Topology and Logic
Asymmetric Topology in Computer Science
Steve Matthews
Department of Computer Science
University of Warwick
Coventry, CV4 7AL, UK
[email protected]
with
Michael Bukatin (Nokia Corporation, Boston) and Ralph Kopperman (CCNY)
AMS Fall Central Sectional Meeting
University of Akron, Akron Ohio
October 20th-21st, 2012
Acknowledgments
I
May I thank the organisers of the Special Session on A
Survey of Lattice-Theory Mathematics and its Applications
for their kind invitation to speak today.
I
On the occasion of his 70th birthday may I acknowledge
the outstanding contribution and role model of Professor
Ralph Kopperman in the progress of asymmetric topology.
Abstract
Let us take topology to be the study of mathematical properties
preserved by continuous deformation, and logic to be the study
of truth through deductive reasoning in a formal language. As
continuity depends on the infinitary notion of limit, and
deductions are finite, there seems to be little commonality
between these subjects. But today’s pervasive influence of
computers in mathematics necessitates that topologists
understand how continuous deformation is to be programmed,
and conversely that computer programmers have more access
to topology to model their computations. Today’s talk will survey
how topology & logic have been steadily growing closer during
the past century, and then present our own research upon
developing a middle way for topology and logic.
Abstract
Contents
Metric spaces
Topological spaces
Logic via Topology (Scott)
Topology via Logic (Vickers)
Many valued truth logic
Non zero self distance
Asymmetry for metric spaces
Partial information
Negation as failure
Failure takes time
No such thing as a free lunch
Discrete partial versus fuzzy metrics
Conclusions and further work
Further reading
c David Kopperman
Metric Spaces
Definition (1)
A metric space (Fréchet, 1906) is a tuple
(X , d : X × X → [0, ∞)) such that,
d(x, x) = 0
d(x, y ) = 0 ⇒ x = y
d(x, y ) = d(y , x)
d(x, z) ≤ d(x, y ) + d(y , z)
Maurice Fréchet
(1878-1973)
Example (1)
| · − · | : (−∞, +∞)2 → [0, ∞) where |x − y | = x − y if
y ≤ x , y − x otherwise.
Metric Spaces
I
As with most mathematics there is an unquestioned
philosophical identification in the theory of metric spaces
between ontology and epistemology. That is, all that exists
in a metric space is presumed knowable (by examining
distances), and all that is knowable (by distance) is
presumed to exist.
I
Hence in the time of Fréchet it was eminently sensible to
axiomatize self distance by d(x, x) = 0 and
d(x, y ) = 0 ⇒ x = y .
I
The past century has taught us that some problems are
undecidable (i.e. there are more truths than are sound
reasoning of proofs), and more recently that fallacies (i.e.
believed but unsound reasoning) such as might arise in
large scale data processing can pass us by unnoticed.
I
How well prepared are metric and topological spaces for
the information age?
Metric Spaces
The ideal of logic for ascertaining perfect truth
in Star Trek’s portrayl of the 23rd century is a
wonderful caricature of how we appreciate
undecidable problems and fallacies in the
information age of the 20-21st centuries.
Captain Kirk arbitrates between his closest
confidants, the totally logical Mr Spock and
the passionate Dr McCoy.
Mr Spock’s reasoning is infallible but as a
result incomplete. Dr McCoy’s reasoning is
emotional, daring, wider reaching, but
as a result fallible.
Just as Kirk needs Spock & McCoy to explore
the galaxy so we need to bring together the
infallibility of logical reasoning with the
abstract expressiveness of (say) topology.
Topological Spaces
Definition (2)
A topological space is a pair (X , τ ⊆ 2X ) such that,
{} ∈ τ and X ∈ τ
∀A, B ∈ τ . A ∩ B ∈ τ (closure under finite intersections)
S
∀Ω ⊆ τ .
Ω ∈ τ (closure under arbitrary unions)
Each A ∈ τ is termed an open set of the topology τ , and
each complement X − A a closed set.
Definition (3)
For each topological space (X , τ ) a basis of τ is a Ω ⊆ τ
such that each member of τ is a union of members of Ω .
Example (2)
The finite open intervals (x, y ) are a basis for the usual
topology on (−∞, +∞) .
Topological Spaces
Definition (4)
For each metric space (X , d) , a ∈ X , and > 0
an open ball B (a) = {x ∈ A : d(x, a) < } .
Lemma (4)
For each metric space (X , d) the open balls form the basis for
a topology τd over X .
Ontology ≡ epistemology in the sense of Hausdorff separation
(i.e. T2 ),
a 6= b ⇒ ∃, δ . B (a) ∩ Bδ (b) = {}
(distinct points can be separated by disjoint neighbourhoods)
Example (3)
Let d be the metric on the set {F , T } of truth values false &
true such that d(F , T ) = 1 . Then τd = 2{F ,T } ,
and {{F }, {T }} is a basis for τd .
Topological Spaces
I
Approximation in point set topology by neighbourhoods is
approximation of a totally known limit point by totally known
open sets.
I
So, is not an open set such as an open ball
B (a) an approximation for each and every
point x ∈ B (a) ? The answer would be YES!
if ontology and epistemology of naive set
theory were consistent & synonymous.
