Changes

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=====1.2.1.1  Observation and Action=====

=====1.2.1.1. Observation and Action=====

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====1.3.2  Inquiry and Education====

====1.3.2. Inquiry and Education====

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2.  Conceptual Framework

2.  Conceptual Framework

2.1  Systems Theory and Artificial Intelligence

2.1. Systems Theory and Artificial Intelligence

If the principles of systems theory are taken seriously in their application to AI,

If the principles of systems theory are taken seriously in their application to AI,

a momentary expedient, a side-effect of their current interpretation.

a momentary expedient, a side-effect of their current interpretation.

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

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

address these problems and to begin settling this frontier.

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?

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

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

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

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

of artifice do not.

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

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

dissolve the treacle of complexity that downs our best theories.

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.

Differences between problem states are not always defined.

the problem space of last resort has no metric definable.

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

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

a differential calculus for logic?

a differential calculus for logic?

2.3  Differential Calculus of Propositions

2.3. Differential Calculus of Propositions

There are two different analogies to keep straight in the following discussion.  First is the comparison of boolean vs. real types with regard to functions and vectors.  These types provide mathematical representation for the qualitative vs. quantitative constituencies, respectively.  Second is the three-part analogy within the qualitative realm.  It relates logical propositions with mathematical functions and sets of vectors, both functions and vectors being of boolean type.

There are two different analogies to keep straight in the following discussion.  First is the comparison of boolean vs. real types with regard to functions and vectors.  These types provide mathematical representation for the qualitative vs. quantitative constituencies, respectively.  Second is the three-part analogy within the qualitative realm.  It relates logical propositions with mathematical functions and sets of vectors, both functions and vectors being of boolean type.

2.3.1  Propositions and Differences

2.3.1. Propositions and Differences

As a first step, I have taken the problem of propositional calculus modeling and viewed it from the standpoint of differential geometry.  In this I exploit an analogy between propositional calculus and the calculus on differential manifolds.  In the qualitative arena propositions may be viewed as boolean functions.  They are associated with areas or arbitrary regions of a Venn diagram, or subsets of an n-dimensional cube.  Logical interpretations, in the technical sense of boolean-valued substitutions in propositional expressions, may be viewed as boolean vectors.  They correspond to single cells of a Venn diagram, or points of an n-cube.  Put altogether, these linkages form a three part analogy between conceptual objects in logic and the two mathematical domains of functions and sets.  In its pivotal location, critical function, and isosceles construction this analogy suggests itself as the pons asinorum of the subject I can see developing.  But I can't tell till I've crossed it.

As a first step, I have taken the problem of propositional calculus modeling and viewed it from the standpoint of differential geometry.  In this I exploit an analogy between propositional calculus and the calculus on differential manifolds.  In the qualitative arena propositions may be viewed as boolean functions.  They are associated with areas or arbitrary regions of a Venn diagram, or subsets of an n-dimensional cube.  Logical interpretations, in the technical sense of boolean-valued substitutions in propositional expressions, may be viewed as boolean vectors.  They correspond to single cells of a Venn diagram, or points of an n-cube.  Put altogether, these linkages form a three part analogy between conceptual objects in logic and the two mathematical domains of functions and sets.  In its pivotal location, critical function, and isosceles construction this analogy suggests itself as the pons asinorum of the subject I can see developing.  But I can't tell till I've crossed it.

2.3.2  Three Part Analogy

2.3.2. Three Part Analogy

For future use it is convenient to label the various elements of the three-part analogy under discussion.

For future use it is convenient to label the various elements of the three-part analogy under discussion.

2.3.2.1  Functional Representation

2.3.2.1. Functional Representation

Functional representation is the link that converts logical propositions into boolean functions.  Its terminus is an important way station for mediating the kinds of computational realizations I hope eventually to reach.  This larger endeavor is the project of declarative functional programming.  It has the goal of giving logical objects a fully operational meaning in software, implementing logical concepts in a functional programming style without sacrificing any of their properly declarative nature.  I have reason to hope this can be a fruitful quest, in part from the reports of more seasoned travelers along these lines, e.g. (Henderson, 1980), (Peyton Jones, 1987), (Field & Harrison, 1988), (Huet, 1990), (Turner, 1990).

Functional representation is the link that converts logical propositions into boolean functions.  Its terminus is an important way station for mediating the kinds of computational realizations I hope eventually to reach.  This larger endeavor is the project of declarative functional programming.  It has the goal of giving logical objects a fully operational meaning in software, implementing logical concepts in a functional programming style without sacrificing any of their properly declarative nature.  I have reason to hope this can be a fruitful quest, in part from the reports of more seasoned travelers along these lines, e.g. (Henderson, 1980), (Peyton Jones, 1987), (Field & Harrison, 1988), (Huet, 1990), (Turner, 1990).

