Topology and Logic - University of Warwick · led to our understanding of incompleteness in logic and computability theory. 11/69. Topological Spaces I And so Russell’s paradox

Topology and LogicAsymmetric Topology in Computer Science

Steve Matthews

Department of Computer ScienceUniversity of Warwick

Coventry, CV4 7AL, [email protected]

withMichael Bukatin (Nokia Corporation, Boston) and Ralph Kopperman (CCNY)

AMS Fall Central Sectional MeetingUniversity of Akron, Akron Ohio

October 20th-21st, 2012

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Acknowledgments

I May I thank the organisers of the Special Session on ASurvey of Lattice-Theory Mathematics and its Applicationsfor their kind invitation to speak today.

I On the occasion of his 70th birthday may I acknowledgethe outstanding contribution and role model of ProfessorRalph Kopperman in the progress of asymmetric topology.

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Abstract

Let us take topology to be the study of mathematical propertiespreserved by continuous deformation, and logic to be the studyof truth through deductive reasoning in a formal language. Ascontinuity depends on the infinitary notion of limit, anddeductions are finite, there seems to be little commonalitybetween these subjects. But today’s pervasive influence ofcomputers in mathematics necessitates that topologistsunderstand how continuous deformation is to be programmed,and conversely that computer programmers have more accessto topology to model their computations. Today’s talk will surveyhow topology & logic have been steadily growing closer duringthe past century, and then present our own research upondeveloping a middle way for topology and logic.

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Abstract

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Contents

Metric spacesTopological spacesLogic via Topology (Scott)Topology via Logic (Vickers)Many valued truth logicNon zero self distanceAsymmetry for metric spacesPartial informationNegation as failureFailure takes timeNo such thing as a free lunchDiscrete partial versus fuzzy metricsConclusions and further workFurther reading

c© David Kopperman

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Metric Spaces

Maurice Fréchet(1878-1973)

Definition (1)A metric space (Fréchet, 1906) is a tuple(X , d : X × X → [0,∞)) such that,

d(x , x) = 0d(x , y) = 0 ⇒ x = yd(x , y) = d(y , x)

d(x , z) ≤ d(x , y) + d(y , z)

Example (1)| · − · | : (−∞,+∞)2 → [0,∞) where |x − y | = x − y ify ≤ x , y − x otherwise.

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Metric SpacesI As with most mathematics there is an unquestioned

philosophical identification in the theory of metric spacesbetween ontology and epistemology. That is, all that existsin a metric space is presumed knowable (by examiningdistances), and all that is knowable (by distance) ispresumed to exist.

I Hence in the time of Fréchet it was eminently sensible toaxiomatize self distance by d(x , x) = 0 andd(x , y) = 0⇒ x = y .

I The past century has taught us that some problems areundecidable (i.e. there are more truths than are soundreasoning of proofs), and more recently that fallacies (i.e.believed but unsound reasoning) such as might arise inlarge scale data processing can pass us by unnoticed.

I How well prepared are metric and topological spaces forthe information age?

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Metric Spaces

The ideal of logic for ascertaining perfect truthin Star Trek’s portrayl of the 23rd century is awonderful caricature of how we appreciateundecidable problems and fallacies in theinformation age of the 20-21st centuries.

Captain Kirk arbitrates between his closestconfidants, the totally logical Mr Spock andthe passionate Dr McCoy.

Mr Spock’s reasoning is infallible but as aresult incomplete. Dr McCoy’s reasoning isemotional, daring, wider reaching, butas a result fallible.

Just as Kirk needs Spock & McCoy to explorethe galaxy so we need to bring together theinfallibility of logical reasoning with theabstract expressiveness of (say) topology.

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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)∀Ω ⊆ τ .

⋃Ω ∈ τ (closure under arbitrary unions)

Each A ∈ τ is termed an open set of the topology τ , andeach complement X − A a closed set.

Definition (3)For each topological space (X , τ) a basis of τ is a Ω ⊆ τsuch 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 usualtopology on (−∞,+∞) .

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Topological Spaces

Definition (4)For each metric space (X ,d) , a ∈ X , and ε > 0an open ball Bε(a) = x ∈ A : d(x ,a) < ε .

