Finding identities for the Noki 5110

Harold Thimbleby - http://www.uclic.ucl.ac.uk/harold

This file is the background for establishing some matrix identities for the Nokia 5110 example used in the papers.

•Define the Nokia 5110 FSM

First we define the Nokia using a tree, and symbolic names taken from the device itself. This tree was then carefully checked against the Nokia itself.

readable[nokia] ^= menu["standby", {menu["phone book", {"search", "service nos", "add entry", "erase", "edit", "send entry", menu["options", {"type of view", "memory status"}], "speed dials"}], menu["messages", {"inbox", "outbox", "write messages", menu["message settings", {menu["set 1", {"message centre number", "messages sent as", "message validity"}], menu["common", {"delivery reports", "reply via same centre"}]}], "info service", "voice mailbox number"}], menu["call register", {"missed calls", "received calls", "dialled numbers", "erase recent calls", menu["show call duration", {"last call duration", "all calls' duration", "received calls' duration", "dialled calls' duration", "clear timers"}], menu["show call costs", {"last call cost", "all calls' cost", "clear counters"}], menu["call costs settings", {"call costs' limit", "show costs in"}]}], menu["settings", {menu["call settings", {"automatic redial", "speed dialling", "call waiting options", "own number sending", "automatic answer"}], menu["phone settings", {menu["language", {"Automatic", "Engish", "Deutsch", "Français", "Nederlands", "Italiano", "Dansk", "Svenska", "Norsk", "Suomi", "Español", "Português", "<Russian>", "Eesti", "Latviesu", "Lietuviu", "<Arabic>", "<Hebrew>"}], "cell info display", "welcome note", "network selection", "lights"}], menu["security settings", {"PIN code request", "fixed dialling", "closed user group", "phone security", "change access codes"}], "restore factory settings"}], menu["call divert", {"divert all calls without ringing", "divert when busy", "divert when not answered", "divert when phone off or no coverage", "cancel all diverts"}], menu["games", {"memory", "snake", "logic"}], "calculator", menu["clock", {"alarm clock", "clock settings"}], menu["tones", {"incoming call alert", "ringing tone", "ringing volume", "message alert tone", "keypad tones", "warning and game tones", "vibrating alert"}]}] ;

•Convert tree to matrix

menuname[menu[name_, _]] := name ;

menuname[name_] := name ;

ToStateNo[device_, s_List] := Map[ToStateNo[device, #] &, s]

ToStateNo[device_, tran[a_, b_, c_]] := tran[ToStateNo[device, a], b, ToStateNo[device, c]]

ToStateNo[device_, s_] := Position[vocabulary[device], s] [[ 1, 1 ]]

FromStateNo[device_, n_] := Part[vocabulary[device], n]

Normalise function inserts self-transitions as needed.

normalise[bm_] := Module[{m = bm}, (* if any row is zero, put an identity transition in it *) Do[If[Plus @@ m[[i]] == 0, m[[i, i]] = 1], {i, Length[m]}] ; Return[m] ]

initialise[device_, fullName_] := Module[{p, b, bm}, vocabulary[device] ^= Union @ Flatten[symbolicTransitions[device] /. tran[a_, _, b_] -> {a, b}] ; numericTransitions[device] ^= ToStateNo[device, symbolicTransitions[device]] ; numericGoals[device] ^= ToStateNo[device, #] & /@ symbolicGoals[device] ; numberOfStates[device] ^= Max @ Flatten[numericTransitions[device] /. tran[a_, _, b_] -> {a, b}] ; buttons[device] ^= Union[symbolicTransitions[device] /. tran[_, but_, _] -> but] ; startStateNo[device] ^= ToStateNo[device, startState[device]] ; For[b = 1, b <= Length @ buttons[device], b ++, bm = Table[0, {numberOfStates[device]}, {numberOfStates[device]}] ; Scan[(bm [[ # [[ 1 ]], # [[ 3 ]] ]] = 1) &, Cases[numericTransitions[device], tran[_, buttons[device][[b]], _]]] ; button[device, b] = normalise[bm] ; ] ; name[device] ^= fullName ; ] ;