This knowledge would consist of axioms, inferential procedures, and a running accumulation of theorems.  A developmental programming system of this sort would permit designers to anticipate many features of contemplated programs before running the risk of risking to run them.  One vital requirement of the ideal system must be provisioned in the most primitive elements of its construction.  The ideal system plus knowledge-base plus intelligence needs to be developmental in the added sense of a developing mentality.  Undistracted by all the positive features that an ideal system must embody, a great absence must also be arranged by its designers.  To the extent foreseeable there must be no foreclosure of interpretive freedom.  The intended programming language, the sans critical koine of the utopian realm, must place as little possible prior value on the primitive tokens that fund its form of expression.  An early implementation of a knowledge-based system for program development, using a refinement tree to search a space of correct programs, is described in (Barstow, 1979).

This knowledge would consist of axioms, inferential procedures, and a running accumulation of theorems.  A developmental programming system of this sort would permit designers to anticipate many features of contemplated programs before running the risk of risking to run them.  One vital requirement of the ideal system must be provisioned in the most primitive elements of its construction.  The ideal system plus knowledge-base plus intelligence needs to be developmental in the added sense of a developing mentality.  Undistracted by all the positive features that an ideal system must embody, a great absence must also be arranged by its designers.  To the extent foreseeable there must be no foreclosure of interpretive freedom.  The intended programming language, the sans critical koine of the utopian realm, must place as little possible prior value on the primitive tokens that fund its form of expression.  An early implementation of a knowledge-based system for program development, using a refinement tree to search a space of correct programs, is described in (Barstow, 1979).

2.3.2.2  Characteristic Relation

2.3.2.2. Characteristic Relation

Characteristic relation denotes the two-way link that relates boolean functions with subsets of their boolean universes, whether pictured as Venn diagram regions or n-cube subsets does not matter.  Indicative conversion describes the traffic or exchange on this link between the two termini.  Given a set A, the function fA which has the value 1 on A and 0 off A is commonly called the characteristic function or indicator function of A.  Since every boolean function f determines a unique set S = Sf of which it is the indicator function f = fS, this forms a convertible relationship between boolean functions and sets of boolean vectors.  This fact is also described as an isomorphism between the function space (U -> B) and the power set P(U) = 2U of the universe U.  The associated set Sf is often called the support of the function f.  Alternatively, it may serve as a helpful mnemonic and a useful handle on this edge of the analogy to call Sf the characteristic region, indicated set, or simply the indication of the function f, and to say that the function characterizes or indicates the set where its value is positive (that is, greater than 0, and therefore equal to 1 in B).

Characteristic relation denotes the two-way link that relates boolean functions with subsets of their boolean universes, whether pictured as Venn diagram regions or n-cube subsets does not matter.  Indicative conversion describes the traffic or exchange on this link between the two termini.  Given a set A, the function fA which has the value 1 on A and 0 off A is commonly called the characteristic function or indicator function of A.  Since every boolean function f determines a unique set S = Sf of which it is the indicator function f = fS, this forms a convertible relationship between boolean functions and sets of boolean vectors.  This fact is also described as an isomorphism between the function space (U -> B) and the power set P(U) = 2U of the universe U.  The associated set Sf is often called the support of the function f.  Alternatively, it may serve as a helpful mnemonic and a useful handle on this edge of the analogy to call Sf the characteristic region, indicated set, or simply the indication of the function f, and to say that the function characterizes or indicates the set where its value is positive (that is, greater than 0, and therefore equal to 1 in B).

2.3.2.3  Indicative Conversion

2.3.2.3. Indicative Conversion

The term indicative conversion and the associated usages are especially apt in light of the ordinary linguistic relationship between declarative sentences and verb forms in the indicative mood, which "represent the denoted act or state as an objective fact" (Webster's).  It is not at all accidental that a fundamental capacity needed to support declarative programming is the pragmatic facilitation of this semantic relation, the ready conversion between propositions as indicator functions and properties in extension over indicated sets.  The computational organism that would function declaratively must embody an interior environment with plenty of catalysts for the quick conversion of symbolically expressed functional specifications into images of their solution sets or sets of models.

The term indicative conversion and the associated usages are especially apt in light of the ordinary linguistic relationship between declarative sentences and verb forms in the indicative mood, which "represent the denoted act or state as an objective fact" (Webster's).  It is not at all accidental that a fundamental capacity needed to support declarative programming is the pragmatic facilitation of this semantic relation, the ready conversion between propositions as indicator functions and properties in extension over indicated sets.  The computational organism that would function declaratively must embody an interior environment with plenty of catalysts for the quick conversion of symbolically expressed functional specifications into images of their solution sets or sets of models.