Lemma (4)For each metric space (X , d) the open balls form the basis fora topology τd over X .Ontology ≡ epistemology in the sense of Hausdorff separation(i.e. T2),

a 6= b ⇒ ∃ε, δ . Bε(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 = 2F ,T ,and F, T is a basis for τd .

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Topological Spaces

I Approximation in point set topology by neighbourhoods isapproximation of a totally known limit point by totally knownopen sets.

I So, is not an open set such as an open ballBε(a) an approximation for each and everypoint x ∈ Bε(a) ? The answer would be YES!if ontology and epistemology of naive settheory were consistent & synonymous.

I Russell’s paradox of 1901 argues thatif we could define R = x |x 6∈ xthen R ∈ R ⇔ R 6∈ R . Russell in 1916

I This paradox (i.e. self contradiction) known to Russell et.al.led to our understanding of incompleteness in logic andcomputability theory.

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Topological Spaces

I And so Russell’s paradox of 1901 was subsequentlyaddressed through restriction to consistent truths (e.g.typed sets) and exclusion of known contradictions.

I Later large scale information processing such as secondworld war codebreaking introduced the possibility ofcontradictions unknowable & unavoidable in practicealthough knowable in theory.

I Topological spaces thus need the following development.Neighbourhood Approximation (i.e. open sets)

⇒ Consistent Approximation (e.g. T0 separable)⇒ Inconsistent Approximation (Topology with mistakes)

For some researchers topology has always been & willalways be the study of Hausdorff separableneighbourhoods. Now we argue a case for the furthergeneralisation of topology through to inconsistency.

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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 separabletopology became meaningful.

Definition (6)The information ordering (in the study of consistentapproximation) is,

a v b ⇔ ( ∀O ∈ τ . a ∈ O ⇒ b ∈ O )

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Topological Spaces

I a v b is read as compute up from a to b. This is amodel for consistent approximation. No mistakes arepossible, 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 timeof paradoxes where contradictions could bemanaged by exclusion. It was just viable inthe 1970s where fallible programmers ofmainframe computers could be maderesponsible for managing their mistakes.Now try telling an iPod user that theycan never make a mistake and there is nosuch thing as an undo option in theirfavourite app.

I Sadly consistent approximation does not scale up to meettoday’s demands for inconsistency.

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Topological Spaces

Hausdorff separability was taken forgranted until the 20th century story ofincompleteness, Bletchley Park codebreaking, and Computer Sciencedeveloped.

Machine Room, Hut 6, 1943I ∀ x . x = x (identity)I ∀ x , y . x = y ⇒ y = x (symmetry)I ∀ x , y , z . x = y ∧ y = z ⇒ x = z (associativity)

T2 separability is consistent with a two valued logic of truth(false and true). As a means of mathematically describingmany structures in our natural 3D world metric spaces remain apowerful tool. However, increasingly within our everyday worldpeople need to be as aware of how such structures are to beeffectively represented within a computer as with what is theirinherent nature.

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Many Valued Truth LogicAdding Unknown and Contradiction to Truth Valued Logic

>

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@I

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F

T

⊥

> (pronounced top in lattice theory and both in four valuedlogic) introduces the possibility of an overdeterminedmathematical value,

F ,T (pronounced false, true) typifies two well definedconsistent distinct values in a mathematical theory.

⊥ (pronounced bottom in lattice theory and either in fourvalued logic) introduces the possibility ofan underdetermined mathematical value.

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>

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@I

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@I

F

T

⊥

> as computing power and apps grow so does the scope forhumans to enter and expect computers to (consistently)process their inconsistent data.

F ,T the traditional idealised realm of mathematics in which allvalues, axioms, theorems, etc. are presumed to belogically consistent.

⊥ perhaps the wait is finite, perhaps infinite, we just cannotknow how long if ever it will take to computethe desired mathematical value.

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>

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@I

@@

@I

F

T

⊥

We assume that information is partially ordered.⊥ @ F @ > and ⊥ @ T @ >.F 6v T and T 6v F (two valued logic is unchanged).x v y ⇒ f (x) v f (y) (functions are monotonic).

We envisage a vertical structure of information processingorthogonal to a given horizontal mathematical

structure.

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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 relationx v y iff x ≥ y .

