Changes

===2.2.  Differential Geometry and Logic Programming===

===2.2.  Differential Geometry and Logic Programming===

In this section I make a quick reconnaissance of the border areas between logic and geometry, charting a beeline for selected trouble spots.  In the following sections I return to more carefully survey the grounds needed to address these problems and to begin settling this frontier.

In this section I make a quick reconnaissance of the border areas between

logic and geometry, charting a beeline for selected trouble spots.  In the

following sections I return to more carefully survey the grounds needed to

address these problems and to begin settling this frontier.

====2.2.1.  Differences and Difficulties====

====2.2.1.  Differences and Difficulties====

Why have I chosen differential geometry and logic programming to try jamming together? A clue may be picked up in the quotation below.  When the foundations of that ingenious duplex, AI and cybernetics, were being poured, one who was present placed these words in a cornerstone of the structure (Ashby, 1956, p. 9).

Why have I chosen differential geometry and logic programming to try jamming together?

A clue may be picked up in the quotation below.  When the foundations of that ingenious

duplex, AI and cybernetics, were being poured, one who was present placed these words

in a cornerstone of the structure (Ashby, 1956, p. 9).

| The most fundamental concept in cybernetics is that of "difference",

| either that two things are recognisably different or that one thing

The most fundamental concept in cybernetics is that of "difference", either that two things are recognisably different or that one thing has changed with time.

| has changed with time.

A deliberate continuity of method extends from this use of difference in

A deliberate continuity of method extends from this use of difference in goal-seeking behavior to the baby steps of AI per se, namely, the use of difference-reduction methods in the form of what is variously described as means-ends analysis, goal regression, or general problem solving.

goal-seeking behavior to the baby steps of AI per se, namely, the use of

difference-reduction methods in the form of what is variously described

as means-ends analysis, goal regression, or general problem solving.

=====2.2.1.1.  Distance and Direction=====

=====2.2.1.1.  Distance and Direction=====

Legend tells us that the primal twins of AI, the strife-born siblings of Goal-Seeking and Hill-Climbing, began to stumble and soon came to grief on certain notorious obstacles.  The typical scenario runs as follows.

Legend tells us that the primal twins of AI, the strife-born siblings of

Goal-Seeking and Hill-Climbing, began to stumble and soon came to grief

on certain notorious obstacles.  The typical scenario runs as follows.

    At any moment in time the following question is posed:

    In this problem space how ought one choose to operate

At any moment in time the following question is posed:<br>

    in order to forge of one's current state a new update

In this problem space how ought one choose to operate<br>

    that has hopes of being nearer to one's engoaled fate?

in order to forge of one's current state a new update<br>

that has hopes of being nearer to one's engoaled fate?

But before Jack and Jill can start up the hill they will need

But before Jack and Jill can start up the hill they will need a whole bucket of prior notions to prime the pump.  There must be an idea of distance, in short, a metric function defined on pairs of states in the problem space.  There must be an idea of direction, a longing toward a goal that informs the moment, that fixes a relation of oriented distances to transition operators on states.  Stated in linguistic terms the directive is a factor that commands and instructs.  It arranges a form of interpretation that endows disparities with a particular sense of operational meaning.

a whole bucket of prior notions to prime the pump.  There must

be an idea of distance, in short, a metric function defined on

pairs of states in the problem space.  There must be an idea of

direction, a longing toward a goal that informs the moment, that

fixes a relation of oriented distances to transition operators on

states.  Stated in linguistic terms the directive is a factor that

commands and instructs.  It arranges a form of interpretation that

endows disparities with a particular sense of operational meaning.

Intelligent systems do not get to prescribe the problem spaces that will

Intelligent systems do not get to prescribe the problem spaces that will be thrown their way by nature, society, and the outside world in general. These nominal problems would hardly constitute problems if this were the case.  Thus it pays to consider how intelligent systems might evolve to cast ever wider nets of competence in the spaces of problems that they can handle.  Striving to adapt the differential strategies of classical cybernetics and of early AI to "soaring" new heights (Newell, 1990), to widening gyres of ever more general problem spaces, there comes a moment when the predicament thickens but the atmosphere of theory and the wings of artifice do not.

be thrown their way by nature, society, and the outside world in general.

These nominal problems would hardly constitute problems if this were the

case.  Thus it pays to consider how intelligent systems might evolve to

cast ever wider nets of competence in the spaces of problems that they

can handle.  Striving to adapt the differential strategies of classical

cybernetics and of early AI to "soaring" new heights (Newell, 1990), to

widening gyres of ever more general problem spaces, there comes a moment

when the predicament thickens but the atmosphere of theory and the wings

of artifice do not.

=====2.2.1.2.  Topology and Metric=====

=====2.2.1.2.  Topology and Metric=====

Topology is the most unconstrained study of spaces, beginning as it does with spaces that have barely enough hope of geometric structure to deserve the name of spaces (Kelley, 1961).  An attention to this discipline inspires caution against taking too lightly the issue of a metric.  There is no longer any reason to consider the question of a metric to be a trivial one, something whose presence and character can be taken for granted.  For each space that can be contemplated there arises a typical suite of questions about the existence and the uniqueness of a possible metric.  Some spaces are not metrizable at all (Munkres, sec. 2-9).  Those that are may have a multitude of different metrics defined on them.  My own sampling of differential methods in AI, both smooth and chunky style, suggests to me that this multiplicity of possible metrics is the ingredient that conditions one of their chief sticking points, a computational viscosity that consistently sticks in the craw of computers.  Unpalatable if not intractable, it will continue to gum up the works, at least until some way is found to dissolve the treacle of complexity that downs our best theories.