2.3.3  Pragmatic Roles

2.3.3. Pragmatic Roles

The part of the analogy that carries propositions into functions combines with the characteristic relation between functions and sets to generate a multitude of different ways to describe essentially the same conceptual objects.  From an information-theoretic point of view "essentially the same" means that the objects in comparison are equivalent pieces of information, parameterized or coded by the same number of bits and falling under isomorphic types.  When assigning characters to individual examples of these entities, I think it helps to avoid drawing too fine a distinction between the logical, functional, and set-theoretic roles that have just been put in correspondence.  Thus, I avoid usages that rigidify the pragmatic dimensions of variation within the columns below:

The part of the analogy that carries propositions into functions combines with the characteristic relation between functions and sets to generate a multitude of different ways to describe essentially the same conceptual objects.  From an information-theoretic point of view "essentially the same" means that the objects in comparison are equivalent pieces of information, parameterized or coded by the same number of bits and falling under isomorphic types.  When assigning characters to individual examples of these entities, I think it helps to avoid drawing too fine a distinction between the logical, functional, and set-theoretic roles that have just been put in correspondence.  Thus, I avoid usages that rigidify the pragmatic dimensions of variation within the columns below:

Though it may be advisable not to reify the practical distinctions among these roles, this is not the same thing as failing to see them or denying their use.  Obviously, these differences may vary in relative importance with the purpose at hand or context of use.  However, the mere fact that a distinction can generally be made is not a sufficient argument that it has any useful bearing on a particular purpose.

Though it may be advisable not to reify the practical distinctions among these roles, this is not the same thing as failing to see them or denying their use.  Obviously, these differences may vary in relative importance with the purpose at hand or context of use.  However, the mere fact that a distinction can generally be made is not a sufficient argument that it has any useful bearing on a particular purpose.

2.3.3.1  Flexible Roles and Suitable Models

2.3.3.1. Flexible Roles and Suitable Models

When giving names and habitations to things by the use of letters and types, a certain flexibility may be allowed in the roles assigned by interpretation.  For example, in the form "p: U -> B", the name "p" may be taken to denote a proposition or a function, indifferently, and the type U may be associated with a set of interpretations or a set of boolean vectors, correspondingly, whichever makes sense in a given context of use.  One dimension that does matter is drawn through these three beads:  propositions, interpretations, and values.  On the alternate line it is produced by the distinctions among collections, individuals, and values.

When giving names and habitations to things by the use of letters and types, a certain flexibility may be allowed in the roles assigned by interpretation.  For example, in the form "p: U -> B", the name "p" may be taken to denote a proposition or a function, indifferently, and the type U may be associated with a set of interpretations or a set of boolean vectors, correspondingly, whichever makes sense in a given context of use.  One dimension that does matter is drawn through these three beads:  propositions, interpretations, and values.  On the alternate line it is produced by the distinctions among collections, individuals, and values.

The interpretations that render a proposition true, i.e. the substitutions for which the proposition evaluates to true, are said to satisfy the proposition and to be its models.  With a doubly modulated sense that is too apt to be purely accidental, the model set is the "content" of the proposition's formal expression (Eulenberg, 1986).  In functional terms the models of a proposition p are the pre-images of truth under the function p.  Collectively, they form the set of vectors in p-1(1).  In another usage the set of models is called the fiber of truth, in other words, the equivalence class [1]p of the value 1 under the mapping p.

The interpretations that render a proposition true, i.e. the substitutions for which the proposition evaluates to true, are said to satisfy the proposition and to be its models.  With a doubly modulated sense that is too apt to be purely accidental, the model set is the "content" of the proposition's formal expression (Eulenberg, 1986).  In functional terms the models of a proposition p are the pre-images of truth under the function p.  Collectively, they form the set of vectors in p-1(1).  In another usage the set of models is called the fiber of truth, in other words, the equivalence class [1]p of the value 1 under the mapping p.

2.3.3.2  Functional Pragmatism

2.3.3.2. Functional Pragmatism

The project of functional programming itself fits within a broader philosophical mission, the pragmatism of C.S. Peirce and John Dewey, which seeks to clarify abstract concepts and occult properties by translating them into operational terms, see (Peirce, Collected Papers) and (Dewey, 1986).  These thinkers had clear understandings of the relation between information and control, giving early accounts of inquiry processes and problem-solving, intelligence and goal-seeking that would sound quite familiar to cyberneticians and systems theorists.  Similar ideas are reflected in current AI work, especially by proponents of means-ends analysis and difference reduction methods (Newell, 1990), (Winston, ch. 5).