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Definition (9)A lattice is a partially ordered set in which each pair of pointsx , y has a unique greatest lower bound (aka infinum or meet)denoted x u y , and unique lowest upper bound (akasupremum or join) denoted x t y .

Example (5)A set is a lattice when partially ordered by set inclusion. Infinumis set intersection, and supremum is set union.

Example (6)The extension of two valued truth logic from F ,T toF ,⊥,>,T where F u T = > and F t T = ⊥ is a lattice.

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Definition (10)A two argument function (in infix notation) op is symmetric ifx op y = y op x for all x and y .

Definition (11)A two argument function (in infix notation) op is associative ifx 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, areassociative, and x t (y u z) = (x t y) u (x t z) andx u (y t z) = (x u y) t (x u z) .

Definition (12)A function f over a lattice is distributive iff (x u y) = (f x) u (f y) and f (x t y) = (f x) t (f y) .

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Many valued truth logic has an important role in the jointprogress of mathematics and computer science. Before the ageof computing, mathematics had to find its own way (for usmetric spaces). When computing gathered pace in the 1960s itwas from the presumption that mathematics was always to becomputed bottom-up upon a single machine architecture fromthe consistent nothing of ⊥ . Now, in today’s world of parallelnetwork based computing, there is an additional increasinglydemanding necessity that pre-computed possibly inconsistentinformation may arrive top-down from other sources (machineor human) to be reconciled consistently with a mathematicalmodel below.

Truth table for negation.P ⊥ F T >¬P ⊥ T F >

Negation is monotonic and distributive.22 / 69


Truth table for sequential and (computing left-to-right).P∧Q ⊥ F T >⊥ ⊥ F ⊥ ⊥F ⊥ F F FT ⊥ 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 ⊥ FF F F F FT ⊥ F T >> F F > >

Parallel and is monotonic, symmetric, distributive,and above sequential and.

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I And so we see that the structure of two valued truth logicF ,T can be generalised in such a way that renders itmore relevant to incorporate concepts which become bothobvious and necessary in the age of computing.

I It is, in effect, sufficient to use our intuition to extend twovalued truth logic. Sadly it is more challenging to see howto extend more sophisticated mathematical structures suchas metric spaces.

I The insight here is to appreciate that the metric concept ofself distance could meaningfully (as in each ofmathematics and CS) be non-zero as well as zero.

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Non zero self distancePartial 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 Bε(a) = x ∈ A : p(x ,a) < ε .

Lemma (5)The open balls form the basis for a topology. This isasymmetric in the sense that there may be x , y suchthat y ∈ clx ∧ x 6∈ cly (i.e. T0 separation).

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Definition (15)For each partial metric space (X , p) the partial ordering isx vp y ⇔ p(x , x) = p(x , y) .

Lemma (6)A metric space is precisely a partial metric space for whicheach self distance is 0 . In such a space the partial ordering isequality.

Since the introduction of partial metric spaces other equivalentor alternative formulations have been suggested.

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Definition (16)A weighted metric space is a tuple(X , d , | · | : X → (−∞,∞)) such that (X , d) is a metricspace.

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.27 / 69


After introducing non zero self distance to metric spaces thequestion of negative distance arises. Suppose we now allowdistance to be negative. Then we can introduce the followingpartial metric for many valued truth logic.

p(>,>) = −1p(F ,F ) = p(F ,>) = p(T ,T ) = p(T ,>) = 0p(F ,T ) = 1

2

p(⊥,⊥) = p(⊥,F ) = p(⊥,>) = p(⊥,T ) = 1

Note: we choose this particular partial metric in order that theinduced weighted metric space has the familiar metricdp(F ,T ) = 1 .

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I A partial metric space (X , p) generalises Fréchet’slandmark notion of metric space (X , d) throughgeneralising the notion of self-distance.

I The initial motivation was provided by computer sciencerequiring the notion of partial ordering & T0 separabilityfor computable functions. However, this was too weak forin comparison to pre-CS mathematics which could affordthe luxury of assuming their topological spaces to beHausdorff separable (i.e. T2 ).

I While it is true that CS motivated non zero self-distanceand partial metric spaces could there be other interestingapplications & interpretations of this research?

