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transcript
CDMTCS
Research
Report
Series
Construmath South 2012
Applications of Non-Classical
Logic
Proceedings of the Workshop
at Westport
Maarten McKubre-Jordens,
Raazesh Sainudiin (Eds.)
University of Canterbury, NZ
CDMTCS-420
19 April 2012
Centre for Discrete Mathematics and
Theoretical Computer Science
Construmath South 2012
Applications of Non-Classical Logic
Proceedings of the Workshop at Westport
26–28 January 2012
Maarten McKubre-Jordens & Raazesh Sainudiin (Eds.)
This meeting was supported by Marie Curie grant PIRSES-GA-2008-230822 from the European Union,
with counterpart funding from the Ministry of Research, Science and Technology of New Zealand, for
the project Construmath. Additional financial and infrastructure support was generously provided by
the University of Canterbury.
Co-Organizers of Construmath South 2012
Maarten McKubre-Jordens & Raazesh Sainudiin
Proceedings Compiled by
Maarten McKubre-Jordens & Raazesh Sainudiin
Administrative Support
Penelope Goode
Technical Support
Steve Gourdie
Copyright c� 2012
The copyright of any material in this booklet (including without limitation the text, computer code, artwork,
photographs, and images) is owned by the respective authors and/or the editors. You may request permission to
use the copyright materials in this booklet by contacting the respective authors and/or the editors.
Foreword
This meeting was held on 26–28 January, 2012, at the Westport Field Station of the Uni-versity of Canterbury, on the South Island of New Zealand. It was aimed at fostering theexchange of ideas between various disciplines, emphasizing links between mathematics,computer science, philosophy and statistics.
Tutorials and talks on various aspects of non-classical logics were run, with a view tousing these aspects in other areas of research. Talks generally ran for about an hour,with a generous period of discussion and questions following. Thanks to the wide-rangingnature of the research interests of the group, this format proved to be very conducive togenerating cross-disciplinary ideas and constructive critique. Participants also had theopportunity to explore the seal colony near Westport and walked from the colony to thelighthouse.
Participants ofConstruMath South 2012: Applications of Non-Classical Logic
and Their Double Pendulum Release Signatures
The participants from left to right in the first image are: Ruriko Yoshida, Cris Calude,Raazesh Sainudiin, Maarten McKubre-Jordens, Elena Calude, Nicholas Duncan, JamesDent, Ty Baen, Zach Weber, Ed Mares and Bruce Burdick. Each participant released amechatronically measurable double pendulum. The remaining eleven images (from leftto right and row by row) show the positions of each arm of the double pendulum throughtime upon release by each participant in the above list order.
Contents
Parameter Estimation in Epistemologically Valid Machine Interval Exper-imentsRaazesh Sainudiin (with Alex Danis and Warwick Tucker)University of Canterbury 1
What’s the Deal with Relevance?An Introduction to Relevant LogicEdwin MaresVictoria University of Wellington 6
Paraconsistent MathematicsZach WeberUniversity of Otago 17
Constructive Methods in MathematicsMaarten McKubre-JordensUniversity of Canterbury 19
A Constructive Approach to the Complexity of Mathematical ProblemsCristian Calude and Elena CaludeUniversity of Auckland / Massey University 21
Abstract Stone Duality - A Logic for TopologyNicholas DuncanUniversity of Canterbury 27
Epistemolog
ically
valid
experim
ent
Outline
Statemen
t(H
ume,
1777
)
“Butwheredi↵eren
te↵
ects
have
beenfoundto
follow
from
causes,
whichareto
appearance
exactly
similar,allthesevariouse↵
ects
must
occurto
themindin
transferringthepast
tothefuture,anden
terinto
ourco
nsideration,when
we
determinetheprobabilityoftheeven
t.”[1]
Epistem
ologically
valid
experim
ent
Datafrom
ado
uble
pendu
lum
Mod
elwithparameter
space⇥⇥
(Rk,k
<1
Actionspaceof
point
estimationA
=⇥⇥
Solution
MLEisCSPwithepistemolog
ically
valid
action
spaceI⇥⇥
⇤
Set-valuedintegrators,(T
,F,?)-basedestimators
Blabb
eron
Ong
oing
Work
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Epistem
ologically
valid
experim
ent
Definition
Epistemologyisthestud
yof
thenature
andgrou
ndsof
know
ledg
eespecially
withreferenceto
itslim
itsandvalidity.
Definition
Astatistica
lex
perim
entE P
isthetriple
(X,F
X,P
)consisting
ofasamplespaceX
ofallpossibleem
pirically
observable
realizations
ofanaturalph
enom
enon
�,asigm
a-algebraFXon
X,anda
family
ofprob
ability
measuresP
={P
✓,✓
2⇥},
where
each
P
✓is
aprob
ability
measure
onthemeasurablespace(X
,FX).
The
✓is
anindexbelon
ging
totheindexset⇥.The
indexmap
d(✓)=
P
✓:⇥!P
associates
every✓2⇥
withP
✓2P,in
anarbitrarymannerthat
even
allowsfortheindexmap
dto
bethe
identity
map
with⇥
=P.
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Param
eter
Estim
ationin
Epistem
ologically
valid
Machine
Interval
Experim
ents
Raazesh
Sainu
diin
Departm
entof
MathematicsandStatistics,
Universityof
Canterbury,Private
Bag
4800
,
Christchu
rch,
New
Zealand
jointworkwith
Alexand
erDanisandWarwickTucker
Departm
entof
Mathematics,
Upp
sala
University,Box
480Upp
sala,Sweden
Con
struMathSou
th2012,Westport,New
Zealand
Janu
ary26-28,
2011
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
The
DualisticContext
(“The
BiggerPicture”)
Tradition:Mod
ernEurop
eanEmpiricism
Internal
Consisten
cy:AristoteleanLogic
Universe
ofHyp
otheses:Pop
per’sFalsifia
bility
Empirical
Resolution:Mechatron
ically
MeasuredDataD
o
Model
:DeterministicODE-IVPs
Param
eter
Spac
e:fin
itedimension
alparameter
space
Approac
h:Statistical
DecisionTheory(set-valuedapproach)
Enginee
ringConstraints:Resou
rce-lim
ited
Info.Proc.
Objective:Epistem
ologically
Valid
Param
eter
Estim
ation
Solution:Com
puter-aidedProofs&
Interval
Analysis
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
1
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
ODEModel:Dam
ped
SinglePendu
lum
Trajectories
Kinetic
energy
ofthearm
consistsof
only
rotation
alkinetic
energy
T=
1 2I'
2,
Thepoten
tial
energy
ofthepen
dulum
iscalculatedby
consideringthege
ometric
positionof
thecentreof
massab
ovetheeq
uilibrium
position,V
=ml cg(1
�co
s')
Lag
rangian
ofthesinglepen
dulum:
L=
T�
V(1)
=1 2I'
2�
ml cg(1
�co
s').
(2)
TheEuler-Lag
range
form
:d
dt
✓@L
@'
◆�
@L
@'
=0,
(3)
givingtheeq
uationof
motionforthesinglepen
dulum
system
,
I'+
mgl csin'=
0(4)
or,
'=
�⇠2sin'
(5)
where⇠=
qml cg
I.
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
ODEModel:Dam
ped
SinglePendu
lum
Trajectories
Tonumerically
integratetheeq
uationof
motion,weco
nvert
(5)into
asystem
offirst
order
equationsby
letting'=
!an
ddi↵eren
tiating,
!=
'.Thuswehavethesystem
offirstorder
equations,
' !
�=
!
�⇠2sin'
�(6)
Frictionmay
bead
ded
tothesystem
byad
dingan
other
term
to(4).
Thefriction
inthis
case
ismodeled
asprop
ortion
alto
thean
gularvelocity,thetorqueproducedis
givenby,
⌧ b=
µ'
giving(4)as,
I'+
µ'+
mgl csin'=
0(7)
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Phenomenon:Dam
ped
DoublePendu
lum
Trajectories
A:DPSchem
atic
B:StreamingDPdata
C:Enclosuresof
twoinitially
closetrajectories
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
ODEModel:Dam
ped
SinglePendu
lum
Trajectories
ϕ
cl
Mod
elthearm
asadistribu
tedmasswithcentre
ofmass
locatedat
adistance
l
cfrom
thepivot,
mom
entof
inertiathearm
isIanditsmassism.
theacceleration
dueto
gravityisg⇡
9.81ms�
2,
'istheangu
larposition,
'istheangu
larvelocity,
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
2
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
ODEModel:Passive
DoublePendu
lum
Trajectories
ϕ
ω
l 3
l 1
l 2
thecentre
ofmassof
theinnerarm
isdistance
l
1
thedistance
betweenpivots
oftheinnerarm
isl
3
thecentre
ofmassof
theou
terarm
isdistance
l
2
toparm
hasmassm
1andmom
entof
inertiaof
I 1similarlyfortheou
terarm
they
arem
2andI 2
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
ODEModel:Passive
DoublePendu
lum
Trajectories
After
somework...
Derivationof
equationsviatheEuler-Lag
range
equationsof
motionfollo
wsin
anman
ner
analog
ousto
that
presen
tedforthepassive
singlepen
dulum...
Wecanthis
param
etricfamily
ofvector
fieldsforou
rstatisticalexperim
entwithdata
{x(t
i)} t
i2Tas
follo
ws: ⇥⇥3
✓,
x(t)=
Zf(x
⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤;
✓)
Here,
x iis
asample
timean
dy i
=('
1,'
2)givesthean
gularpositionsof
each
arm
at
timex i
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Num
erical
Errorsdu
eto
LNR
Overflow
Error
(12!
=479001600,13!6=
1932053504)
Rou
ndingError
(actualresult-compu
tedresult)
Cancellation
Error
(accum
ulated
roun
d-o↵
error)
TruncationError
(from
doingon
lyfin
itelymanyop
erations)
Con
versionError
(decim
alto
finitesetof
binary
numbers)
Heuristic
punctual
localop
timizationisno
trigorous!
-0.2
-0.1
00.1
0.2
0
0.05
0.1
0.15
0.2
-1
01
23
45
6
-4
-3
-2
-1012
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Lim
itson
EmpiricalResolution
Inorderto
makethegrou
ndof
know
ledg
eab
out�
withLER
epistemologically
soun
d,theem
pirically
indiscerniblesets
mustbe
allowed
toenterthestatisticalexperim
entas
data.
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
3
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Epistem
ologically
valid
experim
ent
Definition
Epistemologyisthestud
yof
thenature
andgrou
ndsof
know
ledg
eespecially
withreferenceto
itslim
itsandvalidity.