The project of functional programming itself fits within a broader philosophical mission, the pragmatism of C.S. Peirce and John Dewey, which seeks to clarify abstract concepts and occult properties by translating them into operational terms, see (Peirce, Collected Papers) and (Dewey, 1986).  These thinkers had clear understandings of the relation between information and control, giving early accounts of inquiry processes and problem-solving, intelligence and goal-seeking that would sound quite familiar to cyberneticians and systems theorists.  Similar ideas are reflected in current AI work, especially by proponents of means-ends analysis and difference reduction methods (Newell, 1990), (Winston, ch. 5).

Of all the complex systems that attract human interest, the human mind's own doings, knowing or not, must eventually form a trajectory that ensnares itself in questions and wonderings:  Where will it be off to next?  What is it apt to do next?  How often will it recur to the various things it does?  The mind's orbit traced in these questions has a compelling power in its own right to generate wonder.

Of all the complex systems that attract human interest, the human mind's own doings, knowing or not, must eventually form a trajectory that ensnares itself in questions and wonderings:  Where will it be off to next?  What is it apt to do next?  How often will it recur to the various things it does?  The mind's orbit traced in these questions has a compelling power in its own right to generate wonder.

2.3.4  Abstraction, Behavior, Consequence

2.3.4. Abstraction, Behavior, Consequence

There are many good reasons to preserve the logical features and constraints attaching to computational objects, i.e. programs and data structures.  Chief among these reasons are:  axiomatic abstraction, behavioral coordination, and consequential definition.

There are many good reasons to preserve the logical features and constraints attaching to computational objects, i.e. programs and data structures.  Chief among these reasons are:  axiomatic abstraction, behavioral coordination, and consequential definition.

2.3.4.1  Axiomatic Abstraction

2.3.4.1. Axiomatic Abstraction

The capacity for abstraction would permit an expert system for dynamic simulation to rise above the immediate flux of the process simulated.  Eventually, this could enable the software intelligence to adduce, reason about, and test hypotheses about generic properties of the system under study.  Even short of this autonomy, the resources of abstract representation could at least provide a medium for transmuting embedded simulations into axioms and theories.  For the systems prospector such an interface, even slightly reflective, can heighten the chances of panning some nugget of theory and lifting some glimmer of insight from the running stream of simulations.

The capacity for abstraction would permit an expert system for dynamic simulation to rise above the immediate flux of the process simulated.  Eventually, this could enable the software intelligence to adduce, reason about, and test hypotheses about generic properties of the system under study.  Even short of this autonomy, the resources of abstract representation could at least provide a medium for transmuting embedded simulations into axioms and theories.  For the systems prospector such an interface, even slightly reflective, can heighten the chances of panning some nugget of theory and lifting some glimmer of insight from the running stream of simulations.

2.3.4.2  Behavioral Coordination

2.3.4.2. Behavioral Coordination

The guidelines of pragmatism are remarkably suited as regulative principles for synthesizing AI and systems theory, where it is required to clarify the occult property of intelligence in terms of dynamic activity and behavior.  This involves realizing abstract faculties, like momentum and intelligence, as hypotheses about the organization of trajectories through manifolds of observable features.  In these post-revolutionary times, cognitively and chaotically speaking, it is probably not necessary to be reminded that this effort contains no prior claim of reductionism.  The pragmatic maxim can no more predetermine the mind to be explained by simple reflexes than it can constrain nature to operate by linear dynamics.  If these reductions are approximately true of particular situations, then they have to be discovered on site and proven to fit, not imposed with eyes closed.

The guidelines of pragmatism are remarkably suited as regulative principles for synthesizing AI and systems theory, where it is required to clarify the occult property of intelligence in terms of dynamic activity and behavior.  This involves realizing abstract faculties, like momentum and intelligence, as hypotheses about the organization of trajectories through manifolds of observable features.  In these post-revolutionary times, cognitively and chaotically speaking, it is probably not necessary to be reminded that this effort contains no prior claim of reductionism.  The pragmatic maxim can no more predetermine the mind to be explained by simple reflexes than it can constrain nature to operate by linear dynamics.  If these reductions are approximately true of particular situations, then they have to be discovered on site and proven to fit, not imposed with eyes closed.

2.3.4.3  Consequential Definition

2.3.4.3. Consequential Definition

The ability to deduce consequences of specified/acquired features and generic/imposed constraints would support the ultimate prospects toward unification of several stylistic trends in programming.  Among these are the employment of class hierarchies and inheritance schemes in frame-system and semantic network knowledge bases (Winston, ch. 8), object-oriented programming methodologies (Shriver & Wegner, 1987), and constraint based programming (Van Hentenryck, 1989).  The capacity for deduction includes as a special case the ability to check logical consistency of declarations.  This has applications to compilation type-checking (Peyton Jones, 1987) and deductive data-base consistency (Minker, 1988).