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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 anarbitrarily 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 computergames, where both a 3D space and the view of that space fromany point in space has to be modelled.

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Partial InformationScott Domain Theory

I x vp y iff p(x , x) = p(x , y) models the computingnotion that the data content of x can be increased to thatof y .

I Very relevant to classical problems of recursion in logic,computability theory, and programming language design ofthe 1960s. But, these problems are now resolved, andlanguage design today faces very different challenges.

I The what truth of x vp y is in general insufficient todetermine the how algorithm used to compute from xto y .

I With the benefit of hindsight it is easy to argue that theasymmetric topology of domain theory could not progressbeyond its roots of modelling the what of computation.

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>

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@I

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F

T

⊥In the least fixed point domain theory tradition of Kleene, Tarski,& Scott ¬T = F , ¬F = T , ¬⊥ = ⊥ , ¬> = > , andtn≥0 ¬n(⊥) can be used to define the ideal meaning for ψrecursively defined by ψ = ¬ψ .

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>

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T

⊥BUT! ψ 6= ⊥ in the real world of computing as any attempt tocompute the value of ψ would in some finite time time out(whatever this may mean) with an error message such asControl Stack Overflow. Thus while the above least fixed pointtheory is fundamental to the nature of computing it is far frombeing the whole story.

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In the functional programming languageHaskell we can write the definitionfor negation in two valued logic as follows.

data Bool = True | False

not :: Bool -> Boolnot True = Falsenot False = True

psi :: Bool

psi = not psi

Jeff Crouse "Recursion shirt"

From many artistic works it is clear thatrecursion (and computable loops in general)can only be finitely known. What then isthe cost of knowledge?

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DIMAP logo

I Meanwhile, at Warwick’s Centre for DiscreteMathematics & its Applications (DIMAP)algorithms have truly flourished as avigorous form of combinatorics, a how branchof totally defined mathematics for the studyof countable data structures.

I For example, a finite directed graph is studied to determineits shortest path.

I Like it or not the reality is that DIMAP has no need forasymmetric topology and partial information.

I However, this need not imply the demise of domain theory.Rather, we ask how can domain theory reach out tocontemporary computing concepts?

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The Prisoner (1967)

Patrick McGoohan (1928-2009)

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)

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www.netreach.net/~sixofone

Partial InformationStatic domain theory

The Prisoner (1967)

www.sixofone.co.uk

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] ...

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www.sixofone.co.uk


I As a mathematics undergraduate of metric & topologicalspaces in 1976-7 at Imperial College I became a Prisonerof T2 costless separation.

I Trying to become a free mind of PhD repute in ComputerScience 1980-3 at Warwick I was still a Prisoner of themisnomer that the cost of producing each x in domaintheory could be incorporated within the emerging notion ofnon zero self distance for metric spaces.

I In the 1960s dynamic, interactive, adaptive, or intelligentsystems were few, while in today’s world of the web andcloud 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 thesame fate of domain theory?

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I Introducing the notion of cost to computing a partial metricdistance p(x , y) is an important initial step in developing afuture theory of metric spaces that may be termeddynamic, interactive, adaptive, or intelligent.

I For example, in classical logic we want to retain the doublenegative elimination theorem ¬¬A ↔ A while in additionasserting that ¬¬A is more costly to compute than A .

I Applying this idea to our earlier definition ψ = ¬ψ wouldmean that we could retain the ideal Kleene, Tarski, & Scottdomain theory meaning tn≥0 ¬n(⊥) of least fixed pointsand use cost as a criteria for defining an error ControlStack Overflow.

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Negation as failureNon-monotonic reasoning

"I went to Stanford to study for a PhDin Mathematics, but my real interestwas Logic. I was still looking to findthe truth, and I was sure that Logicwould be the key to finding it. My bestcourse was axiomatic set theory withDana Scott. He gave us lots of theoremsto prove as homework. At first my markswere not very impressive. But Dana Scottmarked the coursework himself, and wrotelots of comments. My marks improvedsignificantly as a result."

from Robert Kowalski: A Short Story of My Life and WorkApril 2002

Dana Scott (photo byB. Otrecht, 1969)

Bob Kowalski

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I Negation as failure (NAF) is a non-monotonic inferencerule in logic programming used to derive a truth for aformula not p from failure to derive the truth of p .

