.
|
||
(define mode? (set/c (set/c variable?)))
|
||
|
||
;;; A network dynamics is a network plus a mode.
|
||
(struct dynamics (network mode)
|
||
#:methods gen:dds
|
||
[(define/match (dds-step-one dyn s)
|
||
[((dynamics network mode) s)
|
||
(for/set ([m mode]) (update network s m))])])
|
||
|
||
;;; Given a list of variables, builds the asynchronous mode (a set of
|
||
;;; singletons).
|
||
(define (make-asyn vars)
|
||
(for/set ([v vars]) (set v)))
|
||
|
||
;;; Given a list of variables, builds the synchronous mode (a set
|
||
;;; containing the set of variables).
|
||
(define (make-syn vars) (set (list->set vars)))
|
||
|
||
;;; Given a network, applies a function for building a mode to its
|
||
;;; variables and returns the corresponding network dynamics.
|
||
(define (make-dynamics-from-func network mode-func)
|
||
(dynamics network (mode-func (hash-keys network))))
|
||
|
||
;;; Creates the asynchronous dynamics for a given network.
|
||
(define (make-asyn-dynamics network)
|
||
(make-dynamics-from-func network make-asyn))
|
||
|
||
;;; Creates the synchronous dynamics for a given network.
|
||
(define (make-syn-dynamics network)
|
||
(make-dynamics-from-func network make-syn))
|