The ability to deduce consequences of specified/acquired features and generic/imposed constraints would support the ultimate prospects toward unification of several stylistic trends in programming.  Among these are the employment of class hierarchies and inheritance schemes in frame-system and semantic network knowledge bases (Winston, ch. 8), object-oriented programming methodologies (Shriver & Wegner, 1987), and constraint based programming (Van Hentenryck, 1989).  The capacity for deduction includes as a special case the ability to check logical consistency of declarations.  This has applications to compilation type-checking (Peyton Jones, 1987) and deductive data-base consistency (Minker, 1988).

2.3.5  Refrain

2.3.5. Refrain

The analogy between propositional calculus and differential geometry is extended as far as possible by continuing to cast propositions and interpretations in roles similar to those exercised by real-valued functions and real-coordinate vectors in the quantitative world.  In a number of reaches tentative trials of the analogy will render fit correspondences.  Beyond these points it is critically important to examine those stretches where the analogy breaks, and there to consider the actual temperament and proper treatment of the qualitative situation in its own right.

The analogy between propositional calculus and differential geometry is extended as far as possible by continuing to cast propositions and interpretations in roles similar to those exercised by real-valued functions and real-coordinate vectors in the quantitative world.  In a number of reaches tentative trials of the analogy will render fit correspondences.  Beyond these points it is critically important to examine those stretches where the analogy breaks, and there to consider the actual temperament and proper treatment of the qualitative situation in its own right.

A text that has been useful to me in relating classical and modern treatments of differential geometry is (Spivak, 1979).  The standard for logic programming via general resolution theorem proving was set by (Chang & Lee, 1973).  A more recent reference is (Lloyd, 1987), which concentrates on Prolog type programming in the Horn clause subset of logic.  My own incursions through predicate calculus theorem proving and my attempts to size up the computational complexity invested there have led me to the following opinions.

A text that has been useful to me in relating classical and modern treatments of differential geometry is (Spivak, 1979).  The standard for logic programming via general resolution theorem proving was set by (Chang & Lee, 1973).  A more recent reference is (Lloyd, 1987), which concentrates on Prolog type programming in the Horn clause subset of logic.  My own incursions through predicate calculus theorem proving and my attempts to size up the computational complexity invested there have led me to the following opinions.

2.4  Logic Programming

2.4. Logic Programming

Militating against the charge of declarative programmers to achieve their goals through logic, a surprising amount of computational resistance seems to reside at the level of purely sentential or propositional operations.  In investigating this situation I have come to believe that progress in logic programming will be severely impeded unless these factors of computational complexity at the level of propositional calculus are addressed and either resolved or alleviated.

Militating against the charge of declarative programmers to achieve their goals through logic, a surprising amount of computational resistance seems to reside at the level of purely sentential or propositional operations.  In investigating this situation I have come to believe that progress in logic programming will be severely impeded unless these factors of computational complexity at the level of propositional calculus are addressed and either resolved or alleviated.

At my current state of understanding I can propose nothing more complicated than to work toward a position of increased knowledge about the practical logistics of this problem domain.  A reasonable approach is to explore the terrain at this simplest level, using the advantages afforded by a propositional calculus interpreter and relevant utilities in software.  A similar strategy of starting from propositional logic and working up in stages to predicate logic is exploited by (Maier & Warren, 1988), in this case building a Prolog interpreter by successive refinement.

At my current state of understanding I can propose nothing more complicated than to work toward a position of increased knowledge about the practical logistics of this problem domain.  A reasonable approach is to explore the terrain at this simplest level, using the advantages afforded by a propositional calculus interpreter and relevant utilities in software.  A similar strategy of starting from propositional logic and working up in stages to predicate logic is exploited by (Maier & Warren, 1988), in this case building a Prolog interpreter by successive refinement.

2.4.1  Differential Aspects

2.4.1. Differential Aspects

The fact that a difference calculus can be developed for boolean functions is well-known (Kohavi, sec. 8-4,), (Fujiwara, 1985) and was probably familiar to Boole, who was a master of difference equations before he turned to logic.  And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's (Menabrea, p. 225-297).  At the very outset of general-purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation (Morrison & Morrison, 1961), (Melzak, ch. 4).

The fact that a difference calculus can be developed for boolean functions is well-known (Kohavi, sec. 8-4,), (Fujiwara, 1985) and was probably familiar to Boole, who was a master of difference equations before he turned to logic.  And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's (Menabrea, p. 225-297).  At the very outset of general-purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation (Morrison & Morrison, 1961), (Melzak, ch. 4).