I not p may be different from ¬p depending upon thecompleteness of the inference algorithm and that of theformal logic system assumed.

I "Although it is in general not complete,its chief advantage is the efficiency ofits implementation. Using it the deductiveretrieval of information can be regardedas a computation" (Negation as Failure,Keith L. Clark, Logic and Databases1978, p. 113-141).

Keith L. Clark

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I My sincerest apologies to Professor Kowalski who taughtme an inspiring introduction to logic programming in Prologat Imperial College 1979-80. Grounded in the costlessidealism of complete logic and T2 separation of metricspaces I totally misunderstood the practical necessity andwisdom for NAF in computing.

I Looking back may I now offer the following thoughts uponNAF. (1) There is an implicit reasonable assumption thatprogrammers use intelligent algorithms that for all intents& purposes progress computationally. Thus completelogic is a fine ideal, but is not a practical necessity incomputing. (2) Failure can be monotonic.

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Failure takes timeMonotonic reasoning

Although ¬¬⊥ = ⊥ in domain theory (as ¬¬A ↔ A in logic)we require a model for which the cost of computing ¬⊥ isdiscernibly greater than that of computing ⊥ . Consider thefollowing sequence p0,p1, . . . of partial metrics.

Tp p p p p p p

@@@

@@@I

Fppppppp

pppppppppppppppppppppppppppppppppppppppppp

ppp p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p p p p p p p p p pp p p

pn(⊥,⊥)

⊥

pn(T ,T ) = 2−n

pn(F ,F ) = 2−n

pn(F ,T ) = 2−n + 2−1

pn(T ,⊥) = 2−n + 1pn(F ,⊥) = 2−n + 1pn(⊥,⊥) = 2−n + 1

p is monotonic in the sense that each pn is a partial metrichaving the usual domain theory ordering ⊥ v F ∧ ⊥ v T , and∀x , y ∀0 ≤ n < m . pn(x , y) > pm(x , y)

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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 ofcomputing 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 iscomputed for some n ≥ ] where ] <∞ is the maximumcost allowed. Then for our earlier example ψ = ¬ψ wecan reason ∀]∃n ≥ ] . |ψ|n . That is, we can reasonmonotonically in finite time that the data content of ψ willnot rise above ⊥ , an approximation of its ideal denotation.

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I Suppose 2 < ] . Then |¬¬T |0 = |¬F |1 = |T |2 wherewe 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 realisticlogic of computation implies that any algorithm for definingfailure will include the possibility of some cost |x |n beingnecessary to compute x where ] < n . If only ] hadbeen bigger is our feeble lament. This is an example ofwhere the ideal of logical completeness must besubstituted by the weaker heuristic of intelligent algorithms.

I In contrast to NAF we do not reason non-monotonically tointerpret a failure to prove some p for the logical falsity ofp . Our approach is to progress correctly monotonicallywith partial information as far as is discernibly feasible.

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Failure takes timeInto the abyss

Bill Wadge

I Wadge is much respected for his PhDUC Berkeley research known as theWadge hierarchy, levels of complexityfor sets of reals in descriptive set theory.

I Wadge’s later insight that a complete object is "one that cannotbe further completed" led from metric spaces (of completeobjects), to Lucid (for programming over metric spaces), topartial metric spaces (domain theory for metric spaces), and nowto 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: amultivalued logic in which there are truth values that are notnearly as true as ’standard’ truth, and others that are not nearlyas false as ’standard’ falsity" (Bill Wadge’s blog, 3/2/11).

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Wadge appreciated that "When you look for long into an abyss, theabyss 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’dfound the promised land. We had an attractive computingmodel that was nevertheless faithful to the declarativesemantics of statements-as-equations. However, there wasa snake in paradise, as David May explained in the fatefulmeeting in the Warwick Arts Center Cafeteria. ... And it’sthe need to discard data that leads to serious trouble. Itcould be that a huge amount of resources were required toproduce ..., resources that were wasted because we threwit out. But a real catastrophe happens if the data to bediscarded requires infinite resources. Then we wait foreverfor a result that is irrelevant, and the computationdeadlocks." (Bill Wadge’s blog, 6/12/11)

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I In a demand driven (as opposed to data driven)programming language some potential catastrophes arenever encountered. But, for the usual decidability reasonsin computation and incompleteness of logic, not allcatastrophes can be so avoided.