Epistem
ological
Con
sideration
s:
Lim
itson
Num
erical
Resolution
Lim
itson
EmpiricalResolution
Lim
itson
Lingu
isticResolution(futurework!)
Lim
itson
...
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Lim
itson
Num
erical
resolution
(LNR)
Com
puters
supp
ortafin
itesetof
fixed
leng
thflo
ating-point
numbersof
theform
x=
±m
·be=
±0.m
1m
2···m
p·b
e
where,m
isthesign
edmantissaof
precisionp,bisthebase
(usually
2)ande,bou
nded
betweeneande,istheexpon
ent.
Whenb=
2,thedigits
ofthemantissam
1=
1and
m
i2{0,1},8i,1
<i
p[3].
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Epistem
ologically
Valid
Experim
ent
Wewantan
epistemologically
valid
experim
entthat
accoun
tsfortheph
ysical
limitson
empiricalresolution
(“show
whatyoucanactually
see”)
numerical
resolution
(“compu
tewhatyouactually
can”
)
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Solution ActionSpace
Aof
theclassicalestimationprob
lem
ismerely
theparameter
space⇥⇥
Epistem
ologically
valid
action
spaceAA
isa
machine-representable
Hausdor↵-extension
theParam
eter
Space
⇥⇥EN
�!I⇥⇥
⇤:=
I⇥⇥[;
I⇥⇥isthesetof
allcompact
boxes
in⇥⇥.
;hasto
beaddedto
ourepistemologically
valid
AAidentifia
bilityof
theextend
edexperim
entindexedby
I⇥⇥in
term
sof
symmetricsetdi↵erence
followsfrom
identifia
bilityof
theoriginal
experim
entindexedby
⇥⇥andinclusionmon
oton
yof
theindexmap
(likelihoo
dor
cond
itionalprob
ability
ofdata
givenparameter)
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
4
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Data
lossless
compression
(minim
alsu�cientstatistic)
ofthe
trajectory
themeasurablediscrete
statetransition
salon
gwiththe
transition
time
timestam
ps,arm-positionstates
areintegers
representing
intervals
sample
number,en
coder1,
enco
der2
2600
1042
-10
1578
-1-1
6752
-2-1
. . . 1222
243-2
2048
012
2933
0-1
2048
0
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Lim
itson
EmpiricalResolution
Wordsof
Vladik
Kreinov
ich(tworecentLos
Alamos
Rep
orts
onMeauremen
tErrors)
Inman
ysuch
situations,
theon
lythingwekn
owis
theupper
bou
nddon
the
measuremen
terror.
Thus,
afterwege
tthemeasuredva
lueX,theon
lyinform
ation
that
wehav
eab
outtheactual
(unkn
own)va
luexis
that
xbelon
gsto
theinterval
[X�
d,X
+d].
Here,
wehav
etw
och
oices:
(a)
wecanaskan
expertan
dco
meupwithasubjectiveprob
ability
distribution
onthis
interval.How
ever,thereis
nogu
aran
teethat
this
distribution
iscorrect,
andthat
thereco
mmen
dationsbased
onthis
subjectiveexpertdistribution
are
valid
fortheactual
(unkn
own)distribution
ofthemeasuremen
terror.
(b)
Another
approa
chis
touse
robust
statistics
–aspecialtypecalledinterval
computation
s.Wedonot
know
theexactdistribution
,weon
lykn
owthat
this
distribution
islocatedon
theinterval.So,
wewan
tto
mak
eco
nclusion
swhich
arevalid
nomatterwhat
this
distribution
is.
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Thanks Manythanks
to:
Piers
Law
renceforcompletingtheph
ysical
doub
lependu
lum
inCivilEng
gDept.’sLathe
(AlanNicho
lson
),Richard
Brown
coordinatedElectronicdesign
andMikeStuartdidit
UCDMSforsupp
orting
thedo
uble
pendu
lum
project
(especially)
BobBroughton(logistics,partsorder,etc)
DavidWall($andkindwords)
Dou
glas
Bridg
eset
al’sCon
struMathGrant
for
Upp
salaCAPA-CanterburyU
CDMSair-tra�
c
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
Epistemolog
ically
valid
experim
ent
Lim
itson
Numerical
resolution
(LNR)
Lim
itson
Empirical
Resolution
Lim
itson
Empirical
Resolution
Bibliography
DHume.
Anen
quiryco
ncerninghuman
understan
ding:
Section
VI-OFPROBABILIT
Y.
InCW
Elio
t,ed
itor,TheHarvard
classics:English
philosophersoftheseventeen
thandeighteen
thcenturies,
1910ed
ition,vo
lume37
.TheCollierPress,17
77.
ANeu
maier.
Intervalmethodsforsystem
sofeq
uations.
Cam
bridge
university
press,
1990
.
IEEETaskP75
4.
ANSI/IEEE754-1985,StandardforBinaryFloating-P
ointArithmetic.
IEEE,New
York,
1985
.
AN
Shiryaev.
Probability.
Springe
r-Verlag,
1989
.
Piers
Law
rence,MichaelStuart,
RichardBrown,WarwickTucker
andRaa
zesh
Sainudiin
.
Amechatronicallymeasurable
double
pen
dulum
formach
ineintervalexperim
ents.
IndianStatistical
Institute
Technical
Rep
ort,
isiban
g/ms/20
10/1
1,Octob
er25
,20
10
A.Dan
is,R.Sainudiin
&W
.Tucker
MLEof
amachineinterval
experim
ent
5
What’s the Deal with Relevance?
An Introduction to Relevant Logic
Edwin MaresVictoria University of Wellington
Relevant Logics are logical systems that reject the so-called paradoxes of material andstrict implication. They also brand certain inferences valid in classical or intuitionistlogic as fallacies of relevance. Consider, for example, the inference
A
) B ! B.
This inference is valid in classical and intuitionist logic because B ! B is provable inany context (read ‘context’ as possible world for classical logic, evidential situation forintuitionist logic). The proof of B ! B need have nothing to do with A, but this doesnot matter according to classical or intuitionist logic. The premise in an inference thatis considered to be deductively valid in relevant logic, on the other hand, has really to beused in the proof of the conclusion. It is this notion of real use that is the key concept ofrelevant logic.
The notion of real use can be understood in various ways. In terms of a Gentzen-stylesequent calculus, for example, it can be understood at least in part in terms of therejection of weakening on the left-hand side of the turnstile. In terms of Fitch-Lemmonstyle natural deduction system, it can be understood in terms of labels that are employedto keep track of the use of hypotheses. For example, the following is a relevant deduction:
1.2.3.4.5.6.7.8.9.10.
�������������������
A ! (B ! C){1}���������������
A ! B{2}�����������
A{3}A ! (B ! C){1}B ! C{1,3}A ! B{2}B{2,3}C{1,2,3}
A ! C{1,2}(A ! B) ! (A ! C){1}
hyp.hyp.hyp.1, reit.3, 4, ! E2, reit.3, 6, ! E5, 7, ! E3� 8, ! I2� 9, ! I
11. (A ! (B ! C)) ! ((A ! B) ! (A ! C); 1� 10, ! I
The treatment of the subscripted labels can be tricky, especially in the rules concerningconjunction (see the slides for the talk), but the basic idea is quite simple. When ahypothesis is introduced, it is given a new number. The hypothesis has to be used in theproof of a conclusion for it to be discharged, and this use is evident from the appearanceof its number in the subscript of the conclusion. Similarly, if we leave a hypothesisundischarged – as a premise in an argument – its number must appear in the subscriptof the conclusion in order for the deduction to be considered relevantly valid.
I interpret the subscripts in terms of the theory of situations. A situation is a partialrepresentation of a universe. A situation need not contain all the information about auniverse in it. For example, as I write this, I have no idea what the weather is in New
6
York; that information is not available to me and so it is not in this situation (I couldbe considered to be also in various other situations, some of which include the currentweather in New York, but I will leave that for now). A step in a relevant proof, say,A{1} says that a particular situation, s
1
, contains the information that A. In the proofabove, we have the hypotheses that the information that A ! (B ! C) is contained ins1
, A ! B is contained in s2
, and A is contained in s3
. We also are assuming that thesethree situations obtain in the same world. On the basis of this, we infer, for example,that B ! C is contained in a situation (labelled in the proof as {1, 3}) in the same world.
The logic described in the foregoing paragraphs is the logic R of relevant implication. Notall relevant logicians accept R as representing the last word on relevance. Many acceptweaker logics. One reason for doing so is that they want a logic to act as a basis for anaive theory of truth or a naive set theory. Here I will only treat theories of truth, sincethe chapter by Zach Weber treats naive set theory. I don’t need to go through all theissues concerning the theory of truth, but I will present the key problem, that is, theCurry paradox. Consider the Curry sentence,
(C) If this sentence is true, then the moon is made of green cheese.
Let p mean ‘the moon is made of green cheese’. We know that, by virtue of the meaningof C that is is logically equivalent to C ! p. So, the following proof is valid in R:
1.2.3.4.
��������
C{1}C $ (C ! p);C ! p{1}p{1}
hyp.stipulation1, 2, $ E1, 3, ! E
5. C ! p; 1� 4, ! I6. C $ (C ! p); stipulation7. C; 5, 6, $ E8. p; 5, 7, ! E
In order to bar this derivation, some relevant logicians to replace the sets in the sub-scripted labels with multisets. In a multiset, the same number can occur twice. Theproof cannot be completed now:
1.2.3.4.
��������
C[1]
C $ (C ! p)[]
C ! p[1]
p[1,1]
hyp.stipulation1, 2, $ E1, 3, ! E
5. C ! p[1]
1� 4, ! I6. ????
We only have one hypothesis to discharge, but it was used twice to prove C ! p. Thuswe have part of a means of banning the derivation of Curry’s paradox. But the questionis: how can we interpret logics with this restriction?
Further Reading
The natural deduction system for the relevant logic R is set out in Anderson and Belnap,Entailment, volume I (Princeton: Princeton University Press, 1975). Natural deductionsystems for alternative relevant logics are set out in Ross Brady (ed.), Relevant Logic andits Rivals, volume 2 (Farnham, Surrey: Ashgate, 2003). Philosophical interpretations ofrelevant logics are found in Stephen Read, Relevant Logic: A Philosophical Interpretation
7
of Inference (Oxford: Blackwell, 1989) and Edwin Mares, Relevant Logic: A PhilosophicalInterpretation (Cambridge: Cambridge University Press, 2004). Greg Restall, Introduc-tion to Substructural Logic (London: Routledge, 2000) places relevant logic in a moregeneral context.