I
Russell’s paradox of 1901 argues that
if we could define R = {x|x 6∈ x}
then R ∈ R ⇔ R 6∈ R .
I
Russell in 1916
This paradox (i.e. self contradiction) known to Russell et.al.
led to our understanding of incompleteness in logic and
computability theory.
Topological Spaces
I
And so Russell’s paradox of 1901 was subsequently
addressed through restriction to consistent truths (e.g.
typed sets) and exclusion of known contradictions.
I
Later large scale information processing such as second
world war codebreaking introduced the possibility of
contradictions unknowable & unavoidable in practice
although knowable in theory.
Topological spaces thus need the following development.
I
Neighbourhood Approximation (i.e. open sets)
⇒ Consistent Approximation (e.g. T0 separable)
⇒ Inconsistent Approximation (Topology with mistakes)
For some researchers topology has always been & will
always be the study of Hausdorff separable
neighbourhoods. Now we argue a case for the further
generalisation of topology through to inconsistency.
Topological Spaces
Definition (5)
A topological space (X , τ ⊆ 2X ) is T0 separable if
a 6= b ⇒ ∃O ∈ τ . (a ∈ O ∧ b 6∈ O) ∨ (b ∈ O ∧ a 6∈ O)
(distinct points can be separated by a neighbourhood)
Lemma (5)
T2 ⇒ T0
Inspired by Dana Scott in the 1960s asymmetric T0 separable
topology became meaningful.
Definition (6)
The information ordering (in the study of consistent
approximation) is,
a v b ⇔ ( ∀O ∈ τ . a ∈ O ⇒ b ∈ O )
Topological Spaces
I
a v b is read as compute up from a to b. This is a
model for consistent approximation. No mistakes are
possible, and there is no way to undo a computation. E.g.
there is no way to return from b down to a .
I
This model is consistent with the 1901 time
of paradoxes where contradictions could be
managed by exclusion. It was just viable in
the 1970s where fallible programmers of
mainframe computers could be made
responsible for managing their mistakes.
Now try telling an iPod user that they
can never make a mistake and there is no
such thing as an undo option in their
favourite app.
I
Sadly consistent approximation does not scale up to meet
today’s demands for inconsistency.
Topological Spaces
Hausdorff separability was taken for
granted until the 20th century story of
incompleteness, Bletchley Park code
breaking, and Computer Science
developed.
I
∀ x . x = x (identity)
I
∀ x, y . x = y ⇒ y = x (symmetry)
I
∀ x, y , z . x = y ∧ y = z ⇒ x = z (associativity)
Machine Room, Hut 6, 1943
T2 separability is consistent with a two valued logic of truth
(false and true). As a means of mathematically describing
many structures in our natural 3D world metric spaces remain a
powerful tool. However, increasingly within our everyday world
people need to be as aware of how such structures are to be
effectively represented within a computer as with what is their
inherent nature.
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
>
I
@
@
@
F
T
I
@
@
@⊥
> (pronounced top in lattice theory and both in four valued
logic) introduces the possibility of an overdetermined
mathematical value,
F , T (pronounced false, true) typifies two well defined
consistent distinct values in a mathematical theory.
⊥ (pronounced bottom in lattice theory and either in four
valued logic) introduces the possibility of
an underdetermined mathematical value.
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
>
I
@
@
@
T
F
I
@
@
@
⊥
> as computing power and apps grow so does the scope for
humans to enter and expect computers to (consistently)
process their inconsistent data.
F , T the traditional idealised realm of mathematics in which all
values, axioms, theorems, etc. are presumed to be
logically consistent.
⊥ perhaps the wait is finite, perhaps infinite, we just cannot
know how long if ever it will take to compute
the desired mathematical value.
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
>
I
@
@
@
F
T
I
@
@
@⊥
We assume that information is partially ordered.
⊥ @ F @ > and ⊥ @ T @ >.
F 6v T and T 6v F
x v y ⇒ f (x) v f (y )
(two valued logic is unchanged).
(functions are monotonic).
We envisage a vertical structure of information processing
orthogonal to a given horizontal mathematical
structure.
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
Definition (7)
A partially ordered set (poset) is a relation (X , v ⊆ X × X )
such that
x v x (reflexivity)
x v y ∧ y v x ⇒ x = y (antisymmetry)
x v y ∧ y v z ⇒ x v z (transitivity)
Definition (8)
For each poset (X , v) , (X , @) is the relation such that,
x @ y ⇔ x v y ∧ x 6= y .
Example (4)
The real numbers (−∞, ∞) are partially ordered by the relation
x v y iff x ≥ y .
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
Definition (9)
A lattice is a partially ordered set in which each pair of points
x, y has a unique greatest lower bound (aka infinum or meet)
denoted x u y , and unique lowest upper bound (aka
supremum or join) denoted x t y .
Example (5)
A set is a lattice when partially ordered by set inclusion. Infinum
is set intersection, and supremum is set union.
Example (6)
The extension of two valued truth logic from {F , T } to
{F , ⊥, >, T } where F u T = > and F t T = ⊥ is a lattice.
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
Definition (10)
A two argument function (in infix notation) op is symmetric if
x op y = y op x for all x and y .