General :: spell : Possible spelling error: new symbol name \" button \" is similar to existing symbols {Button, buttons} .

build[nokia] := Module[{auxconvert, transition, p = {}, goals = {}}, transition[from_, button_, to_] := AppendTo[p, tran[menuname[from], button, menuname[to]]] ; auxconvert[menu[name_, items_]] := Module[{i}, transition[name, "Select", items [[ 1 ]]] ; For[i = 1, i <= Length @ items, i ++, transition[items [[ i ]], "Down", items [[ If[i == Length @ items, 1, i + 1] ]]] ; transition[items [[ i ]], "Up", items [[ If[i == 1, Length @ items, i - 1] ]]] ; transition[items [[ i ]], "C", name] ; If[Head @ items [[ i ]] === menu, auxconvert[items [[ i ]]], AppendTo[goals, operation @ menuname[items [[ i ]]]] ; transition[items [[ i ]], "Doit", Last @ goals] ; transition[Last @ goals, "Doit", items [[ i ]]] ; ] ] ] ; auxconvert[readable[nokia]] ; symbolicGoals[nokia] ^= goals ; startState[nokia] ^= "standby" ; symbolicTransitions[nokia] ^= p ; initialise[nokia, "Nokia 5110 navigable menu"] ; ]

build[nokia] ;

Utilities for creating state vectors from state names.

vec[s_] := Table[If[i == ToStateNo[nokia, s], 1, 0], {i, numberOfStates[nokia]}]

devec[s_] := FromStateNo[nokia, Position[s, 1][[1, 1]]]

•Matrix definitions

These are the button names in the device.

buttons[nokia]

{C, Doit, Down, Select, Up}

Number of states.

numberOfStates[nokia]

188

Define useful matrix names. The Nokia's actual Navi button is a combination of what we've called Doit and Select.

i = IdentityMatrix[numberOfStates[nokia]] ;

u = button[nokia, 5] ; c = button[nokia, 1] ; s = button[nokia, 4] ; d = button[nokia, 3] ; doit = button[nokia, 2] ; navi = Sign[doit + s] ;

•Find identities and interesting facts

Module[{i, j, k, v, name}, v = {n, c, u, d} ; name[1] = "navi" ; name[2] = "c" ; name[3] = "up" ; name[4] = "down" ; For[i = 1, i <= 4, i ++, For[j = 1, j <= 4, j ++, For[k = 1, k <= 4, k ++, If[v[[i]] . v[[j]] == v[[k]], Print[name[i], ".", name[j], "==", name[k]] ] ] ] ] ]

up . c == c

down . c == c

Module[{zz}, For[zz = 0, zz < 20, zz ++, If[MatrixPower[c, zz] == MatrixPower[c, zz + 1], Print["c^", zz, "==c^", zz + 1] ; Break[]]] ]

c^ 4 ==c^ 5

MatrixPower[c, 4] == MatrixPower[c, 5]

True

c . c . c . c . c == c . c . c . c

True

r = c . c . c . c ;

r . c == r

True

s . s . c == s

False

n . d . d . d . d . d . c == n . c

True

r . s . s . c == r . s

True

r . s . c == r

True

r . s . s . s . c == r . s . s

False

rr = c . c . c ; rr . s . s . c == rr . s

False

r . navi . navi . c == r . navi

False

doit . doit == i

True

Inverse[doit] == doit

True

Det[s]