8
RelevantLogicalsorejectstheassociatedinferences
B)A!B
A)B_¬B
¬A
)A!B
...Thesearecalledthefallaciesofrelevance.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
3/32
TheProof-TheoreticFramework:NaturalDeduction
FollowingAndersonandBelnap,IuseaFitch-stylenaturaldeduction
system.ConsideranNDproofofoneoftheparadoxes:
1. 2. 3. 4.
A B A
B!A
hyp
hyp
1,reit
23,!I
5.A!(B!A)
14,!I
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
4/32
SoWhat’stheDealWithRelevance?
AnIntroductiontoRelevantLogic
EdMares
PhilosophyProgramme
and
TheCentreforLogic,Language,andComputation
VictoriaUniversityofWellington
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
1/32
RelevantLogic
RelevantLogicisasubsystemofclassicallogiccreatedtoavoidthe
so-calledparadoxesofmaterialandstrictimplication,suchas
p!(q!p)
(p^¬p)!q
p!(q_¬q)
p!(q!q)
(p!q)_(q!r)
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
2/32
9
Implicationelimination
A!B
a
Aa
Ba[
b!E
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
7/32
RealUse
Thekeynotionthatisaddedtotheclassicalsysteminordertomakeit
relevantisthatoftherealuseofhypotheses.
Thisconceptisnotexplicitlydefined,butweuseanintuitive
understandingofrealusetoalloworbancertainrules.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
8/32
Relevantlogicsstopthisbyaddingsubscriptstostepsintheproofand
addingrestrictionsthatutilizethesubscripts.
Whenweintroduceahypothesis,wegiveitanumber(thatisnewtothe
proof).Wekeeptrackofthehypothesesthatareusedtoproduceagiven
lineofaproof.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
5/32
Implicationintroduction
A{k}. . .
Ba
A!B
a{k}!I
wherek2
a.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
6/32
10
TwoSortsofConjunction:1.ExtensionalConjunction
Aa
Ba )A^B
a^I
A^B
a
)A
a^E
A^B
a
)B
a
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
11/32
2.IntensionalConjunction(Fusion)
Aa
Bb )AB
a[
bI
AB
a
A!(B!C) b
)C
a[
bE
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
12/32
Considertheclassicalconjunctionrules:
Aa
Bb )A^B
a[
b^I
A^B
a
)A
a^E
A^B
a
)B
a
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
9/32
1. 2. 3. 4. 5. 6.
A
B A A^B
AB!A
hyp
hyp
1,reit
2,3,^I
4,^E
25,!I
7.A!(B!A)
16,!I
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
10/32
11
NegationElimination:
¬A
a
Ab )f a[
b¬E1
¬A{k}
. . .f a
Aa{k}
¬E2
wherek2
a.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
15/32
RelevantLogicsdonotcontain
f a )A
a
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
16/32
(A1...AnB)!Ca`(A1...An)!(B!C)
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
13/32
NegationRules
NegationIntroduction:
A{k}. . .
f a¬A
a{k}¬I
wherek2
a.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
14/32
12
Ourworryisnotaboutwhetherthepremisecanbetrueortheconclusions
canbefalse.
Rather,itistheworrythattheconclusionsdonotfollowfrom
the
premises.
Onewayofunderstandingthisistosaythatthepremisesdonotcontain
theinformationthattheconclusionshold.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
19/32
ContainingInformation
Whatisitforasituationtocontaininformation?
InformationisarelationalnotionWhatcountsasinformationinan
environmentrelativetoanagentarethefeaturesofthatenvironmentthat
shecouldknowaboutgivenhercognitiveandsensorycapacities.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
20/32
TheSemantics
Thesemanticsisaframetheory,inasensesimilartoKripke’s
semanticsformodalandintuitionistlogic.
Thepointsintherelevantframearesituations(inthesenseof
BarwiseandPerry).
Aconcretesituationisapartofaworld.Forexample,thisroom
from
3-3:45pm
today.
Thissituationcontainscertaininformation(e.g.whatcolourthese
chairsarenow,whatiscurrentlyonthescreen,...)
Anditfailstocontainotherinformation(e.g.whetheritisrainingin
Wellingtonrightnow,...).
Thereistrueinformationthatthissituationdoesnotcontain.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
17/32
ARelevantProblem
withTruth
Whatiswrongwiththefollowinginferences?
p^¬p
)q p
)q!q
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
18/32
13
Logicalvalidityisnottruthpreservation,butinformationpreservation.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
23/32
InformationConditions,NotTruthConditions.—JonBarwise
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
24/32
AbstractSituations
Ontheinformationalinterpretation,thepointsoftherelevantsemantics
areabstractsituations.
Weabstractfrom
thesalientfeaturesofrealsituationstocreateageneral
notionofasituation,andthenusethesefeaturestodeterminewhetheror
nottheycontainparticularinformation.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
21/32
InformationConditions
Theinformationconditionassociatedwithaconnectiveistobe
distinguishedfrom
itstruthcondition.
Aninformationconditionisaconditionunderwhichsomeinformation
ofagiventypeisinasituation.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
22/32
14
ExtensionalConjunction
Theinformationconditionforextensionalconjunctionisstraightforward:
s|=A^B
i§
s|=Aands|=B
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
27/32
IntensionalConjunction
Buttheinformationconditionforintensionalconjunctionisabitmore
di¢cult:
s|=AB
i§
scontainsalltheconsequencesofasituationthatcontainsAputtogether
withasituationthatcontainsB.(I.e.,therearesituationstandusuch
thatt|=Aandu|=B,andifweweretohypothesizethattanduwere
tocoexistinsomeworld,wecouldinferthatasubsituationofswouldalso
existinthatworld.)
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
28/32
ConnectingtheNaturalDeductionSystem
toFrames
Astepinaproof
Aa
isreadassayingthatasituations aissuchthat
s a|=A.
WhenwewriteahypothesisA{k},wearesayingine§ect,“supposethat
thereissomesituations ksatisifiesA”.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
25/32
Implication
Whenwehave
A!B
a
inalineinaproof,wearesayingthat,ats awehaveavailableinformation
thatperfectlyreliablyallowsustoinferfrom
therebeingasituationinthe
sameworldass athatcontainstheinformationthatAtotherealsobeing
asituationinthatworldthatcontainstheinformationthatB.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
26/32
15
Thepresentsystem
validatescontraction:
X,A,A,Y
`C
X,A,Y
`C
andsomepeoplethinkthisisbad.(Butit’snot.)
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
31/32
Butifwechangethenatureofthesubscripts,wealsohavetocomeup
withadi§erentinterpretationofthesystem.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
32/32
TheFalsum
s|=f
i§
sisanimpossiblesituation.
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
29/32
Wecanmodifythenaturaldeductionsystem
inseveral
ways
Oneoftheeasiestisbychangingthenatureofthesubscripts:
Wecanmakethesubscriptsmultisetsratherthansets
Wecanmakethesubscriptssequences
Wemakethesubscriptsbinarytrees(structures,inthesenseof
Slaney-Restall)
Mares
(VictoriaUniversityofWellington)SoWhat’stheDealWithRelevance?AnIntroductiontoRelevantLogic
30/32
16
Paraconsistent Mathematics
Zach WeberUniversity of Otago
Overview
When we practice mathematics, we make some very intuitive assumptions that can triggercontradictions. Well known examples include the original infinitesimal calculus and naiveset theory, the latter based on naive comprehension:
9y8x(x 2 y $ A(x))
Paraconsistency is a method for preserving our original mathematical intuitions, by con-trolling for inconsistency with a weaker logical consequence relation, `.
‘Classical’ inferences
In a paraconsistent setting, classical inferences like ex falso quodlibet (A,¬A ` B) anddisjunctive syllogism (A,¬A _ B ` B) are not in general valid. Nevertheless, becauseparaconsistent theories are not trivial (i.e. some sentences are not satisfied), these infer-ences can be restored in appropriate forms. An absurdity constant is defined
? := 8x8yx 2 y
yielding the property that ? ` A for any sentence A. Then ex falso and disjunctivesyllogism are both valid when ¬A is replaced by the property that A entails ?. If wefurther identify
0 := {x : ?}, 1 := {0}then we find a consistency point at the bottom of the number line: 0 = 1 is absurd,and thus so is any sentence that implies 0 = 1. Using this consistency point, we canconfirm some structural facts that are very ‘far away’, such as N being unbounded inR, Konig’s Lemma (and Brouwer’s Fan Theorem), and the Heine-Borel Theorem. Acomplementary consistency point is generated at the top of the number line, at theuniversal set V = {x : 9yx 2 y}.
Connections with other areas
Paraconsistent mathematics thus o↵ers a way to control arguments in a more nuancedway (especially when the underling logic is a relevant logic). The logic makes ‘intensional’distinctions’, which is especially clear when we look at non-equivalent definitions of emptysets, such as {x : ?} and {x : x 6= x}. (The latter may have some members, even thoughit has no members.)
Paraconsistency is a natural dual to constructive mathematics, but it is not opposed toconstructivisim – in fact, constructive techniques are particularly powerful in paracon-sistent settings. The goals of the program are to recapture classical results, and extendthem into the study of the inconsistent, which is intrinsically interesting and beautiful inits own right, and which may yet find applications in any domain where inconsistency ispossible.
17
References
Getting started:
Inconsistent Mathematics, Internet Encyclopedia of Philosophy: http://www.iep.utm.edu/math-inc/
Recent papers:
Weber, Zach (2010). Transfinite Numbers in Paraconsistent Set Theory. Review of Symbolic Logic 3(1): 71-92.
McKubre-Jordens, Maarten and Weber, Zach (2012). Real Analysis in Paraconsistent Logic. Journal of Philo-sophical Logic, to appear.
See also:
Brady, Ross (2006). Universal Logic, CSLI. [****This book includes the classical model theoretic proofs that show
paraconsistent mathematics is not trivial.****]
Mortensen, Chris (1995). Inconsistent Mathematics. Kluwer Academic Publishers.
Priest, Graham (2006). In Contradiction: A Study of the Transconsistent. Oxford University Press. second
edition.