Definition (11)
A two argument function (in infix notation) op is associative if
x op (y op z) = (x op y ) op z for all x, y , & z .
Example (7)
In a distributive lattice, meet & join exist, are symmetric, are
associative, and x t (y u z) = (x t y ) u (x t z) and
x u (y t z) = (x u y ) t (x u z) .
Definition (12)
A function f over a lattice is distributive if
f (x u y ) = (f x) u (f y ) and f (x t y ) = (f x) t (f y ) .
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
Many valued truth logic has an important role in the joint
progress of mathematics and computer science. Before the age
of computing, mathematics had to find its own way (for us
metric spaces). When computing gathered pace in the 1960s it
was from the presumption that mathematics was always to be
computed bottom-up upon a single machine architecture from
the consistent nothing of ⊥ . Now, in today’s world of parallel
network based computing, there is an additional increasingly
demanding necessity that pre-computed possibly inconsistent
information may arrive top-down from other sources (machine
or human) to be reconciled consistently with a mathematical
model below.
Truth table for negation.
P
⊥ F T >
¬P ⊥ T F >
Negation is monotonic and distributive.
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
Truth table for sequential and (computing left-to-right).
P∧Q ⊥ F T >
⊥
⊥ F ⊥ ⊥
F
⊥ F F F
T
⊥ F T >
>
⊥ F > >
Sequential and is monotonic, not symmetric as ⊥ ∧ F 6= F ∧ ⊥ ,
and ⊥ ∧ Q = ⊥ for each Q .
Truth table for parallel and (Belnap logic).
P∧Q ⊥ F T >
⊥
⊥ F ⊥ F
F
F F F F
T
⊥ F T >
>
F F > >
Parallel and is monotonic, symmetric, distributive,
and above sequential and.
Many Valued Truth Logic
Adding Unknown and Contradiction to Truth Valued Logic
I
And so we see that the structure of two valued truth logic
{F , T } can be generalised in such a way that renders it
more relevant to incorporate concepts which become both
obvious and necessary in the age of computing.
I
It is, in effect, sufficient to use our intuition to extend two
valued truth logic. Sadly it is more challenging to see how
to extend more sophisticated mathematical structures such
as metric spaces.
I
The insight here is to appreciate that the metric concept of
self distance could meaningfully (as in each of
mathematics and CS) be non-zero as well as zero.
Non zero self distance
Partial metric spaces
Definition (13)
A partial metric space (Matthews, 1992) is a tuple
(X , p : X × X → [0, ∞)) such that,
p(x, x) ≤ p(x, y ) (small self-distance)
p(x, x) = p(y , y ) = p(x, y ) ⇒ x = y
(indistancy implies equality)
p(x, y ) = p(y , x) (symmetry)
p(x, z) ≤ p(x, y ) + p(y , z) − p(y , y ) (triangularity)
Definition (14)
An open ball B (a) = {x ∈ A : p(x, a) < } .
Lemma (5)
The open balls form the basis for a topology. This is
asymmetric in the sense that there may be x, y such
that y ∈ cl{x} ∧ x 6∈ cl{y } (i.e. T0 separation).
Non zero self distance
Partial metric spaces
Definition (15)
For each partial metric space (X , p) the partial ordering is
x vp y ⇔ p(x, x) = p(x, y ) .
Lemma (6)
A metric space is precisely a partial metric space for which
each self distance is 0 . In such a space the partial ordering is
equality.
Since the introduction of partial metric spaces other equivalent
or alternative formulations have been suggested.
Non zero self distance
Partial metric spaces
Definition (16)
A weighted metric space is a tuple
(X , d, | · | : X → (−∞, ∞)) such that (X , d) is a metric
space.
Lemma (7)
If (X , d, | · |) is a weighted metric space then
p(x, y ) =
d(x, y ) + |x| + |y |
2
is a partial metric such that p(x, x) = |x| for each x ∈ X .
If (X , p) is a partial metric space and
dp (x, y ) = 2 × p(x, y ) − p(x, x) − p(y , y ) , |x|p = p(x, x)
then (X , dp , | · |p ) is a weighted metric space.
Non zero self distance
Partial metric spaces
After introducing non zero self distance to metric spaces the
question of negative distance arises. Suppose we now allow
distance to be negative. Then we can introduce the following
partial metric for many valued truth logic.
p(>, >) = −1
p(F , F ) = p(F , >) = p(T , T ) = p(T , >) = 0
p(F , T ) =
1
2
p(⊥, ⊥) = p(⊥, F ) = p(⊥, >) = p(⊥, T ) = 1
Note: we choose this particular partial metric in order that the
induced weighted metric space has the familiar metric
dp (F , T ) = 1 .
Non zero self distance
Partial metric spaces
I
A partial metric space (X , p) generalises Fréchet’s
landmark notion of metric space (X , d) through
generalising the notion of self-distance.
I
The initial motivation was provided by computer science
requiring the notion of partial ordering & T0 separability
for computable functions. However, this was too weak for
in comparison to pre-CS mathematics which could afford
the luxury of assuming their topological spaces to be
Hausdorff separable (i.e. T2 ).
I
While it is true that CS motivated non zero self-distance
and partial metric spaces could there be other interesting
applications & interpretations of this research?