Topology is the most unconstrained study of spaces, beginning as it does

with spaces that have barely enough hope of geometric structure to deserve

the name of spaces (Kelley, 1961).  An attention to this discipline inspires

caution against taking too lightly the issue of a metric.  There is no longer

any reason to consider the question of a metric to be a trivial one, something

whose presence and character can be taken for granted.  For each space that can

be contemplated there arises a typical suite of questions about the existence

and the uniqueness of a possible metric.  Some spaces are not metrizable at

all (Munkres, sec. 2-9).  Those that are may have a multitude of different

metrics defined on them.  My own sampling of differential methods in AI,

both smooth and chunky style, suggests to me that this multiplicity of

possible metrics is the ingredient that conditions one of their chief

sticking points, a computational viscosity that consistently sticks

in the craw of computers.  Unpalatable if not intractable, it will

continue to gum up the works, at least until some way is found to

dissolve the treacle of complexity that downs our best theories.

=====2.2.1.3.  Relevant Measures=====

=====2.2.1.3.  Relevant Measures=====

Differences between problem states are not always defined. And even when they are, relevant differences are not always defined in the manner that would form the most obvious choice. Relevant differences are differences that make a difference, in the well-known pragmatist phrase, bearing on the problem and the purpose at hand.  The qualification of relevance adds information to the abstractly considered problem space.  This extra information has import for the selection of a relevant metric, but nothing says it will ever determine a unique metric suited to a given situation. Relevant metrics are generally defined on semantic features of the problem domain, involving pragmatic equivalence classes of objects. Measures of distinction defined on syntactic features, in effect, on the language that is used to discuss the problem domain, are subject to all of the immaterial differences and the accidental collision of expression that acts to compound the computational confusion and distraction.

Differences between problem states are not always defined.

And even when they are, relevant differences are not always

defined in the manner that would form the most obvious choice.

Relevant differences are differences that make a difference, in

the well-known pragmatist phrase, bearing on the problem and the

purpose at hand.  The qualification of relevance adds information

to the abstractly considered problem space.  This extra information

has import for the selection of a relevant metric, but nothing says

it will ever determine a unique metric suited to a given situation.

Relevant metrics are generally defined on semantic features of the

problem domain, involving pragmatic equivalence classes of objects.

Measures of distinction defined on syntactic features, in effect,

on the language that is used to discuss the problem domain, are

subject to all of the immaterial differences and the accidental

collision of expression that acts to compound the computational

confusion and distraction.

When the problem of finding a fitting metric develops the intensity

When the problem of finding a fitting metric develops the intensity to cross a critical threshold, a strange situation is constellated. The new level of problemhood is noticed as an afterthought but may have a primeval reality about it in its own right, its true nature. The new circle of problem states may circumscribe and underlie the initial focus of attention.  Can the problem of finding a suitable metric for the original problem space be tackled by the same means of problem solving that worked on the assumption of a given metric? A reduction of that sort is possible but is hardly ever guaranteed. The problem of picking the best metric for the initial problem space may be as difficult as the problem first encountered.  And ultimately there is always the risk of reaching a level of circumspection where the problem space of last resort has no metric definable.

to cross a critical threshold, a strange situation is constellated.

The new level of problemhood is noticed as an afterthought but may

have a primeval reality about it in its own right, its true nature.

The new circle of problem states may circumscribe and underlie the

initial focus of attention.  Can the problem of finding a suitable

metric for the original problem space be tackled by the same means

of problem solving that worked on the assumption of a given metric?

A reduction of that sort is possible but is hardly ever guaranteed.

The problem of picking the best metric for the initial problem space

may be as difficult as the problem first encountered.  And ultimately

there is always the risk of reaching a level of circumspection where

the problem space of last resort has no metric definable.

====2.2.2.  Logic with a Difference====

====2.2.2.  Logic with a Difference====

In view of the importance of differential ideas in systems theory and against the background of difficulties just surveyed, I have thought it worthwhile to carefully pursue this quest:  to extend the concepts of difference and due measure to spaces that lack the obvious amenities and expedients.  The limits of rational descriptive capacity for any conceivable sets of states have their ultimate horizon in logic. This is what must be resorted to when only qualitative characterizations of a problem space are initially available.  Therefore I am led to ask what will be a guiding question throughout this work:  What is the proper form of a differential calculus for logic?

In view of the importance of differential ideas in systems theory and against the

background of difficulties just surveyed, I have thought it worthwhile to carefully

pursue this quest:  to extend the concepts of difference and due measure to spaces

that lack the obvious amenities and expedients.  The limits of rational descriptive

capacity for any conceivable sets of states have their ultimate horizon in logic.

This is what must be resorted to when only qualitative characterizations of a

problem space are initially available.  Therefore I am led to ask what will

be a guiding question throughout this work:  What is the proper form of

a differential calculus for logic?

===2.3.  Differential Calculus of Propositions===

===2.3.  Differential Calculus of Propositions===