18
Constructive Methods in Mathematics
Maarten McKubre-JordensUniversity of Canterbury
In Brief
The point of using constructive methods in mathematics is to explicitly exhibit anyobject or algorithm that the mathematician claims exists; so constructive proof provides,in principle, a mechanical method. Loosely speaking, one replaces the absolute notion oftruth in mathematics, with (algorithmic) provability. Constructive proofs:
1. embody (in principle) an algorithm (for computing objects, converting other algo-rithms, etc.), and
2. prove that the algorithm they embody is correct (i.e. that it meets its design speci-fication).
Constructive techniques
Upon adopting only constructive methods, we lose some powerful proof tools in ourarsenal, such as unrestricted use of the Law of Excluded Middle (LEM) and anythingwhich validates it, such as double negation elimination and unrestricted use of proof bycontradiction1. We cannot, in general, constructively prove 9xP (x) by assuming ¬9xP (x)and deriving a contradiction; that doesn’t compute the required x.
However the news isn’t all bad. In a lot of cases, constructive alternatives to non-constructive classical principles in mathematics, leading to some very strong results. Forexample, the classical least upper bound principle is not constructively provable.
LUB Any nonempty set of reals that is bounded from above has a least upper bound.
However the constructive least upper bound principle is provable.
CLUB Any order-located nonempty set of reals that is bounded from above has a leastupper bound.
A set is order-located if given any real x, the distance from x to the set is computable. Itis quite common for a constructive alternative to be classically equivalent to the classicalprinciple; and, indeed, classically every nonempty set of reals is order-located.
To see why LUB is not provable, we may consider a so-called Brouwerian counterexample(or weak counterexample), such as the set
S = {x 2 R : (x = 2) _ (x = 3 ^ P )}
where P is some as-yet unproven statement, such as Goldbach’s conjecture. If the setS had a computable LUB, then we would have a quick proof of the Goldbach conjec-ture’s truth or of its unprovability. A Brouwerian counterexample is an example which
1Which is not to say that LEM is false. Both Russian recursive mathematics, in which LEM is provably false, and
classical mathematics, in which it is logically true, are models of constructive mathematics—so in a way, LEM is independentof constructive mathematics, and hence non-constructive.
19
shows that if a certain property holds, then it is possible to constructively prove a non-constructive principle (such as LEM); and thus the property itself must be essentiallynon-constructive.
It is often the case that a classical theorem becomes more enlightening when seen fromthe constructive viewpoint2. For example, in the least upper bound principle the extracomputational information provided by being order-located is enough to guarantee thecomputability of the least upper bound.
Within constructive mathematics a number of methods has been developed, enrichingthe subject to a degree where it is comparable to its classical counterpart in complexity,and often exceeds it in computational informativity.
Connections with other disciplines
The connection of constructive mathematics with computer science and programming isclear. A major upshot of the constructive approach is to identify with relative ease thesorts of things that computers cannot do (it is usually easier to prove a negative result),and so to guide the programmer to focus on what is achievable.
Like paraconsistency, constructivism brings out finer-grained details of proof that areoften casually dismissed in classical proofs. In fact, a single classical theorem can lead toseveral constructively discernible di↵erent theorems, where the constructive techniquesbring to the fore extra computational strength required in the hypotheses, or furtherinformation contained in the conclusion.
References
For a more in-depth introduction:
Bridges, D.S. Constructive Mathematics. Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/
entries/mathematics-constructive/
Further reading:
Aberth, O. (1980) Computable Analysis. New York: McGraw-Hill.
Aczel, P., and Rathjen, M. (2001) Notes on Constructive Set Theory. Report No. 40, Institut Mittag-Le✏er,Royal Swedish Academy of Sciences.
Beeson, M.J. (1985) Foundations of Constructive Mathematics. Heidelberg: Springer-Verlag.
Bishop, E. and Bridges, D.S. (1985) Constructive Analysis. Grundlehren der math. Wissenschaften, Heidelberg:Springer-Verlag.
Bridges, D.S. and Richman, F. (1987) Varieties of Constructive Mathematics. Cambridge: Cambridge UniversityPress.
Bridges, D.S. and Vıta, L.S. (2006) Techniques of Constructive Analysis. Universitext, Heidelberg: Springer-Verlag.
Dummett, M. (2000) Elements of Intuitionism. Oxford Logic Guides 39, Oxford: Clarendon Press.
Martin-Lof, P. (1968) Notes on Constructive Analysis. Stockholm: Almquist & Wixsell.
Weirauch, K. (2000) Computable Analysis. EATCS Texts in Theoretical Computer Science, Heidelberg: Springer-
Verlag.
2Although it would be unfair to say that constructive mathematics is revisionist in nature. Indeed, Brouwer proved his
fan theorem intuitionistically in 1927, but the first proof of Konig’s lemma (its classical equivalent) was published in 1933.
20
Dothefollowingstatem
ents
Ithefour
colour
theorem,
IFermat’sgreattheorem,
ItheRiemannhypothesis,
ItheCollatz
conjecture
shareacommon
mathematical
prop
erty?
And
,ifthereissuch
aprop
erty,how
canweuseitforabetter
understand
ingof
thesestatem
ents?
3/19
9
Com
putabilityandCom
plexity
1
Universalitytheo
rem.There
exists
(and
canbeconstructed)
a(Turing)
machine
U—calleduniversal—such
that
foreverymachine
Vthereexists
aconstant
c=
c
U,V
such
that
foreveryprogram
�thereexists
a�0forwhich
thefollowingtwocond
itions
hold:
IU(�
0 )=
V(�),
I|�
0 |
|�|+
c.
4/19
9
Aconstructiveapproachtothe
complexityofmathematical
problems
C.S.Calud
e(U
oA)andE.Calud
e(M
asseyU)
Con
struMathSou
th2012,Wesport
1/19
9
Thistalk
presents
onoverview
ofresultsob
tained
withan
algorithmic
uniform
metho
dto
measure
thecomplexityof
alarge
classof
mathematical
prob
lemsanddiscussesafew
open
prob
lems.
2/19
9
21
Anexam
pleof
aprog
ram
forU
The
followingprogram
compu
tesin
dtheprod
uctof
two
non-negative
integers
stored
inaandb:
number
instruction
0&h,e
1&d,0
2=b,0,8
3&e,1
4+d,a
5=b,e,8
6+e,1
7=a,a,4
8&e,h
9=a,a,c
7/19
9
Com
putabilityandCom
plexity
2
The
haltingproblem
foramachine
Visthefunction
⇤V
defin
edby
⇤V(�)=
⇢1,
ifV(�)=
1,
0,otherwise.
Undecidab
ility
theo
rem.IfU
isun
iversal,then⇤
Uis
incompu
table,
i.e.thehaltingprob
lem
foraun
iversalmachine
isun
decidable.
8/19
9
Aun
iversalTuringmachine
1
Asimple,
minim
al(eachinstructionisessential)un
iversalTuring
machine
Ucanbedesign
edusingthefollowingfiveinstructions:
=r1,r2,r3(branching
instruction)
&r1,r2(assigning
instruction)
+r1,r2(sum
)!r1(readon
ebit)
%(halt)
5/19
9
Aun
iversalTuringmachine
2
Aregister
machine
program
consists
ofafin
itelistof
labeled
instructions
from
theab
ovelist,withtherestrictionthat
thehalt
instructionappears
only
once,as
thelast
instructionof
thelist.
The
inpu
tdata
(abinary
string
)followsim
mediately
afterthehalt
instruction.
Aprogram
notreadingthewho
ledata
orattempting
toread
past
thelast
data-bitresultsin
arun-timeerror.
Som
eprograms(astheon
espresentedin
thispaper)have
noinpu
tdata;
theseprogramscann
othaltwithan
under-read
error.
6/19
9
22
Com
plexity
Complexity
C
U(⇡)=
min{|⇧P|:⇡=
8nP(n)}.
Invarian
cetheo
rem.IfU,U
0areun
iversal,then
thereexists
aconstant
c=
c
U,U
0such
that
forall⇡=
8nP(n),P
compu
table:
|CU(⇡)�C
U0 (⇡)|
c.
Inco
mputability
theo
rem.IfU
isun
iversal,then
C
Uis
incompu
table.
11/19
9
Com
puting
thesize
oftheprog
ram
MULT
number
instruction
code
leng
th0
&h,e
010001001
00110
141
&d,0
0100101
100
102
=b,0,8
00011
100
1110010
153
&e,1
0100110
101
104
+d,a
111
00101
010
115
=b,e,8
00011
00110
1110010
176
+e,1
111
00110
101
117
=a,a,4
00010
010
11010
138
&e,h
0100110
0001001
149
=a,a,c
00010
010
00100
13
Total
leng
th:128.
12/19
9
⇧1–problem
s
Aprob
lem
⇡of
theform
8�P(�),
where
Pisacompu
tablepredicateiscalleda⇧
1–problem.
IAny⇧
1–problem
isfin
itelyrefutable.
IFor
every⇧
1–problem
⇡=
8�P(�)weassociatetheprogram
⇧P=
inf{n:P(n)=
false}
which
satisfies:
⇡istrue
i↵U(⇧
P)=
1.
ISolving
thehaltingprob
lem
forU
solves
all⇧
1–problem
s.
9/19
9
Examples
The
prob
lems
Ithefour
colour
theorem,
IFermat’sgreattheorem,
ItheRiemannhypothesis,
ItheCollatz’sconjecture
areall⇧
1–problem
s.
Ofcourse,no
tallprob
lemsare⇧
1–problem
s.For
exam
ple,
the
twin
prim
econjecture.
10/19
9
23
Riemannhypothesispredicate
The
negation
oftheRiemannhypothesisisequivalent
tothe
existenceof
positiveintegers
k,l,m
,nsatisfying
thefollowing:
1.n�
600,
2.8y
<n[(y+1)
|m],
3.m
>0&8y
<m[y
=0_9x
<n[¬
[(x+1)
|y]]],
4.explog(m
�1,l),
5.explog(n
�1,k),
6.(l�n)2
>4n
2k
4,
where
x|z
means
“xdividesz”andexplog(a,b)isthepredicate
9x[x
>b+1&(1
+1/
x)x
b
a+1<
4(1+1/x)x
b].
15/19
9
The
Collatz
conjecture
Given
apositiveintegera
1thereexists
anaturalN
such
that
a
N=
1,where
a
n+1=
⇢a
n/2,
ifa
niseven,
3an+1,
otherwise.
The
Collatz
conjecture
isa⇧
1-statement,bu
ttheproo
fis
non-constructive!Writing
The
Collatz
conjecture
asa
⇧2-statement(i.e.of
theform
8n9i
R(n,i),where
R(n,i)isa
compu
tablepredicate)
iseasy
andconstructive.