Non zero self distance
Partial metric spaces
Definition (17)
A based metric space is a tuple (X , d, φ ∈ X ) such that
(X , d) is a metric space.
That is, a based metric space is a metric space with an
arbitrarily chosen base point.
By defining |x| = d(x, φ) we make a weighted metric space
(X , d, | · |) in which φ = > and |φ| = 0 .
Alternatively, assuming d to be bounded above by some a ,
define |x| = a − d(x, φ) to make a weighted metric space
(X , d, | · |) in which φ = ⊥ and |φ| = a .
A potential use of based metric spaces in CS is in computer
games, where both a 3D space and the view of that space from
any point in space has to be modelled.
Partial Information
Scott Domain Theory
I
x vp y iff p(x, x) = p(x, y ) models the computing
notion that the data content of x can be increased to that
of y .
I
Very relevant to classical problems of recursion in logic,
computability theory, and programming language design of
the 1960s. But, these problems are now resolved, and
language design today faces very different challenges.
I
The what truth of x vp y is in general insufficient to
determine the how algorithm used to compute from x
to y .
I
With the benefit of hindsight it is easy to argue that the
asymmetric topology of domain theory could not progress
beyond its roots of modelling the what of computation.
Partial Information
Scott Domain Theory
>
I
@
@
@
T
F
I
@
@
@⊥
In the least fixed point domain theory tradition of Kleene, Tarski,
& Scott ¬T = F , ¬F = T , ¬⊥ = ⊥ , ¬> = > , and
tn≥0 ¬n (⊥) can be used to define the ideal meaning for ψ
recursively defined by ψ = ¬ψ .
Partial Information
Scott Domain Theory
>
I
@
@
@
T
F
I
@
@
@⊥
BUT! ψ 6= ⊥ in the real world of computing as any attempt to
compute the value of ψ would in some finite time time out
(whatever this may mean) with an error message such as
Control Stack Overflow. Thus while the above least fixed point
theory is fundamental to the nature of computing it is far from
being the whole story.
Partial Information
Scott Domain Theory
In the functional programming language
Haskell we can write the definition
for negation in two valued logic as follows.
data Bool = True | False
not :: Bool -> Bool
not True = False
not False = True
psi :: Bool
psi = not psi
From many artistic works it is clear that
recursion (and computable loops in general)
can only be finitely known. What then is
the cost of knowledge?
Jeff Crouse "Recursion shirt"
Partial Information
Scott Domain Theory
I
Meanwhile, at Warwick’s Centre for Discrete
Mathematics & its Applications (DIMAP)
algorithms have truly flourished as a
vigorous form of combinatorics, a how branch
of totally defined mathematics for the study
of countable data structures.
DIMAP logo
I
For example, a finite directed graph is studied to determine
its shortest path.
I
Like it or not the reality is that DIMAP has no need for
asymmetric topology and partial information.
I
However, this need not imply the demise of domain theory.
Rather, we ask how can domain theory reach out to
contemporary computing concepts?
Partial Information
Scott Domain Theory
Number 6: Where am I?
Number 2: In the Village.
Number 6: What do you want?
Number 2: We want information.
Number 6: Whose side are you on?
Number 2: That would be telling.
We want information... information...
information.
Number 6: You won’t get it.
Number 2: By hook or by crook, we will.
Number 6: Who are you?
Number 2: The new Number 2.
Number 6: Who is Number 1?
Number 2: You are Number 6.
Number 6: I am not a number,
I am a free man!
Number 2: [Sinister laughing] ...
(www.netreach.net/~sixofone)
The Prisoner (1967)
Patrick McGoohan (1928-2009)
Partial Information
Static domain theory
Number 6: Where am I?
Number 2: In DIMAP.
Number 6: What do you want?
Number 2: We want algorithms.
Number 6: Whose side are you on?
Number 2: That would be telling.
We want algorithms... algorithms...
algorithms.
Number 6: You won’t get it.
Number 2: By hook or by crook, we will.
Number 6: Who are you?
Number 2: The new Number 2.
Number 6: Who is Number 1?
Number 2: You are Number 6.
Number 6: I am not a number,
I am a discrete domain theorist!
Number 2: [Sinister laughing] ...
The Prisoner (1967)
www.sixofone.co.uk
Partial Information
Scott Domain Theory
I
As a mathematics undergraduate of metric & topological
spaces in 1976-7 at Imperial College I became a Prisoner
of T2 costless separation.
I
Trying to become a free mind of PhD repute in Computer
Science 1980-3 at Warwick I was still a Prisoner of the
misnomer that the cost of producing each x in domain
theory could be incorporated within the emerging notion of
non zero self distance for metric spaces.
I
In the 1960s dynamic, interactive, adaptive, or intelligent
systems were few, while in today’s world of the web and
cloud computing there are few that are not.
I
And so, how can the concepts of non zero self distance &
asymmetric topology of partial metric spaces avoid the
same fate of domain theory?
Partial Information
Scott Domain Theory
I
Introducing the notion of cost to computing a partial metric
distance p(x, y ) is an important initial step in developing a
future theory of metric spaces that may be termed
dynamic, interactive, adaptive, or intelligent.