I Our monotonic treatment of Failure takes time may thus bepartially correct in the sense of domain theory, but is it soweak as to be useless in practice?

I A sequence p0,p1, . . . of consistent partial metrics as justdescribed is an interesting step forward, but hardly acomputable notion of partial metric. That is, is there anotion of partial metric that can express the best and worstof computation?

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Definition (18)A discrete partial metric space is a set (X ,pm)|m ≥ 0of related partial metric spaces in which evolving self distancescan be associated to represent computational costs defined byan intelligent form of discrete mathematics.

Example (8)Let (X , pm : X × X → am

0 ,am1 ,a

m2 , . . . )|m ≥ 0 be such

that,

∀m,n ≥ 0 . amn > am

n+1∀n ≥ 0 . a0

n > a1n > a2

n > . . . > a0n+1

∀m,n ≥ 0 . amn = a0

n+1 + (a0n − a0

n+1)× 2−m

Then (X , pm)|m ≥ 0 is a discrete partial metric space.

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I Suppose x0 v x1 v x2 v . . . to be a domain theorychain 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 byunnoticed in domain theory. Suppose that for either a finiteor infinite period an xi (and hence |xi | ) remain constantas i increases. Then this has no significance in domaintheory, but may or may not be a computational catastrophe.

I That is, when computational resources (including humanintelligence) just happen to be sufficient no one notices orcares about (what may we term) Wadge’s abyss , butwhen they are not a catastrophe may well arise.

I With discrete partial metric spaces we can now do more toexpress Wadge’s abyss within domain theory.

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I Suppose that xi v xi+1 for some i in our domain theorychain, and that xi = xi+1 . Then there is no data increasefrom xi to xi+1 , but it is reasonable to assume acomputational pause of whatever kind does take place(which may or may not terminate).

I Suppose our domain theory chain to be definable by adiscrete 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 lengthm > 0 . Then the cost of producing the value xi+1 canbe defined as am

j

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I Interestingly the partial metric notion of size|xi | = p(xi , xi) (which is fixed for all time) now becomesdynamic. From xi it is rarely predictable what resources(affordably finite, unaffordably finite, or infinite) are neededto compute xi+1 .

I What we can do with a discrete partial metric is todynamically keep track of a chain of values,

a0j = am0

j > am1j > am2

j > . . .

as long as is affordably finite and until (if ever) the valuexi+1 is computed.

I Thus Wadge’s abyss is observable by means of keepingtrack of a chain (i.e. x ) of chains; bringing domain theorya step closer to the complexity of algorithms.

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I So far in our work we have omitted Wadge’s earlier notionof complete object in Lucid, the Dataflow ProgrammingLanguage. In the sense of partial metrics a completelydefined (data content) object is precisely one whoseself-distance is zero. It has proved to be a very usefulassumption for contraction mapping style fixed pointtheorems over partial metrics.

I Our work today upon Wadge’s abyss uses the range[0,∞) for counting both increasing data content andincreasing cost. In our cost-conscious research "one thatcannot be further completed" may turn out to be anunaffordable computation.

I We have yet to consider how contraction mappingtheorems might be formed for discrete partial metricspaces.

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Dana Scott’s inspired work has givenus the T0 topology to model partialinformation. But, a computablefunction could still be an impossibleobject as depicted in Escher’sWaterfall which appears to producean unending flow of water.

Equivalent to the paradoxof the Penrose Triangleimpossible object.

The supposed paradox disappearsin discrete partial metric spaceswhere (dynamic) cost can becomposed with (static) data content.

WaterfallLithograph by M.C. Escher (1961)

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Failure takes timeHiatons

I Ashcroft & Wadge (1977) introduced a programminglanguage called Lucid in which each input or output is adiscrete (i.e. finite or countably infinite) sequence of datavalues. In domain theory terms,

〈〉 @ 〈a〉 @ 〈a, b〉 @ 〈a, b, c〉 @ 〈a, b, c, d〉 @ . . .