How
togeneralisethecomplexitymetho
dfor⇧
2-statements?
16/19
9
Com
plexityClasses
Because
oftheincompu
tabilitytheorem,weworkwithup
per
bou
ndsforC
U.Astheexactvalueof
C
Uisno
tim
portant,we
classify⇧
1–problem
sinto
thefollowingclasses:
CU,n=
{⇡:⇡isa⇧
1–problem
,CU(⇡)
nkb
it}.
13/19
9
Som
eResults
ICU,1:Legendre’sconjecture(there
isaprim
enu
mber
between
n
2and(n
+1)
2,foreverypositiveintegern),Fermat’slast
theorem
(there
areno
positiveintegers
x,y
,zsatisfying
the
equation
x
n+y
n=
z
n,foranyintegervaluen>
2)and
Goldbach’sconjecture(every
even
integergreaterthan
2can
beexpressedas
thesum
oftwoprim
es)
ICU,2:Dyson’sconjecture(the
reverseof
apow
erof
twois
neverapow
erof
five)
ICU,3:theRiemannhypothesis(allno
n-trivialzerosof
the
Riemannzeta
function
have
real
part
1/2),Euler’sinteger
partitiontheorem
(the
number
ofpartitions
ofan
integerinto
oddintegers
isequalto
thenu
mber
ofpartitions
into
distinct
integers).
ICU,4thefourcolourtheorem
(the
vertices
ofeveryplanar
graphcanbecoloured
withat
mostfour
coloursso
that
notwoadjacent
vertices
receivethesamecolour)
14/19
9
24
Indu
ctivecompu
tation
oftheCollatz
conjecture
Define
thefunction C(n)=
⇢n,
if9i(F
i (n)=
1),
1,otherwise,
where
F(x)=
⇢x/2,
ifxiseven,
3x+1,
otherwise,
andF
iistheith
iterationof
F.
Nextwedefin
etheindu
ctiveTuringmachine
M
ind,2
Collatz
by
M
ind,2
Collatz
=
⇢0,
if8n
�1,C(n)=
n,
1,otherwise.
19/19
9
The
Collatz
conjecture
isin
theclassCind,2
U,1
label
instruction
label
instruction
label
instruction
&OR,1
&D,0
L9
+G,F
&T,1
&E,1
+G,F
L1
&OC,1
&F,1
+G,1
&N,1
L6
=F,G,L8
=E,E,L3
L2
&G,T
+E,1
L10
=G,1,L11
&K,N
+F,1
+N,1
L3
=K,1,L10
=E,2,L7
=E,E,L2
&E,1
=E,E,L6
L11
&OC,T
L4
&F,E
L7
&E,0
=OC,T,L12
+F,1
+D,1
&OR,0
=F,K,L5
=E,E,L6
=E,E,L13
+E,1
L8
=E,1,L9
L12
+T,1
=E,E,L4
&G,D
=E,E,L1
L5
&K,E
=E,E,L3
L13
%
Indu
ctiveprogram
fortheCollatz
conjecture
20/19
9
Indu
ctiveCom
plexityandCom
plexityClasses
ofFirst
Order
Bytransformingeach
program⇧
PforU
into
aprogram⇧
ind,1
Pfor
U
ind(U
working
in“ind
uctive
mod
e”)wecandefin
etheindu
ctive
complexityof
first
orderby
C
ind,1
U(⇡)=
min{|⇧ind,1
P|:⇡=
8nP(n)},
theindu
ctivecomplexityclassesof
orderon
eby
Cind,1
U,n
={⇡
:⇡isa⇧
1–statement,C
ind,1
U(⇡)
nkb
it},
andprovethat
CU,n=
Cind,1
U,n
.
17/19
9
Indu
ctiveCom
plexityandCom
plexityClasses
ofHigherOrders
Byallowingindu
ctiveprogramsof
order1as
routines
weget
indu
ctiveprogramsof
order2,
sowecandefin
etheindu
ctive
complexityof
second
order(for
morecomplex
prob
lems)
C
ind,2
U(⇢)=
min{|M
ind,2
R|:⇢=
8n9iR(n,i)},
andtheindu
ctivecomplexityclassof
second
order:
Cind,2
U,n
={⇢
:⇢=
8n9iR(n,i),C
ind,2
U(⇢)
nkb
it}.
18/19
9
25
References2
C.S.Calud
e,E.Calud
e,K.Svozil.The
complexityof
provingchaoticity
andtheChu
rch-TuringThesis,Chaos20
0371
03(201
0),1–
5.
C.S.Calud
e,M.J.
Dinneen.Exact
approxim
ations
ofom
eganu
mbers,
Int.JournalofBifurcation&
Chaos17
,6(200
7),19
37–1
954.
E.Calud
e.The
complexityof
Riemann’sHyp
othesis,Journalfor
Multiple-ValuedLogicandSoftComputing,(201
2),to
appear.
E.Calud
e.Fermat’sLastTheorem
andchaoticity,NaturalComputing,
(201
1),DOI:10
.100
7/s110
47-011
-928
2-9.
M.J.
Dinneen.A
prog
ram-sizecomplexitymeasure
formathematical
prob
lemsandconjectures,in
M.J.
Dinneen,B.Kho
ussainov,A.Nies
(eds.).Computation,PhysicsandBeyond,Springer,Heidelberg,
2012
.
J.Hertel.OntheDi�
cultyof
Goldb
achandDyson
Con
jectures,
CDMTCSResearchReport36
7,20
09,15
pp.
23/19
9
References3
J.Lagarias(ed.),TheUltimateChallenge:The3x
+1Problem,AMS,
2010
.
C.Moo
re,S.Mertens.TheNatureofComputation,OxfordUniversity
Press,Oxford,
2011
.
G.Perelman.Ricci
Flow
andGeometrization
ofThree-M
anifolds,
Massachusetts
Instituteof
Techn
olog
y,Departm
entof
Mathematics
Sim
onsLecture
Series,September
23,20
04.
24/19
9
Twoop
enprob
lems
Whatisthecomplexityof
IPvs
NPprob
lem?
IPoincare’stheorem
(Perelman)?
21/19
9
References1
M.Burgin,
C.S.Calud
e,E.Calud
e.Indu
ctivecomplexitymeasuresfor
mathematical
prob
lems,in
preparation.
C.S.Calud
e,E.Calud
e,M.J.
Dinneen.A
new
measure
ofthedi�culty
ofprob
lems,JournalforMultiple-ValuedLogicandSoftComputing12
(200
6),28
5–30
7.
C.S.Calud
e,E.Calud
e.Evaluatingthecomplexityof
mathematical
prob
lems.
Part1ComplexSystems,18
-3(200
9),26
7–28
5.
C.S.Calud
e,E.Calud
e.Evaluatingthecomplexityof
mathematical
prob
lems.
Part2ComplexSystems,18
-4,(201
0),38
7–40
1.
C.S.Calud
e,E.Calud
e.The
complexityof
theFou
rColou
rTheorem
,LMSJ.Comput.Math.13
(201
0),41
4–42
5.
C.S.Calud
e,E.Calud
e,M.Queen.The
complexityof
theinteger
partitiontheorem,TheoreticalComputerScience,accepted.
22/19
9
26
Abstract Stone Duality - A Logic for Topology
Nicholas DuncanUniversity of Canterbury
Abstract
Abstract Stone Duality (ASD) is a logical system for reasoning and computing with topologicalspaces. Compared to other systems of logic the quantifiers 8 and 9 are restricted to compact and overtspaces, respectively, in ASD. The concept of overt spaces is not seen in topology, as all topologicalspaces are overt, but they play an important computational role in ASD.
In this paper we start with the definition of topology and relate it to computability. Then wefocus on the construction of the type theory underlying ASD, starting with the types, which representspaces, moving on to terms, which represent continuous functions, and finally reaching judgements,which deal with proving results in the calculus.
Next we will consider how local compactness allows computation in the calculus and the con-nection to interval arithmetic. Finally we compare this system to other computable systems, likeRecursive Analysis.
Introduction
Abstract Stone Duality (ASD) is a logical system created by Paul Taylor for reasoningand computing with topological spaces. It is named after Stone’s duality between Booleanalgebras and Stone spaces. This duality manifests itself in the fact that spaces are alsoalgebras, so we can avoid the use of sets by using the spaces as carriers of those algebras.However, in this article we will not dwell on this important aspect of ASD, instead wewill focus on the logic. This system is given as an example of a logic that is suitable fora specific domain in mathematics.
Instead of starting with a large all-encompassing system, such as a set theory, and thenplacing continuous and computable structures on sets, we begin with a logical system inwhich everything is computable and continuous and the operations of the system preservethese properties.
This system is a type theory whose types represent topological spaces, and the terms ofthe calculus represent continuous functions. Everything that can be constructed preservescontinuity and computability.
A major restriction in this system is that quantifiers may only range over certain kindsof spaces. The universal quantifier 8 can only quantify over compact spaces, like theunit interval [0, 1], but not N or Q. The existential quantifier 9 can only quantify overovert spaces. Classically all topological spaces are overt, however in some constructivetopological settings like locale theory not all spaces are overt. Many of the spaces in ASDare overt, like N, Q, and R, however not all spaces are overt. Indeed those that are overtembody some computational process to access their elements. The subspace of all zerosof a function is often not overt, since deciding whether a real number is equal to zero isnot computable.
Parts of this calculus, like the real numbers, can be transformed into programs usinginterval arithmetic. The calculus insures that these programs succeed and return aninterval solution to within a given tolerance. This way results can be extracted from thecalculus.
27
In comparison with other approaches to computable topology the closed interval [0, 1]is compact in ASD. This allows us to use the universal quantifier over closed boundedintervals, which is vital for the translation to interval arithmetic.
Topology and Observability
We start with the definition of a topological space.
Definition 1. A topological space consists of a set X of points along with a set T ofsubsets of X, whose elements are called open subsets, such that
• The subsets ; and X are open subsets.
• If U and V are open subsets so is U \ V .
• If {Ui}i2I is a collection of open subsets thenS
i2I Ui is an open subset.
Example 2. The real numbers R has the Euclidean topology where the open sets arearbitrary unions of open intervals (a, b):
U =[
i
(ai, bi), where (ai, bi) = {x 2 R | ai < x < bi}
We compare the definition of open subsets with the concept of observable properties ofa set. An observable property is some subset in which membership is semi-decidable. Anon-rigorous definition is the following:
Definition 3. Let X be a set. A subset S of X is observable if there is a computerprogram, such as a Turing machine, which when given an encoding of an element x of X,halts if x 2 S, or loops (runs forever) otherwise.