I
For example, in classical logic we want to retain the double
negative elimination theorem ¬¬A ↔ A while in addition
asserting that ¬¬A is more costly to compute than A .
I
Applying this idea to our earlier definition ψ = ¬ψ would
mean that we could retain the ideal Kleene, Tarski, & Scott
domain theory meaning tn≥0 ¬n (⊥) of least fixed points
and use cost as a criteria for defining an error Control
Stack Overflow.
Negation as failure
Non-monotonic reasoning
"I went to Stanford to study for a PhD
in Mathematics, but my real interest
was Logic. I was still looking to find
the truth, and I was sure that Logic
would be the key to finding it. My best
course was axiomatic set theory with
Dana Scott. He gave us lots of theorems
to prove as homework. At first my marks
were not very impressive. But Dana Scott
marked the coursework himself, and wrote
lots of comments. My marks improved
significantly as a result."
Dana Scott (photo by
B. Otrecht, 1969)
from Robert Kowalski: A Short Story of My Life and Work
April 2002
Bob Kowalski
Negation as failure
Non-monotonic reasoning
I
Negation as failure (NAF) is a non-monotonic inference
rule in logic programming used to derive a truth for a
formula not p from failure to derive the truth of p .
I
not p may be different from ¬p depending upon the
completeness of the inference algorithm and that of the
formal logic system assumed.
I
"Although it is in general not complete,
its chief advantage is the efficiency of
its implementation. Using it the deductive
retrieval of information can be regarded
as a computation" (Negation as Failure,
Keith L. Clark, Logic and Databases
1978, p. 113-141).
Keith L. Clark
Negation as failure
Non-monotonic reasoning
I
My sincerest apologies to Professor Kowalski who taught
me an inspiring introduction to logic programming in Prolog
at Imperial College 1979-80. Grounded in the costless
idealism of complete logic and T2 separation of metric
spaces I totally misunderstood the practical necessity and
wisdom for NAF in computing.
I
Looking back may I now offer the following thoughts upon
NAF. (1) There is an implicit reasonable assumption that
programmers use intelligent algorithms that for all intents
& purposes progress computationally. Thus complete
logic is a fine ideal, but is not a practical necessity in
computing. (2) Failure can be monotonic.
Failure takes time
Monotonic reasoning
Although ¬¬⊥ = ⊥ in domain theory (as ¬¬A ↔ A in logic)
we require a model for which the cost of computing ¬⊥ is
discernibly greater than that of computing ⊥ . Consider the
following sequence p0 , p1 , . . . of partial metrics.
pn (T , T )
pn (F , F )
pn (F , T )
pn (T , ⊥)
pn (F , ⊥)
pn (⊥, ⊥)
=
=
=
=
=
=
p ppp p p p
ppp ppppp
p ppppppp
2−n
2−n
2−n + 2−1
2−n + 1
2−n + 1
2−n + 1
F
ppppp
pppppp
@
I
@
ppppp
pppppp
ppppppppppp
pppp
pppppp
ppppp
pppp
pppp
p p pp pp pp p p
p
p
p
p
p
ppppp
T
pn (⊥, ⊥)
@
@
@
@
⊥
p is monotonic in the sense that each pn is a partial metric
having the usual domain theory ordering ⊥ v F ∧ ⊥ v T , and
∀x, y ∀0 ≤ n < m . pn (x, y ) > pm (x, y )
Failure takes time
Monotonic reasoning
I
Let our cost function | · | : X → (ω → <) be such that
|x|n = pn (x, x) .
I
Let |¬x|n = |x|n+1 be our way of defining the cost of
computing negation. Then we can reason
|¬F |n = |T |n+1 , |¬T |n = |F |n+1 , and |¬⊥|n = |⊥|n+1 .
I
Suppose now we define (for the purposes of this example)
that a formula x fails if from n = 0 a cost |x|n is
computed for some n ≥ ] where ] < ∞ is the maximum
cost allowed. Then for our earlier example ψ = ¬ψ we
can reason ∀]∃n ≥ ] . |ψ|n . That is, we can reason
monotonically in finite time that the data content of ψ will
not rise above ⊥ , an approximation of its ideal denotation.
Failure takes time
Monotonic reasoning
I
Suppose 2 < ] . Then |¬¬T |0 = |¬F |1 = |T |2 where
we totally compute our result before reaching failure. Thus
¬¬T is discernible from T by means of cost.
I
The reality of the lack of completeness for any realistic
logic of computation implies that any algorithm for defining
failure will include the possibility of some cost |x|n being
necessary to compute x where ] < n . If only ] had
been bigger is our feeble lament. This is an example of
where the ideal of logical completeness must be
substituted by the weaker heuristic of intelligent algorithms.
I
In contrast to NAF we do not reason non-monotonically to
interpret a failure to prove some p for the logical falsity of
p . Our approach is to progress correctly monotonically
with partial information as far as is discernibly feasible.
Failure takes time
Into the abyss
I
Wadge is much respected for his PhD
UC Berkeley research known as the
Wadge hierarchy, levels of complexity
for sets of reals in descriptive set theory.