I Let p(x , y) = 2−n where n is the largest integer suchthat ∀i < n . xi = yi . Then p is a partial metric inducingthe above ordering.

I An unavoidable implication here is that each data value ina sequence takes the same amount of time to input (resp.output).

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I 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.

〈∗, 2, 3, ∗, 5, ∗, 7, ∗, ∗, ∗, 11, . . . 〉

I But ∗ is neither a well defined data value (such as 0) noris say 〈∗, 2〉 a partial value comparable to 〈2〉 indomain theory. So, what is a hiaton ? Frustratingly thetemporal intuition of a pause appears to be sound, but willnot fit into domain theory.

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Time Data Hiatons Discrete Partial Metric0 〈〉〈〉 2−1 + (2−0 − 2−1)× 2−0

1 v 〈1〉〈1〉 2−2 + (2−1 − 2−2)× 2−0

2 v 〈1〉〈1, ∗〉 2−2 + (2−1 − 2−2)× 2−1

3 v 〈1, 2〉〈1, ∗, 2〉 2−3 + (2−2 − 2−3)× 2−0

4 v 〈1, 2〉〈1, ∗, 2, ∗〉 2−3 + (2−2 − 2−3)× 2−1

5 v 〈1, 2, 3〉〈1, ∗, 2, ∗, 3〉 2−4 + (2−3 − 2−4)× 2−0

6 v 〈1, 2, 3〉〈1, ∗, 2, ∗, 3, ∗〉 2−4 + (2−3 − 2−4)× 2−1

7 v 〈1, 2, 3〉〈1, ∗, 2, ∗, 3, ∗, ∗〉 2−4 + (2−3 − 2−4)× 2−2

. . . . . . . . . . . .

As Escher’s Waterfall paradox is resolved by separating timefrom data so we resolve the ambiguity of each hiaton by makingit a time only concept.

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The above example of a data sequence, an associated domaintheory, and hiatons are all observable from a single discretepartial metric.

Data Discrete Partial Metric〈〉 2−1 + (2−0 − 2−1)× 2−0

v 〈1〉 2−2 + (2−1 − 2−2)× 2−1

v 〈1, 2〉 2−3 + (2−2 − 2−3)× 2−1

v 〈1, 2, 3〉 2−4 + (2−3 − 2−4)× 2−2

. . . . . .

Now though this discrete partial metric is non deterministicallydiscovered at the run time of computation, not in generalknowable in advance determined by only the properties of data.Note how the ontology of data is consistent with but (ingeneral) less than our epistemology of computation.

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The embedding retraction relationship from (a suitable) partialmetric to a discrete partial metric works as follows.

an → a0n = an

amn → dam

n e = an

an is (as with a partial metric) a predictable distance for partiallydefined values, but one that can now dynamically change at therun time of computation to pass through the observablediscrete distances am0

n , am1n , . . . > an+1 .

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No such thing as a free lunchMoore’s Law ?

Colossus (1944) was invented to decipherencrypted messages too complex for humansto manage by hand. Ironically computerssoon became too powerful to be valued byhumans. It is as if we are forever goinground in circles, the functionalities ofcomputers sooner or later are overtaken bythe reality of their cost. Why let ourselvesbecome trapped in this pointless infinite loop?Why not ensure our mathematics is alwaysin one-to-one correspondence with its cost?The traditional answer is Moore’s Law,which states that the number of transistorswhich can be placed inexpensively upon anintegrated circuit doubles every two years.According to Moore’s Law there is a freelunch in computing! Really?

"In 1994, a team led by Tony Sale (right)began a reconstruction of a Colossuscomputer at Bletchley Park" (Wikipedia)

Steve Jobs launches the iPad tablet (2010).

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Even if Moore’s Law accurately describes the reality of hardwaredevelopment there appears to be little correlation with humandemand upon computers, which oscillates between seeing machinesas a free lunch and asking the impossible.

Now it’s official! "The current programme of information andcommunications technology (ICT) study in England’s schools will bescrapped from September, the education secretary has announced. Itwill be replaced by an ’open source’ curriculum in computer scienceand 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) "Childrenare being forced to learn how to use applications, rather than to makethem. They are becoming slaves to the user interface and are totallybored by it, ..."