Example 4. The standard example of observable subsets are the recursively enumerablesubsets of N. Every recursively enumerable subset is given by a computer program, andan element x is in a recursively enumerable subset if and only if this computer programhalts on input x.
There are some similarities between open subsets of a topological space and observablesubsets:
• The subset ; is observable, just use a program which runs forever. This programnever halts, so it does not accept any element of X.
• The subset X is observable, use the program which immediately halts.
• If U and V are observable with programs u and v, then U \ V is observable. Takethe program which runs u until it halts, then it runs v. If x is in U and V thenboth programs halt, so x is in U \ V . If x is not in both U and V then one of theprograms will loop on input x. This shows that U \ V is observable.
• Let (Ui)i2N be a sequence of observable subsets, where the sequence of acceptingprograms is also computable (e.g. if the computer programs are encoded by numbersui, then the function i 7! ui should be computable). Then U =
Si2N Ui is also
observable.
28
To see this let (ui) be the sequence of computer programs in which ui accepts Ui.We construct a computer program u that takes an input x and interleaves the com-putations of each ui(x), terminating as soon as one ui(x) terminates. This programcould perform one step of u
0
(x), then one step of u0
(x) and u1
(x), then another ofu0
(x), u1
(x) and u2
(x), and so on. Each loop we introduce a new program in the se-quence. Since the sequence is computable this method of introducing new programsis also computable.
If x is in some Ui then eventually the program ui will halt. If x is not in any of thesubsets Ui then each program ui will loop on input x, so the program u above willalso loop on x. Hence the union is observable.
So we see that observable properties have binary intersections, but only some unions,and the indexing set of these unions must somehow be computable. If the sequence (Ui)was not computable then we cannot construct a new program in each iteration of theprogram u above. Furthermore, if the indexing set of the union is not countable then wecannot interleave the operations like we did in u, and we would miss some elements ofthe indexing set.
In ASD we do not have all unions, but we do have computable unions like in the observablesubsets case. The indexing spaces of these unions will be overt spaces, which we will defineshortly. If a collection of open subspaces is indexed by an overt space then the union isopen. However not all spaces are overt, so we do not have all unions.
Open Subsets vs Predicates vs Functions
Instead of treating observable properties as subsets we will treat them as continuousfunctions of a special kind. This will reduce the number of primitive concepts that wehave to consider. First we give the definition of a continuous function.
Definition 5. Given spaces X and Y a continuous function f : X ! Y is a functionsuch that for every open subset U of Y the inverse image f�1(U) is an open subset ofX. The inverse image of U is the set:
f�1(U) = {x 2 X | f(x) 2 U}
To treat open subsets as continuous functions we need a special topological space, calledthe Sierpinski space.
Definition 6. The Sierpinski space ⌃ classically consists of two points, which we call Tand F , and the topology consists of the three subsets ;, ⌃ and {T}. Note that {F} isnot open.
Given a continuous function � : X ! ⌃ we have an open subset ��1({T}) of X. Con-versely, given any open subset U of X we can define a continuous function X ! ⌃, whichclassically is defined by f(x) = T if x 2 U , or f(x) = F otherwise.
This construction gives us a correspondence between continuous functions X ! ⌃ andopen subsets of X. This can be further extended to closed subsets of X, which classicallyare the set complements of open subsets. A closed subset is given by the inverse imageof the closed set {F}.Now let us consider the continuous functions⌃ ! ⌃. We know that these correspond toopen subsets of⌃ , of which classically there are three: ;, ⌃ and {T}. They correspond
29
to the continuous functions F (constantly false), T (constantly true), and the identityfunction.
Notice that there is no continuous function which swaps T and F . This has importantcomputational significance. The space ⌃ can be thought of as the space of terminationpossibilities of a computer program - either a program terminates (T ), or a program loops(F ). If we think of a space X as the space of inputs of a program, and f : X ! ⌃ asa computer program recognising an observable property, then f(x) = T if the programhalts, or f(x) = F if the program loops.
If we had a function ¬ : ⌃ ! ⌃ which swaps T and F then the set complement ofany observable property would also be observable, just take the corresponding functionf : X ! ⌃ and post-compose with ¬. In the recursively enumerable subset examplethis would mean that co-recursively enumerable subsets would also be observable, sothe halting problem would be decidable. This is not computable, so we would losecomputational ability if ¬ was an acceptable function. Luckily the topology of ⌃ preventsthis behaviour.
Objects of ASD
We have seen that open subsets of a space X can be represented by a continuous functionX ! ⌃. The calculus allows us to abstract away from the set theoretic nature of topolo-gies, and so instead of considering open subsets of a space X we will consider continuousfunctions X ! ⌃. Since we also want to abstract away from the set-theoretic nature offunctions we use the word morphism instead of continuous function.
ASD is a type theory whose types represent spaces. How do we represent the topology ona space X? Classically it is given by a collection of subsets of X, but we have seen thatthese subsets may be represented by morphisms X ! ⌃. To avoid the use of sets thiscollection of morphisms, which we denote⌃ X , should itself be a space. So the topologyof a space X in ASD is itself a space,⌃ X . Classically, for this to be a suitable space Xmust be a locally compact topological space, and then we can give⌃ X the Scott topology.Locally compact spaces will be considered later on in this article, but we will not coverthe Scott topology. See [6] for details. So the classical model of ASD will interpret thetypes as locally compact topological spaces.
Suppose we have interpretations for the types X and⌃ X , how do we ensure that ⌃X is thetopology on X? For this we use a notion from category theory called a monad. If you donot know about monads then feel free to skip this paragraph. Monads allow us to definealgebras whose carriers are not necessarily sets, and the arity of the operations in thealgebra do not need to be indexed by sets. We require that the adjunction (⌃(�) a ⌃(�))be monadic, where⌃ (�) is the exponential functor. This makes the objects ⌃-algebras,and this method bypasses any requirements of underlying sets. For more details see [5].
This leads us to our first axiom, which gives the types of ASD. The calculus of ASDconsists of four syntactic elements: types, terms, statements, and judgements. Many ofthese depend on each other, so the formal definition of the calculus requires a mutuallyinductive definition. We will give parts of the calculus independently, and some stagesmay refer to future stages of definition. This is not intrinsic to ASD itself, as other typetheories have this di�culty.
Axiom 1. The types of ASD consist of the following:
• The basic types 1, ⌃ and N.
30
• If X is a type then so is⌃ X .
• If X and Y are types, so is X ⇥ Y .
• Technical condition: If X is a type then any ⌃-split subspace is also a type. Thisconstruction comes from the monad above, and allows the construction of a varietyof derived types. These types are denoted by {X |E}, where E is a special term oftype⌃ X⇥⌃
Xcalled a nucleus. See [4] for details on this construction.
The derived types of ASD can be constructed from the type constructors above, and theyinclude the empty space 0, Q, R, [0, 1], and many other spaces.
In a model of ASD these types are sent to certain objects, but note that the derived typeR need not necessarily be interpreted as the real numbers. In certain constructive settingsthe closed interval [0, 1] is not compact, so R can not be interpreted as the standard realnumbers in such a setting. Also note that the classical interpretation of Q is with thediscrete topology, not the order topology. We will see more of this later.
Logical Terms of the Calculus
Now we will consider the logical terms of the calculus, which will represent logical prop-erties and subspaces. The terms of type ⌃ are called propositions, and the terms of type⌃X are called predicates.
Axiom 2. The logical terms of ASD consist of the following:
• Variables: The types ⌃ and⌃ X all have a countable supply of variables, oftendenoted �,⌧ for propositions and �, for predicates. Each variable has an associatedtype.
• Constants: >,? are terms of type⌃ , which represent true and false.
• Connectives: if � and ⌧ are terms of type⌃ , the connectives � _ ⌧ and � ^ ⌧ areterms of type ⌃, representing disjunction and conjunction, respectively.
• �-abstraction: if �(x) is a term of type ⌃ with a free variable x of type X then�x.�(x) is a term of type ⌃X . �-abstraction is used to construct functions in typetheory.
• �-application: if � is a term of type⌃ X and a is a term of type X, then �(a) is aterm of type ⌃. This term is also denoted �a.
• Equality: if N is a discrete space and n and m are terms of type N , then n =N mis a term of type ⌃.
• Inequality: if H is a Hausdor↵ space and n and m are terms of type H, then n 6=H mis a term of type ⌃.
• Universal quantification: If X is compact and �(x) is a term of type ⌃ with a freevariable x of type X, then 8x : X.�x is a term of type ⌃.
• Existential quantification: Similarly, if X is overt, then 9x : X.�x is a term of type⌃.
31
Note that we do not have the connectives ¬ or ! of type⌃ . These would lead tonon-computability, as we have seen earlier.
These logical terms require us to consider certain kinds of spaces. The discrete, Hausdor↵and compact spaces have their regular interpretation in topology. However, classicallyall spaces are overt, so the concept does not show up in classical topology.
First we consider the discrete spaces. These are spaces in which equality is observable.Classically this corresponds to spaces whose diagonal subset
{(x, x) 2 X ⇥X | x 2 X}
is open, which implies that all subsets are open. In ASD the spaces N,Z and Q arediscrete. Note that R is not discrete. If we consider real numbers as infinite decimalexpansions then to check equality we are required to check the entire expansion, which isnot observable as it would take an infinite amount of time.
Next are the Hausdor↵ spaces. In these spaces inequality is observable. Classicallythese correspond to Hausdor↵ topological spaces, where the diagonal is closed. In ASDthe spaces N,Z,Q,R, and [0, 1] are all Hausdor↵. Classically all discrete spaces areHausdor↵, but this is not so in ASD. Open subsets are not closed under arbitrary unions,so the classical proof that discrete implies Hausdor↵ does not apply.
The compact spaces correspond to the compact spaces in topology, which classically aregiven by the finite subcover property:
Definition 7. A topological space X is compact if for any family of open subsets {Ui}i2Iwhose union is the whole space X, there is a finite subset J of I such that the subfamily{Uj}j2J also covers the whole space X.
Note that we do not require compact spaces to be Hausdor↵. In ASD the bounded closedintervals [a, b] are compact, as well as the Sierpinski space ⌃ .
The overt spaces are invisible in classical topology. A topological space X is overt if theunique continuous function X ! 1 sends open subsets to open subsets. Classically alltopological spaces are overt, however in constructive locale theory not all spaces havethis property. Earlier terminology from locale theory called overt spaces open spaces,as the unique map X ! 1 is open. However this clashes with the terminology for opensubspaces, so overt spaces are the preferred terminology.