Bill Wadge
I
Wadge’s later insight that a complete object is "one that cannot
be further completed" led from metric spaces (of complete
objects), to Lucid (for programming over metric spaces), to
partial metric spaces (domain theory for metric spaces), and now
to discrete partial metric spaces (complexity theory for domains).
I
"I don’t know if infinitesimal logic is the best idea I’ve ever had,
but it’s definitely the best name. So here’s the idea: a
multivalued logic in which there are truth values that are not
nearly as true as ’standard’ truth, and others that are not nearly
as false as ’standard’ falsity" (Bill Wadge’s blog, 3/2/11).
Failure takes time
Into the abyss
Wadge appreciated that "When you look for long into an abyss, the
abyss also looks into you" (attributed to Friedrich Nietzsche).
"When we discovered the dataflow interpretation of Lucid
(see post, Lucid the dataflow language) we thought we’d
found the promised land. We had an attractive computing
model that was nevertheless faithful to the declarative
semantics of statements-as-equations. However, there was
a snake in paradise, as David May explained in the fateful
meeting in the Warwick Arts Center Cafeteria. ... And it’s
the need to discard data that leads to serious trouble. It
could be that a huge amount of resources were required to
produce ..., resources that were wasted because we threw
it out. But a real catastrophe happens if the data to be
discarded requires infinite resources. Then we wait forever
for a result that is irrelevant, and the computation
deadlocks." (Bill Wadge’s blog, 6/12/11)
Failure takes time
Into the abyss
I
In a demand driven (as opposed to data driven)
programming language some potential catastrophes are
never encountered. But, for the usual decidability reasons
in computation and incompleteness of logic, not all
catastrophes can be so avoided.
I
Our monotonic treatment of Failure takes time may thus be
partially correct in the sense of domain theory, but is it so
weak as to be useless in practice?
I
A sequence p0 , p1 , . . . of consistent partial metrics as just
described is an interesting step forward, but hardly a
computable notion of partial metric. That is, is there a
notion of partial metric that can express the best and worst
of computation?
Failure takes time
Into the abyss
Definition (18)
A discrete partial metric space is a set {(X , pm )|m ≥ 0}
of related partial metric spaces in which evolving self distances
can be associated to represent computational costs defined by
an intelligent form of discrete mathematics.
Example (8)
Let {(X , pm : X × X → {a0m , a1m , a2m , . . . })|m ≥ 0} be such
that,
m
∀m, n ≥ 0 . anm > an+1
0
1
0
∀n ≥ 0 . an > an > an2 > . . . > an+1
0
0 ) × 2−m
∀m, n ≥ 0 . anm = an+1
+ (an0 − an+1
Then {(X , pm )|m ≥ 0} is a discrete partial metric space.
Failure takes time
Into the abyss
I
Suppose x0 v x1 v x2 v . . . to be a domain theory
chain of approximations. Then we have their (data)
content weights |xi | = p(xi , xi ) for each i ≥ 0 .
I
The computational abyss observed by Wadge can pass by
unnoticed in domain theory. Suppose that for either a finite
or infinite period an xi (and hence |xi | ) remain constant
as i increases. Then this has no significance in domain
theory, but may or may not be a computational catastrophe.
I
That is, when computational resources (including human
intelligence) just happen to be sufficient no one notices or
cares about (what may we term) Wadge’s abyss , but
when they are not a catastrophe may well arise.
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With discrete partial metric spaces we can now do more to
express Wadge’s abyss within domain theory.
Failure takes time
Into the abyss
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Suppose that xi v xi+1 for some i in our domain theory
chain, and that xi = xi+1 . Then there is no data increase
from xi to xi+1 , but it is reasonable to assume a
computational pause of whatever kind does take place
(which may or may not terminate).
I
Suppose our domain theory chain to be definable by a
discrete partial metric space (X , p : X × X → {a0 , . . . }) .
Then there exists j ≥ 0 such that |xi | = aj .
I
Now assume the pause from xi to xi+1 to be of length
m > 0 . Then the cost of producing the value xi+1 can
be defined as ajm
Failure takes time
Into the abyss
I
Interestingly the partial metric notion of size
|xi | = p(xi , xi ) (which is fixed for all time) now becomes
dynamic. From xi it is rarely predictable what resources
(affordably finite, unaffordably finite, or infinite) are needed
to compute xi+1 .
I
What we can do with a discrete partial metric is to
dynamically keep track of a chain of values,
aj0 = ajm0 > ajm1 > ajm2 >
...
as long as is affordably finite and until (if ever) the value
xi+1 is computed.
I
Thus Wadge’s abyss is observable by means of keeping
track of a chain (i.e. x ) of chains; bringing domain theory
a step closer to the complexity of algorithms.
Failure takes time
Into the abyss
I
So far in our work we have omitted Wadge’s earlier notion
of complete object in Lucid, the Dataflow Programming
Language. In the sense of partial metrics a completely
defined (data content) object is precisely one whose
self-distance is zero. It has proved to be a very useful
assumption for contraction mapping style fixed point
theorems over partial metrics.
I
Our work today upon Wadge’s abyss uses the range
[0, ∞) for counting both increasing data content and
increasing cost. In our cost-conscious research "one that
cannot be further completed" may turn out to be an
unaffordable computation.