2.4.2  Algebraic Aspects

2.4.2. Algebraic Aspects

Finally, there is a body of mathematical work that investigates algebraic and differential geometry over finite fields.  This usually takes place at such high levels of abstraction that the field of two elements is just another special case.  In this work the principal focus is on the field operations of sum (+) and product (.), which correspond to the logical operations of exclusive disjunction (xor, neq) and conjunction (and), respectively.  The stress laid on these special operations creates a covert bias in the algebraic field.  Unfortunately for the purposes of logic, the totality of boolean operations is given short shrift on the scaffold affecting this algebraic slant.  For example, there are sixteen operations just at the level of binary connectives, not to mention the exploding population of k-ary operations, all of which deserve in some sense to be treated as equal citizens of the logical realm.

Finally, there is a body of mathematical work that investigates algebraic and differential geometry over finite fields.  This usually takes place at such high levels of abstraction that the field of two elements is just another special case.  In this work the principal focus is on the field operations of sum (+) and product (.), which correspond to the logical operations of exclusive disjunction (xor, neq) and conjunction (and), respectively.  The stress laid on these special operations creates a covert bias in the algebraic field.  Unfortunately for the purposes of logic, the totality of boolean operations is given short shrift on the scaffold affecting this algebraic slant.  For example, there are sixteen operations just at the level of binary connectives, not to mention the exploding population of k-ary operations, all of which deserve in some sense to be treated as equal citizens of the logical realm.

Moreover, from an algebraic perspective the dyadic or boolean case exhibits several features peculiar to itself.  Binary addition (+) and subtraction (-) amount to the same operation, making each element its own additive inverse.  This circumstance in turn exacts a constant vigilance to avert the verbal confusion between algebraic negatives and logical negations.  The property of being invertible under products (.) is neither a majority nor a typical possession, since only the element 1 has a multiplicative inverse, namely itself.  On account of these facts the strange case of the two element field is often set aside, or set down as a "degenerate" situation in algebraic studies.  Obviously, in turning to take it up from a differential standpoint, any domain that confounds "plus" and "minus" and "not equal to" is going to play havoc with our automatic intuitions about difference operators, linear approximations, inequalities and thresholds, and many other critical topics.

Moreover, from an algebraic perspective the dyadic or boolean case exhibits several features peculiar to itself.  Binary addition (+) and subtraction (-) amount to the same operation, making each element its own additive inverse.  This circumstance in turn exacts a constant vigilance to avert the verbal confusion between algebraic negatives and logical negations.  The property of being invertible under products (.) is neither a majority nor a typical possession, since only the element 1 has a multiplicative inverse, namely itself.  On account of these facts the strange case of the two element field is often set aside, or set down as a "degenerate" situation in algebraic studies.  Obviously, in turning to take it up from a differential standpoint, any domain that confounds "plus" and "minus" and "not equal to" is going to play havoc with our automatic intuitions about difference operators, linear approximations, inequalities and thresholds, and many other critical topics.

2.5  Differential Geometry

2.5. Differential Geometry

One of the difficulties I've had finding guidance toward the proper form of a differential calculus for logic has been the variety of ways that the classical subjects of real analysis and differential geometry have been generalized.  As a first cut, two broad philosophies may be discerned, epitomized by their treatment of the differential df of a function f: X -> R.  Everyone begins with the idea that df ought to be a locally linear approximation dfu(v) or df(u,v) to the difference function Dfu(v) = Df(u,v) = f(u+v) - f(u).  In this conception it is understood that "local" means in the vicinity of the point u and that "linear" is meant with respect to the variable v.

One of the difficulties I've had finding guidance toward the proper form of a differential calculus for logic has been the variety of ways that the classical subjects of real analysis and differential geometry have been generalized.  As a first cut, two broad philosophies may be discerned, epitomized by their treatment of the differential df of a function f: X -> R.  Everyone begins with the idea that df ought to be a locally linear approximation dfu(v) or df(u,v) to the difference function Dfu(v) = Df(u,v) = f(u+v) - f(u).  In this conception it is understood that "local" means in the vicinity of the point u and that "linear" is meant with respect to the variable v.

2.5.1  Local Stress and Linear Trend

2.5.1. Local Stress and Linear Trend

But one school of thought stresses the local aspect, to the extent of seeking constructions that can be meaningful on global scales in spite of coordinate systems that make sense solely on local scales, being allowed to vary from point to point, e.g. (Arnold, 1989).  The other trend of thinking accents the linear feature, looking at linear maps in the light of their character as representations or homomorphisms (Loomis & Sternberg, 1968).  Extenuations of this line of thinking go to the point of casting linear functions under the headings of the vastly more general morphisms and abstract arrows of category theory (Manes & Arbib, 1986), (MacLane, 1971).

But one school of thought stresses the local aspect, to the extent of seeking constructions that can be meaningful on global scales in spite of coordinate systems that make sense solely on local scales, being allowed to vary from point to point, e.g. (Arnold, 1989).  The other trend of thinking accents the linear feature, looking at linear maps in the light of their character as representations or homomorphisms (Loomis & Sternberg, 1968).  Extenuations of this line of thinking go to the point of casting linear functions under the headings of the vastly more general morphisms and abstract arrows of category theory (Manes & Arbib, 1986), (MacLane, 1971).