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http://www.bbc.co.uk/news/education-16493929


The UK has a challenging economic imperative to rescue it’s fadingheritage as a leading creative nation in the world of science andengineering. The supposed post-Thatcherite position is that eachperson should pay their own way through education, and the profitmotive of capitalism should drive one’s interests (academic fees aretypically GBP 9,000 per year from 2012-13). Are we dear friends,experts in logic and mathematics, really to believe that our researchcan be determined by a single economic theory?

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"Freegans are people who employ alternativestrategies for living based on limitedparticipation in the conventional economyand minimal consumption of resources.Freegans embrace community, generosity,social concern, freedom, cooperation,and sharing in opposition to a societybased on materialism, moral apathy,competition, conformity, and greed. ...Freeganism is a total boycott of aneconomic system where the profit motivehas eclipsed ethical considerations andwhere massively complex systems ofproductions ensure that all the productswe buy will have detrimental impactsmost of which we may never even consider."(freegan.info)

integral-options.blogspot.com/2007/01/scavenging-to-live-updated.html

"Freegans rummage through trash bagsoutside Jefferson Market on 6th Ave"

(posted by T.M. Meinch on The Trowel)

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freegan.info

integral-options.blogspot.com

/2007/01/scavenging-to-live-

updated.html

Discrete Partial versus Fuzzy Metrics

I A longstanding interesting question is in what sense canwe say fuzzy metric is synonomous with partial metric?

I There is (as I understand) no presumption that fuzzydistance is necessarily computable distance. That is, fuzzydistance could be ontologically sound but epistemologicallyincomplete.

I Thus discrete partial metric strengthens our notion of fuzzydistance where fuzzy is primarily an epistemologicalconcept, but weakens a primarily ontological concept.

I If fuzzy can ignore the cost of epistemology then thereseems to be little connection between fuzzy metric andpartial metric.

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Discrete Partial versus Fuzzy Metrics

I This brings us back to where we began. Cost is axiomaticin 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 definedin combinatorics a total separate from any denotationalpartial model.

I Partial metrics are hard to justify in either the total world ofmetric topology or in the combinatorics of complexitytheory, but where reconciling the two is axiomatic theyhave yet to be explored in detail.

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Conclusions and further workTopology and logic

Episode 1 of The Prisoner.Arrival. The man angrily resigns.

c© /dcs/acad/sgm/dpm/BuddhaOfCompassion.org

I Computer Science has becomean overly specialised, selfish,highly stressful, career path.I resign! Now I am a Prisonerin research and teaching.

I We need more compassion forcreative time on discrete domaintheory, not publish or perish mania.

I Computability, topology, partiality,& complexity theory can all bereconciled. How?

I Are discrete partial metric spacesa way forward? Or, is Wadge’sinfinitesimal logic better?

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/dcs/acad/sgm/dpm/BuddhaOfCompassion.org

Conclusions and further workTopology and logic

"It has long been my personal view that the separation ofpractical and theoretical work is artificial and injurious. Much ofthe practical work done in computing, both in software and inhardware design, is unsound and clumsybecause the people who do it have not anyclear understanding of the fundamentaldesign principles of their work. Most of theabstract mathematical and theoretical workis sterile because it has no point of contactwith real computing. One of the centralaims of the Programming Research Groupas a teaching and research group hasbeen to set up an atmosphere in whichthis separation cannot happen."

Christopher Strachey (1916-75)Founder of the Oxford ProgrammingResearch Group (1965)

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Further reading

I E.A. Ashcroft & W.W. Wadge. Lucid, a NonproceduralLanguage 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).

I Martin Campbell-Kelly. Christopher Strachey, 1916-75 ABiographical Note. Annals of the History of Computing,Volume 7, Number 1, January 1985.

. . .

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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 ofcomputation. Technical Monograph PRG-2, OxfordUniversity Computing Laboratory, Oxford, England,November 1970.

I S. Vickers. Topology Via Logic, Cambridge Tracts inTheoretical Computer Science 5, 1989.

I W.W. Wadge & E.A. Ashcroft. Lucid, the DataflowProgramming Language, APIC Studies in Data ProcessingNo 22, 1985.

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www.doc.ic.ac.uk/~rak/

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