I
We have yet to consider how contraction mapping
theorems might be formed for discrete partial metric
spaces.
Failure takes time
Into the abyss
Dana Scott’s inspired work has given
us the T0 topology to model partial
information. But, a computable
function could still be an impossible
object as depicted in Escher’s
Waterfall which appears to produce
an unending flow of water.
Equivalent to the paradox
of the Penrose Triangle
impossible object.
The supposed paradox disappears
in discrete partial metric spaces
Waterfall
where (dynamic) cost can be
Lithograph by M.C. Escher (1961)
composed with (static) data content.
Failure takes time
Hiatons
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Ashcroft & Wadge (1977) introduced a programming
language called Lucid in which each input or output is a
discrete (i.e. finite or countably infinite) sequence of data
values. In domain theory terms,
h i @ hai @ ha, bi @ ha, b, ci @ ha, b, c, di @ . . .
I
Let p(x, y ) = 2−n where n is the largest integer such
that ∀i < n . xi = yi . Then p is a partial metric inducing
the above ordering.
I
An unavoidable implication here is that each data value in
a sequence takes the same amount of time to input (resp.
output).
Failure takes time
Hiatons
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Wadge & Ashcroft (1985) tried to add a pause to Lucid.
I
For example, the following would-be ’sequence’ tries to add
pauses termed hiatons to static domain theory.
h∗, 2, 3, ∗, 5, ∗, 7, ∗, ∗, ∗, 11, . . . i
I
But ∗ is neither a well defined data value (such as 0) nor
is say h∗, 2i a partial value comparable to h2i in
domain theory. So, what is a hiaton ? Frustratingly the
temporal intuition of a pause appears to be sound, but will
not fit into domain theory.
Failure takes time
Hiatons
Time
0
1
2
3
4
5
6
7
...
Data
hi
v h1i
v h1i
v h1,
v h1,
v h1,
v h1,
v h1,
...
Hiatons
hi
h1i
h1, ∗i
2i
h1, ∗, 2i
2i
h1, ∗, 2,
2, 3i h1, ∗, 2,
2, 3i h1, ∗, 2,
2, 3i h1, ∗, 2,
...
∗i
∗,
∗,
∗,
Discrete Partial Metric
2−1 + (2−0 − 2−1 ) × 2−0
2−2 + (2−1 − 2−2 ) × 2−0
2−2 + (2−1 − 2−2 ) × 2−1
2−3 + (2−2 − 2−3 ) × 2−0
2−3 + (2−2 − 2−3 ) × 2−1
3i
2−4 + (2−3 − 2−4 ) × 2−0
3, ∗i
2−4 + (2−3 − 2−4 ) × 2−1
3, ∗, ∗i 2−4 + (2−3 − 2−4 ) × 2−2
...
As Escher’s Waterfall paradox is resolved by separating time
from data so we resolve the ambiguity of each hiaton by making
it a time only concept.
Failure takes time
Hiatons
The above example of a data sequence, an associated domain
theory, and hiatons are all observable from a single discrete
partial metric.
Data
Discrete Partial Metric
hi
2−1 + (2−0 − 2−1 ) × 2−0
v h1i
2−2 + (2−1 − 2−2 ) × 2−1
v h1, 2i
2−3 + (2−2 − 2−3 ) × 2−1
v h1, 2, 3i 2−4 + (2−3 − 2−4 ) × 2−2
...
...
Now though this discrete partial metric is non deterministically
discovered at the run time of computation, not in general
knowable in advance determined by only the properties of data.
Note how the ontology of data is consistent with but (in
general) less than our epistemology of computation.
Failure takes time
Hiatons
The embedding retraction relationship from (a suitable) partial
metric to a discrete partial metric works as follows.
an → an0 = an
anm → danm e = an
an is (as with a partial metric) a predictable distance for partially
defined values, but one that can now dynamically change at the
run time of computation to pass through the observable
discrete distances anm0 , anm1 , . . . > an+1 .
No such thing as a free lunch
Moore’s Law ?
Colossus (1944) was invented to decipher
encrypted messages too complex for humans
to manage by hand. Ironically computers
soon became too powerful to be valued by
humans. It is as if we are forever going
round in circles, the functionalities of
computers sooner or later are overtaken by
the reality of their cost. Why let ourselves
become trapped in this pointless infinite loop?
Why not ensure our mathematics is always
in one-to-one correspondence with its cost?
The traditional answer is Moore’s Law,
which states that the number of transistors
which can be placed inexpensively upon an
integrated circuit doubles every two years.
According to Moore’s Law there is a free
lunch in computing! Really?
"In 1994, a team led by Tony Sale (right)
began a reconstruction of a Colossus
computer at Bletchley Park" (Wikipedia)
Steve Jobs launches the iPad tablet (2010).
No such thing as a free lunch
Moore’s Law ?
Even if Moore’s Law accurately describes the reality of hardware
development there appears to be little correlation with human
demand upon computers, which oscillates between seeing machines
as a free lunch and asking the impossible.