0

u . d == i

True

d . u == i

True

u . c == c

True

d . c == c

True

doit . doit == doit

False

devec[vec["standby"] . s . s . c]

phone book

s . u . s . c == s . u

False

s . u == s . u . s . c

False

s . c == i

False

c == u . c

True

u . d == i

True

d . u == i

True

s . s . s . s . s == s . s . s . s . s

True

devec[vec["standby"] . s . u]

tones

Module[{i, sc = s . s . c}, For[i = 1, i <= numberOfStates[nokia], i ++, theState = Table[If[j == i, 1, 0], {j, numberOfStates[nokia]}] ; If[theState . sc != theState . s, Print["s.s.c not identity from ", devec[theState]]] ] ]

s.s.c not identity from add entry

s.s.c not identity from alarm clock

s.s.c not identity from all calls' cost

s.s.c not identity from all calls' duration

s.s.c not identity from <Arabic>

s.s.c not identity from Automatic

s.s.c not identity from automatic answer

s.s.c not identity from automatic redial

s.s.c not identity from calculator

s.s.c not identity from call costs' limit

s.s.c not identity from call costs settings

s.s.c not identity from call divert

s.s.c not identity from call register

s.s.c not identity from call settings

s.s.c not identity from call waiting options

s.s.c not identity from cancel all diverts

s.s.c not identity from cell info display

s.s.c not identity from change access codes

s.s.c not identity from clear counters

s.s.c not identity from clear timers

s.s.c not identity from clock

s.s.c not identity from clock settings

s.s.c not identity from closed user group

s.s.c not identity from common

s.s.c not identity from Dansk

s.s.c not identity from delivery reports

s.s.c not identity from Deutsch

s.s.c not identity from dialled calls' duration

s.s.c not identity from dialled numbers

s.s.c not identity from divert all calls without ringing

s.s.c not identity from divert when busy

s.s.c not identity from divert when not answered

s.s.c not identity from divert when phone off or no coverage

s.s.c not identity from edit

s.s.c not identity from Eesti

s.s.c not identity from Engish

s.s.c not identity from erase

s.s.c not identity from erase recent calls

s.s.c not identity from Español

s.s.c not identity from fixed dialling

s.s.c not identity from Français

s.s.c not identity from games

s.s.c not identity from <Hebrew>

s.s.c not identity from inbox

s.s.c not identity from incoming call alert

s.s.c not identity from info service

s.s.c not identity from Italiano

s.s.c not identity from keypad tones

s.s.c not identity from language

s.s.c not identity from last call cost

s.s.c not identity from last call duration

s.s.c not identity from Latviesu

s.s.c not identity from Lietuviu

s.s.c not identity from lights

s.s.c not identity from logic

s.s.c not identity from memory

s.s.c not identity from memory status

s.s.c not identity from message alert tone

s.s.c not identity from message centre number

s.s.c not identity from messages

s.s.c not identity from messages sent as

s.s.c not identity from message validity

s.s.c not identity from missed calls

s.s.c not identity from Nederlands

s.s.c not identity from network selection

s.s.c not identity from Norsk

s.s.c not identity from options

s.s.c not identity from outbox

s.s.c not identity from own number sending

s.s.c not identity from phone book

s.s.c not identity from phone security

s.s.c not identity from PIN code request

s.s.c not identity from Português

s.s.c not identity from received calls

s.s.c not identity from received calls' duration

s.s.c not identity from reply via same centre

s.s.c not identity from restore factory settings

s.s.c not identity from ringing tone

s.s.c not identity from ringing volume

s.s.c not identity from <Russian>

s.s.c not identity from search

s.s.c not identity from security settings

s.s.c not identity from send entry

s.s.c not identity from service nos

s.s.c not identity from set 1

s.s.c not identity from show call costs

s.s.c not identity from show call duration

s.s.c not identity from show costs in

s.s.c not identity from snake

s.s.c not identity from speed dialling

s.s.c not identity from speed dials

s.s.c not identity from Suomi

s.s.c not identity from Svenska

s.s.c not identity from tones

s.s.c not identity from type of view

s.s.c not identity from vibrating alert

s.s.c not identity from voice mailbox number

s.s.c not identity from warning and game tones

s.s.c not identity from welcome note

s.s.c not identity from write messages

Converted by Mathematica  (May 2, 2003)