2.5.1.1  Analytic View

2.5.1.1. Analytic View

The first group, more analytic, strives to get intrinsic definitions of everything, defining tangent vectors primarily as equivalence classes of curves through points of phase space.  This posture is conditioned to the spare frame of physical theory and is constrained by the ready equation of physics with ante-metaphysics.  In short they regard physics as a practical study that is prior to any a priori.  Physics should exert itself to save the phenomena and forget the rest.  The dynamic manifold is the realm of phenomena, the locus of all knowable reality and the focus of all actual knowledge.  Beyond this, even attributes like velocity and momentum are epiphenomenal, derivative scores attached to a system's dynamic point from measurements made at other points.

The first group, more analytic, strives to get intrinsic definitions of everything, defining tangent vectors primarily as equivalence classes of curves through points of phase space.  This posture is conditioned to the spare frame of physical theory and is constrained by the ready equation of physics with ante-metaphysics.  In short they regard physics as a practical study that is prior to any a priori.  Physics should exert itself to save the phenomena and forget the rest.  The dynamic manifold is the realm of phenomena, the locus of all knowable reality and the focus of all actual knowledge.  Beyond this, even attributes like velocity and momentum are epiphenomenal, derivative scores attached to a system's dynamic point from measurements made at other points.

This incurs an empire of further systems of ranking and outranking, teams and leagues and legions of commissioners, all to compare and umpire these ratings.  When these circumspect systems are not sufficiently circumscribed to converge on a fixed point or a limiting universal system, it seems as though chaos has broken out.  The faith of this sect that the world is a fair game for observation and intelligence seems dissipated by divergences of this sort.  It wrecks their hope of order in phenomena, dooms what they deem a fit domain, a single rule of order that commands the manifold to appear as it does.  To share the universe with several realities, to countenance a real diversity?  It ruins the very idea they most favor of a cosmos, one that favors them.

This incurs an empire of further systems of ranking and outranking, teams and leagues and legions of commissioners, all to compare and umpire these ratings.  When these circumspect systems are not sufficiently circumscribed to converge on a fixed point or a limiting universal system, it seems as though chaos has broken out.  The faith of this sect that the world is a fair game for observation and intelligence seems dissipated by divergences of this sort.  It wrecks their hope of order in phenomena, dooms what they deem a fit domain, a single rule of order that commands the manifold to appear as it does.  To share the universe with several realities, to countenance a real diversity?  It ruins the very idea they most favor of a cosmos, one that favors them.

2.5.1.2  Algebraic View

2.5.1.2. Algebraic View

The second group, more algebraic, accepts the comforts of an embedding vector space with a less severe attitude, one that belays and belies the species of anxiety that worries the other group.  They do not show the same phenomenal anguish about the uncertain multiplicity or empty void of outer spaces.  Given this trust in something outside of phenomena, they permit themselves on principle the luxury of relating differential concepts to operators with linear and derivation properties.  This tendency, ranging from pious optimism to animistic hedonism in its mathematical persuasions, demands less agnosticism about the reality of exterior constructs.  Its pragmatic hope allows room for the imagination of supervening prospects, without demanding that these promontory contexts be uniquely placed or set in concrete.

The second group, more algebraic, accepts the comforts of an embedding vector space with a less severe attitude, one that belays and belies the species of anxiety that worries the other group.  They do not show the same phenomenal anguish about the uncertain multiplicity or empty void of outer spaces.  Given this trust in something outside of phenomena, they permit themselves on principle the luxury of relating differential concepts to operators with linear and derivation properties.  This tendency, ranging from pious optimism to animistic hedonism in its mathematical persuasions, demands less agnosticism about the reality of exterior constructs.  Its pragmatic hope allows room for the imagination of supervening prospects, without demanding that these promontory contexts be uniquely placed or set in concrete.

2.5.1.3  Compromise

2.5.1.3. Compromise

In attempting to negotiate between these two philosophies, I have arrived at the following compromise.  On the one hand, the circumstance that provides a natural context for a manifold of observable action does not automatically exclude all possibility of other contexts being equally natural.  On the other hand, it may happen that a surface is so bent in cusps and knots, or otherwise so intrinsically formed, that it places mathematical constraints on the class of spaces it can possibly inhabit.

In attempting to negotiate between these two philosophies, I have arrived at the following compromise.  On the one hand, the circumstance that provides a natural context for a manifold of observable action does not automatically exclude all possibility of other contexts being equally natural.  On the other hand, it may happen that a surface is so bent in cusps and knots, or otherwise so intrinsically formed, that it places mathematical constraints on the class of spaces it can possibly inhabit.