In locale theory an overt locale has a positivity predicate Pos(a), which holds if a isinhabited. Constructively not all non-empty subsets are inhabited, i.e. have an element,so overt spaces have a way of recognizing when an open is inhabited.
32
Here is a chart taken from [1], Examples 4.26, which gives a variety of di↵erent types ofASD and the properties they have:
space overt discrete compact Hausdor↵
N⇥ ⌃ X 7 7 7
R,Rn X 7 7 X⌃ X 7 X 7
[a, b], 2N X 7 X Xfree SK-algebra X X 7 7
N,Z,Q X X 7 XK-finite X X X 7
finite X X X Xset of codes ofnon-terminatingprograms
7 X 7 X
The free SK-algebra is the free algebra with a non-associative operation x ·y and symbolsS and K such that the two equalities:
((S · x) · y) · z = (x · z) · (y · z) and (K · x) · y = x
hold. This represents combinatory logic with the combinators S and K. Equality isobservable, as we can loop through all possible equalities. However, inequality is notobservable, as programs can be represented as combinators.
K-finite spaces correspond to spaces in which all the elements can be finitely listed, butthere may be repetitions, as inequality is not observable. Some examples are subspacesof a finite space given by a semi-decidable predicate.
Other Terms of the Calculus
Now that we have considered the logical terms we move on to the other terms of thecalculus.
Axiom 3. Non-logical terms of ASD.
Variables: Every term has a countable supply of variables. Each variable has a distin-guished type.
Product terms:
• If s is a term of type S, and t is a term of type T , then hs, ti is a term of type S⇥T ,representing an ordered pair.
• If x is a term of type S ⇥ T , then ⇡1
x is a term of type S and ⇡2
x is a term of typeT . These represent projections from an ordered pair to one of its components.
Numerical terms:
• Zero: 0 is a term of type N.
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• Successor: If n is a term of type N, then Sn is a term of type N.
• Definition by description: If �(n) is a term of type ⌃ with a free variable n of typeN then the n.�n is a term of type N if the two statements:
�n ^ �m ) n =N m and 9n : N.�n , >
hold. Since we have not covered statements yet these can be read as the uniquenessand existence properties of definition by description.
• We also have terms for primitive recursion over any type.
Note that definition by description turns logical predicates into terms of type N. Thisallows unbounded minimization to be represented in the calculus, so we can representall partial recursive functions. However unbounded minimization implies that equality inthe calculus is no longer decidable, but we wanted to represent all computable functions,so this is not a problem.
We can derive more predicate terms from the ones above, like ,�, <,> for N and Q,by using primitive recursion and equality.
There are also derived terms for the derived types. For example the real numbers havethe following derived terms, which we give as an axiom:
Axiom 4. The type R has the following terms:
• Constants: 0 and 1 are terms of type R. Note that 0 is di↵erent from the term 0 oftype N, so we use context to determine which term we mean.
• Operators: If x and y are terms of type R, then x+ y, x⇥ y, x� y are terms of typeR. If we have the judgement y > 0 _ y < 0 , >, then x÷ y is a term of type R.
• Dedekind cuts: Given predicates � and ⌫ of type⌃ Q, or even of type ⌃R, thencut du.�d ^ ⌫u is a term of type R if the following six judgements hold:
9e.(d < e) ^ �e , �d 9t.⌫t ^ (t < u) , ⌫u
9d.�d , > 9e.⌫e , >�d ^ ⌫u ) d < u � d_ ⌫u ( (d < u)
These judgements have been organised into two columns to illuminate the symmetrybetween them. The first line states that the cuts are rounded, so have no maximumor minimum elements. The next line states that the cuts are inhabited. The fifthjudgement states that cuts are disjoint, and the final judgement states that cuts areorder-located.
This axiom is not strictly necessary, as the reals can be constructed in the calculus, butit is useful to see the properties of the real numbers. See [1] for the construction of Rfrom the basic types.
Statements and Judgements
We have given many of the terms of the calculus, the next thing to consider are thestatements. These describe a relationship, such as equality, between two terms.
Definition 8. There are two types of statements :
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• If a and b are terms of type X, then a = b is a statement. Note we have no subscripton the equality sign. Such a statement expresses the fact that a and b represent thesame element of X.
• If ↵ and � are terms of type⌃ X , then ↵ ) �, ↵ ( � and ↵ , � are statements.The statement ) states that the open subspace corresponding to ↵ is included inthe open subspace corresponding to �. The statement , is the same as =, but wewill use the form , for predicates.
We stated earlier that ¬ and ! are not terms of type⌃ , so are not propositional connec-tives. However we can use statements to give some form of negation or implication. Theproposition a ! b can be represented as a ) b, and ¬a can be represented by a ) ?.However the logic is limited, in which we can only have one implication or negation, andit must occur as the outermost connective.
Finally we reach judgements, which are used to express some logical truth in our system.
Definition 9. A context � for a judgement consists of two lists. One list of variabledeclarations, e.g n : N, and another list of statements, where the free variables of thestatements occur in the first list.
Example 10. The following is a valid context:
n : N,� : ⌃N,�n, >
Definition 11. There are three types of judgements:
• Valid type formation: This has the form ` X : type, and states that X is a validtype.
• Term formation: This has the form� ` a : X, which states that a is a valid term oftype X in the context�.
• Statement formation: Similarly this has the form� ` s : X, where s is a statementbetween terms of type X. This states that the statement s holds in the context�.
Note that type formation does not have a context, so we cannot form types which dependon terms, or dependent types as they are called in type theory. Future extensions of thecalculus may allow such types.
We have given a large part of the syntax of the calculus, so now we will consider how tointerpret this syntax. A type judgement can just be interpreted as a space of some sort,such as a topological space, a locale, or a more exotic object. We will assume that weare interpreting the types of the calculus as topological spaces.
The representation of a context is similar to the type judgement. The variable declarationsare represented by a product of topological spaces, one for each variable. The list ofstatements then forms a subspace of that topological space. For the context in Example10 above the representation will be the subspace
{(n,� ) 2 N⇥ ⌃N | �(n) = >}.
A term judgement � ` a : X is represented by a continuous function from the representa-tion of � to the representation of X. So in this calculus terms are continuous functions.
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The empty context is interpreted as the one element topological space {⇤}, so termjudgements of the form ` a : X represent continuous functions {⇤}! X. But suchfunctions correspond to elements of X, hence term judgements with empty contextsrepresent elements of a space.
Finally for statement judgements we have to consider the structure of ⌃. Every topolog-ical space is equipped with a preorder called the specialization preorder, denoted by thebinary relation 6 (rather than ). The specialization preorder on ⌃is F 6 T , and on⌃X it is
f 6 g if and only if 8x 2 X.f(x) 6⌃
g(x)
The judgement� ` ↵ ) � : ⌃X states that the functions f and g which represent ↵ and�, respectively, satisfy f 6 g. The judgement �` ↵ , � : ⌃X similarly state that thefunctions f and g are equal.
The specialization preorder on any Hausdor↵ spaces is trivial, i.e. x 6 y implies x = y,so this preorder is not often seen in topology.
Aside. From the category theory viewpoint the category of topological spaces is anenriched category over preorders. This means that every homset carries a preorder, andfunction composition preserves this order. For the category of locally compact topologicalspaces the preorder is a partial order, and the category is enriched over posets.
Example 12. An example of a judgement is one for ✏-� continuity. Let f be the repre-sentation of a term
x : R ` f(x) : RClassically the function f is continuous at x if for all ✏ > 0 there exists a � > 0 suchthat for all y, if |x � y| < � then |f(x) � f(y)| < ✏. To convert this into the ASDcalculus we run into a few problems. First ✏ is given by quantifying over all positive realnumbers. However the space (0,1) or even R is not compact, so we cannot perform thisquantification. This can be fixed by converting it to a statement, but note that we areno longer allowed to use implication or negation in the rest of the term.
The existence of � is fine, but the quantification over the interval (x� �, x + �) leads toanother problem. We can fix this by quantifying over the compact interval [x� �, x+ �]instead. We end up with the judgement:
x : R, ✏ : R ` ✏ > 0 ) 9� : R. �> 0 ^ 8y : [x� �, x+ �]. |f(x)� f(y)| < ✏
One slight problem with this formulation is that the type [x � �, x + �] depends on theterms x and � but we have not included dependent types in the calculus. A futureadjustment of the calculus may allow such types, but for the cases of quantifying overbounded intervals in the reals we have the following translation: Given a predicate � oftype⌃ R we transform
8x : [a, b].� xinto the following acceptable form:
8x : [0, 1].�(ax+ b(1� x))
This translation is straightforward, so we prefer to use the version above.
The Logical Axioms of ASD
We have given the syntax of the calculus and how to interpret the syntax, so now weshow how to reason with the calculus. This is done through proof rules, which have thetwo forms
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Judgement 1Judgement 2
Judgement 1
Judgement 2
The left rule states that if Judgement 1 holds then we can assert that Judgement 2 holdsas well. The double line version states that if Judgement 1 holds then we can assert thatJudgement 2 holds, and vice-versa.
We start with the proof rules for propositions, which are the terms of type ⌃. We havethe connectives ^ and _ representing conjunction and disjunction.
Axiom 5. The logical connectives ^ and _ satisfy the following proof rules:
�, � , > ` ↵ ) �
� ` � ^ ↵ ) �
�, � , ? ` � ) ↵
� ` � ) � _ ↵
If we interpret the spaces as sets then the first rule is constructively valid, but the secondis not. This proof rule can be interpreted as the equivalence between ¬� ! (� ! ↵) and� ! (� _↵). If we take � to be true, and ↵ to be ¬�, then we get (� _¬�). However wecannot express negation in ASD, so we do not have the law of excluded middle. In factthis proof rule represents a relationship between topological properties of spaces. Whilethe second rule is not constructively valid if interpreted as sets, it is constructively validif we interpret spaces as locales.
Next we consider the logical properties of the space ⌃. We have seen that classicallythere are three open subsets of ⌃. We want to avoid mentioning subsets, so instead weconsider continuous functions⌃ ! ⌃. Classically these functions are determined by theirvalues on T and F . The Phoa principle represents this property, and is a major axiomin the system.
Axiom 6. The Phoa principle: Functions from ⌃ to ⌃ are determined by their valueson ? and >:
�, F : ⌃⌃, � : ⌃ ` F� , F? _ � ^ F>
This has the consequence that F is monotone: F? ) F>, therefore we cannot representa negation function.