Now it’s official! "The current programme of information and
communications technology (ICT) study in England’s schools will be
scrapped from September, the education secretary has announced. It
will be replaced by an ’open source’ curriculum in computer science
and programming designed with the help of universities and industry."
http://www.bbc.co.uk/news/education-16493929 (11/1/12)
The same report contains the following quote from Ian Livingstone
(an advisor to the UK government’s education secretary) "Children
are being forced to learn how to use applications, rather than to make
them. They are becoming slaves to the user interface and are totally
bored by it, ..."
No such thing as a free lunch
Moore’s Law ?
The UK has a challenging economic imperative to rescue it’s fading
heritage as a leading creative nation in the world of science and
engineering. The supposed post-Thatcherite position is that each
person should pay their own way through education, and the profit
motive of capitalism should drive one’s interests (academic fees are
typically GBP 9, 000 per year from 2012-13). Are we dear friends,
experts in logic and mathematics, really to believe that our research
can be determined by a single economic theory?
No such thing as a free lunch
Moore’s Law ?
"Freegans are people who employ alternative
strategies for living based on limited
participation in the conventional economy
and minimal consumption of resources.
Freegans embrace community, generosity,
social concern, freedom, cooperation,
and sharing in opposition to a society
based on materialism, moral apathy,
competition, conformity, and greed. ...
Freeganism is a total boycott of an
economic system where the profit motive
has eclipsed ethical considerations and
where massively complex systems of
productions ensure that all the products
we buy will have detrimental impacts
most of which we may never even consider."
(freegan.info)
integral-options.blogspot.com
/2007/01/scavenging-to-liveupdated.html
"Freegans rummage through trash bags
outside Jefferson Market on 6th Ave"
(posted by T.M. Meinch on The Trowel)
Discrete Partial versus Fuzzy Metrics
I
A longstanding interesting question is in what sense can
we say fuzzy metric is synonomous with partial metric?
I
There is (as I understand) no presumption that fuzzy
distance is necessarily computable distance. That is, fuzzy
distance could be ontologically sound but epistemologically
incomplete.
I
Thus discrete partial metric strengthens our notion of fuzzy
distance where fuzzy is primarily an epistemological
concept, but weakens a primarily ontological concept.
I
If fuzzy can ignore the cost of epistemology then there
seems to be little connection between fuzzy metric and
partial metric.
Discrete Partial versus Fuzzy Metrics
I
This brings us back to where we began. Cost is axiomatic
in our studies.
I
Cost is consistent with but stronger than computability.
I
Our approach is to derive cost from domain theory, in
contrast to complexity theory which is traditionally defined
in combinatorics a total separate from any denotational
partial model.
I
Partial metrics are hard to justify in either the total world of
metric topology or in the combinatorics of complexity
theory, but where reconciling the two is axiomatic they
have yet to be explored in detail.
Conclusions and further work
Topology and logic
I
Computer Science has become
an overly specialised, selfish,
highly stressful, career path.
I resign! Now I am a Prisoner
in research and teaching.
I
We need more compassion for
creative time on discrete domain
theory, not publish or perish mania.
I
Computability, topology, partiality,
& complexity theory can all be
reconciled. How?
I
Are discrete partial metric spaces
a way forward? Or, is Wadge’s
infinitesimal logic better?
Episode 1 of The Prisoner.
Arrival. The man angrily resigns.
c /dcs/acad/sgm/dpm/BuddhaOf
Conclusions and further work
Topology and logic
"It has long been my personal view that the separation of
practical and theoretical work is artificial and injurious. Much of
the practical work done in computing, both in software and in
hardware design, is unsound and clumsy
because the people who do it have not any
clear understanding of the fundamental
design principles of their work. Most of the
abstract mathematical and theoretical work
is sterile because it has no point of contact
with real computing. One of the central
aims of the Programming Research Group
as a teaching and research group has
Christopher Strachey (1916-75)
been to set up an atmosphere in which
Founder of the Oxford Programming
Research Group (1965)
this separation cannot happen."
Further reading
I
E.A. Ashcroft & W.W. Wadge. Lucid, a Nonprocedural
Language with Iteration, Communications of the ACM, pp.
519-526, Vol 20, No 7, July 1977.
I
M. Bukatin, R. Kopperman, S. Matthews & H. Pajoohesh.
Partial Metric Spaces, American Mathematical Monthly,
Volume 116 Number 8, pp 708-718, October 2009.
I
W. Byers. How Mathematicians Think. Using Ambiguity,
Contradiction, and Paradox to Create Mathematics,
Princeton University Press, 2007 (paperback, 2010).
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Martin Campbell-Kelly. Christopher Strachey, 1916-75 A
Biographical Note. Annals of the History of Computing,
Volume 7, Number 1, January 1985.
...
Further reading
...
I
R.D. Kopperman. Asymmetry and duality in topology,
Topology Appl. 66 (1995) 1-39.
I
Bob Kowalski. www.doc.ic.ac.uk/~rak/
I
Dana S. Scott. Outline of a mathematical theory of
computation. Technical Monograph PRG-2, Oxford
University Computing Laboratory, Oxford, England,
November 1970.
I
S. Vickers. Topology Via Logic, Cambridge Tracts in
Theoretical Computer Science 5, 1989.
I
W.W. Wadge & E.A. Ashcroft. Lucid, the Dataflow
Programming Language, APIC Studies in Data Processing
No 22, 1985.