Now we consider the equational axioms of the system. Many of the equality statementsare � or ⌘ rules for type constructors. See [3] chapter 7.2 for an overview of these typesof equalities in type theories.
Axiom 7. Equational axioms:
• Lattice structure: The predicates of a type⌃ X form a distributive lattice. Theconjunction of two predicates � and is the predicate �x.�x^ x and the definitionfor disjunction is similar. We have the following proof rules linking the specializationorder with these connectives:
�,�, : ⌃X` �)
�,�, : ⌃X` � ^ , �
�,�, : ⌃X` � _ ,
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• Application: The � and ⌘ rules hold for �-abstraction and application.
(�x.�)a = �[x := a] �x.�x = �
where the notation �[x := a] means that we have replaced all the free-variables xwith the term a.
• Projections: The � and ⌘ rules hold for projections and pairing:
⇡1
ha, bi = a,⇡2
ha, bi = b, and h⇡1
p,⇡2
pi = p
• Recursion: The � and ⌘ rules hold for recursion over N.
Now we will consider the logical axioms for discrete and Hausdor↵ spaces.
Definition 13. A space X is discrete if the left proof rule holds, and it is Hausdor↵ ifthe right proof rule holds:
� ` n = m : X� ` (n =X m) , > : ⌃
� ` n = m : X� ` (n 6=X m) , ? : ⌃
These proof rules allow us to pass from the equality/inequality predicate to an equalitystatement and vice-versa.
For compact and overt spaces we also require proof rules.
Definition 14. A space X is overt if the left proof rule holds, and it is compact if theright rule holds:
�, x : X ` �x ) �
� ` 9X.�x ) �
�, x : X ` � ) �x
� ` � ) 8X.�x
With these proof rules we can use the quantifiers in similar ways to how they are normallyused. For example, if we have the judgement ` 8x : X.�x , > : ⌃, and we have a termjudgement ` a : X, then we may assert ` �a , > : ⌃ .
In constructive mathematics if we have a term of the form 9x : X.�x then there exists aterm a : X such that �a holds. This result does not hold for ASD. Even in the term model,whose types and terms only consist of those that can be constructed from the axioms ofthe system, we do not have this property. However if the context � only consists of overtspaces then in the term model if� ` 9x : X.�x , > : ⌃ holds we can construct a terma such that� ` �a , > : ⌃ holds.
Furthermore, even if we prove ` �a , > : ⌃ for every term a of a compact space Xit does not imply that we can assert ` 8x : X.�x , > : ⌃, even in the term model.E↵ectively the spaces represented by the types in this system consist of more than theirdefinable elements. This corresponds with the localic point of view where spaces are notdetermined by their points.
Now we will move on to a numerical axiom. The Archimedean axiom prevents non-standard models of the rationals, and allows us to extract numerical results from thecalculus up to a given tolerance.
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Axiom 8. The spaces Q and R satisfy the Archimedean principle. For p and q of typeQ or R we have the judgement
q > 0 ) 9n : Z. q(n� 1) < p < q(n+ 1)
There are a few remaining axioms which we will not spend much time on, but they doplay an important role in ASD. The Scott continuity axiom is a topological one and isbased on the idea that continuous functions⌃ X ! ⌃ preserve directed unions.
Another collection of axioms involve the term focus which we have not mentioned. Thisterm is based on the monadic principle, and can be used to construct points of a spaceX from certain points of⌃ ⌃
X. For more details see the papers [4] and [5].
Local Compactness and Interval Arithmetic
The classical model of ASD is given by locally compact topological spaces, and a con-structive model is given by locally compact locales. The property of being locally compactallows us to perform some form of computation on the space. We start with the topolog-ical definition.
Definition 15. A topological space X is locally compact if for every x 2 X and everyopen subset U containing x, there exists an open subset V and a compact subset K suchthat
x 2 V ✓ K ✓ U ✓ X
A locally compact space is computably generated if the above open and compact subsetscan by computed by a computer program. For the precise definition of computablygenerated locally compact space see [6].
Example 16. For the real numbers we can take the open subsets to be open intervals withrational endpoints, that is the open intervals (a, b) with a, b 2 Q, and the correspondingcompact subset can then be taken to be [a, b]. As the open subsets of R are given byunions of open intervals, each x is inside some open interval (a, b). Now take rationalnumbers between a and x, and then another between x and b, to get an open intervalwith rational endpoints containing x.
The computable open subsets and compact subsets coming from local compactness willbe called cells, and will be denoted with a bold symbol like x. We will now convert thetopological definition of locally compact into an ASD statement. We assume that allspaces are computably generated, so the cells will be indexed by an overt discrete spaceN . For example, the cells of R can be indexed by Q⇥Q or even N.Local compactness states:
a : X,� : ⌃X ` �a , 9x : N. a 2 x ^ 8x : x.� x: ⌃where a 2 x means that a is in the open interval subspace corresponding to the cell x.In other words this statement says that a is in the open subset corresponding to � if andonly if there is a cell x such that a is in the open subset corresponding to that cell, andfor all elements x in the compact subset corresponding to the cell x, the element x is inthe open subset corresponding to �.
So how do we compute with locally compact spaces? The cells are indexed by an overtdiscrete space, so they can be represented on a computer. The locally compact property
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allows us to replace terms �a with the right hand side above, which initiates a search fora cell which satisfies the necessary conditions.
The condition a 2 x is observable, as x represents an open subset, so a non-deterministicsearch will find cells which satisfy this. The other condition, 8x : x.� x is a bit harderto satisfy. However, if the cell is chosen to be su�ciently small then this quantifier canbe replaced with a di↵erent term involving interval arithmetic, which can be verifiedby computer. Any predicate � can be transformed into one which only involves logicalconnectives, interval operations, and existential quantification over an overt discrete spacelike N. The transformed predicate does not involve quantification over R or [a, b]. Fordetails on this translation see [2].
The process of converting a term like � into an interval arithmetic program is involved,and one method requires a form of Prolog which includes non-deterministic branching, �-calculus and interval constraints. The running time of such a program is not yet known,but the calculus guarantees that the program will eventually halt if given a requiredprecision.
Comparison to Other Systems
There are a number of models of ASD, with the classical one being locally compacttopological spaces. Constructive locale theory and formal topology can also providemodels. Furthermore it is possible to use other meta-theories, like Bishop’s ConstructiveMathematics or Martin-Lof’s Type Theory to construct the term model of ASD.
First we consider the soundness and completeness of the calculus. This system is soundsince the axioms are derived from topological properties of locally compact spaces. Forcompleteness we need to consider the term model of ASD, which consists of only thetypes and terms which can be constructed from the axioms alone. The term model canbe characterised in terms of certain topological spaces:
Theorem 17 (Theorem 17.5 in [6]). The category of types and terms of the term modelof ASD is equivalent to the category of computably generated locally compact spaces andcomputable continuous functions.
This implies that any computably generated locally compact space can be representedby a type in ASD, and any computable continuous function can be represent by a term,up to homeomorphism. However, one slight problem with this theorem is that it requiresclassical logic, as it uses the proof that locally compact locales and locally compacttopological spaces agree.
Next we compare ASD to other systems which involve computation with real numbers.One such system is Recursive Analysis. A fundamental property of ASD is that the unitinterval is compact. This does not hold in Recursive Analysis due to the the existenceof singular covers. However ASD can be interpreted in Recursive Analysis, which mayseem to cause a problem. The reason why there is no di�culty is that the real numbersin ASD di↵er from the real numbers in the meta-theory. Interpreting the reals from ASDwill give a di↵erent object than the reals in Recursive Analysis.
Even though the two objects are di↵erent there still could be a problem with singularcovers. In the term model of ASD constructed in Recursive Analysis it is possible to definea sequence of intervals (dn, en) with rational endpoints, whose total length is bounded by1
2
, and it is possible to prove in the meta-theory that (Remark 15.4 in [1])
if ` t : [0, 1] then ` 9n : N. dn < t < en
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This would seem to say that every element of the unit interval is covered by one of theintervals in the sequence, therefore the unit interval is covered by intervals whose totallength is less than 1
2
. However it is not possible to pass from these judgements to
t : [0, 1] ` 9n : N. dn < t < bn
The proof in the meta-theory only talks about definable elements of the unit interval,but this is not enough in ASD. Like locale theory, the unit interval consists of more thanjust its definable elements. In this way singular covers do not cause any di�culties forASD, and there is no contradiction with a compact unit interval.
Conclusion
The logical calculus of Abstract Stone Duality is an interesting system from a logicalperspective as it involves a fairly weak logic. The use of implication and negation isseverely restricted, but we have seen that this is necessary to ensure continuity andcomputability. The quantifiers are also restricted to certain types, the compact and theovert types. Finally equality and inequality are also restricted, to the discrete and theHausdor↵ types, respectively.
These restricted types all correspond to various properties of spaces: discrete, Hausdor↵,compact, and overt. The overt spaces are not visible classically, as all topological spacesare overt. However in constructive settings overtness is a very useful property for aspace to have. In this calculus overtness embodies a computational principle. Due tothe computational properties of this system results can be computed and extracted, in aform which involves interval arithmetic.
For an example application of ASD to the Intermediate Value Theorem see the paper[7] where two versions of the Intermediate Value Theorem are given. One version iscomputational and the other is not. This is because the space of computable solutions isovert, whereas the space of non-computable solutions is not overt. The calculus has theability to distinguish between the two types of solutions.
Abstract Stone Duality is a prime example of a foundational system which involvesa restricted logic which reflects the domain that it models. Instead of using an all-encompassing system, which may have di�culties joining computation and continuity,a smaller system which involves only those principles which preserve computability andcontinuity may be a more appropriate system in various domains.
References
[1] Andrej Bauer and Paul Taylor. The dedekind reals in abstract stone duality. Mathe-matical Structures in Computer Science, 19(04):757–838, 2009.
[2] Paul Taylor. Interval analysis without intervals.http://www.paultaylor.eu/ASD/intawi/intawi.pdf.
[3] Paul Taylor. Practical foundations of mathematics, volume 59 of Cambridge Studiesin Advanced Mathematics. Cambridge University Press, Cambridge, 1999.
[4] Paul Taylor. Sober spaces and continuations. Theory and Applications of Categories,10:No. 12, 248300, 2002.
[5] Paul Taylor. Subspaces in abstract stone duality. Theory and Applications of Cate-gories, 10:No. 13, 301368, 2002.
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