Thus a manifold can embody information that bears on the notion of a larger reality.  By dint of this interpretation the form of the manifold becomes the symbol of its implicated unity.  But what I think I can fathom seems patent enough, that the chances of these two alternatives, plurality and singularity, together make a bet that is a toss up and open to test with each new shape of manifold encountered.  It is likely that the outcome, if at all decidable, falls in accord with no general law but is subject to proof on a case by case basis.

Thus a manifold can embody information that bears on the notion of a larger reality.  By dint of this interpretation the form of the manifold becomes the symbol of its implicated unity.  But what I think I can fathom seems patent enough, that the chances of these two alternatives, plurality and singularity, together make a bet that is a toss up and open to test with each new shape of manifold encountered.  It is likely that the outcome, if at all decidable, falls in accord with no general law but is subject to proof on a case by case basis.

2.5.2  Prospects for a Differential Logic

2.5.2. Prospects for a Differential Logic

Pragmatically, the "proper" form of a differential logic is likely to be regulated by the purposes to which it is intended to be put, or determined by the uses to which it is actually, eventually, and suitably put.  With my current level of uncertainty about what will eventually work out, I have to be guided by my general intention of using this logic to describe the dynamics of inquiry and intelligence in systematic terms.  For this purpose it seems only that many different types of "fiber bundles" or systems of "spaces at points" will have to be contemplated.

Pragmatically, the "proper" form of a differential logic is likely to be regulated by the purposes to which it is intended to be put, or determined by the uses to which it is actually, eventually, and suitably put.  With my current level of uncertainty about what will eventually work out, I have to be guided by my general intention of using this logic to describe the dynamics of inquiry and intelligence in systematic terms.  For this purpose it seems only that many different types of "fiber bundles" or systems of "spaces at points" will have to be contemplated.

Although the limited framework of propositional calculus seems to rule out this higher level of generality, the exigencies of computation on symbolic expressions have the effect of bringing in this level of arbitration by another route.  Even though we use the same alphabet for the joint basis of coordinates and differentials at each point of the manifold, one of our intended applications is to the states of interpreting systems, and there is nothing a priori to determine such a program to interpret these symbols in the same way at every moment.  Thus, the arbitrariness of local reference frames that concerns us in physical dynamics, that makes the arbitrage or negotiation of transition maps between charts (qua markets) such a profitable enterprise, raises its head again in computational dynamics as a relativity of interpretation to the actual state of a running interpretive program.

Although the limited framework of propositional calculus seems to rule out this higher level of generality, the exigencies of computation on symbolic expressions have the effect of bringing in this level of arbitration by another route.  Even though we use the same alphabet for the joint basis of coordinates and differentials at each point of the manifold, one of our intended applications is to the states of interpreting systems, and there is nothing a priori to determine such a program to interpret these symbols in the same way at every moment.  Thus, the arbitrariness of local reference frames that concerns us in physical dynamics, that makes the arbitrage or negotiation of transition maps between charts (qua markets) such a profitable enterprise, raises its head again in computational dynamics as a relativity of interpretation to the actual state of a running interpretive program.

2.6  Reprise

2.6. Reprise

In summing up this sample of literature bearing on my present aims, there is much to suggest a deep relationship between the topics of systems, differentials, logic, and computing, especially when considered in the accidental but undeniable stream of historical events.  I have not come across any strand of inquiry that plainly, explicitly, and completely weaves differential geometry and propositional logic in a computational context.  But I hope to see one day a scintilla of a program that can weld them together in a logically declarative, functionally dynamic platform for intelligent computing.

In summing up this sample of literature bearing on my present aims, there is much to suggest a deep relationship between the topics of systems, differentials, logic, and computing, especially when considered in the accidental but undeniable stream of historical events.  I have not come across any strand of inquiry that plainly, explicitly, and completely weaves differential geometry and propositional logic in a computational context.  But I hope to see one day a scintilla of a program that can weld them together in a logically declarative, functionally dynamic platform for intelligent computing.

3.  Instrumental Focus

3.  Instrumental Focus

3.1  Propositional Calculus

3.1. Propositional Calculus

A symbolic calculus is needed to assist our reasoning and computation in the realm of propositions.  With an eye toward efficiency of computing and ease of human use, while preserving both functional and declarative properties of propositions, I have implemented an interpreter and assorted utilities for one such calculus.  The original form of this particular calculus goes back to the logician C.S. Peirce, who is my personal favorite candidate for the grand-uncle of AI.  Among other things, Peirce discovered the logical importance of NAND/NNOR operators (CP 4.12 ff, 4.264 f), (NE 4, ch. 5), inspired early ideas about logic machines (Peirce, 1883), is credited with "the first known effort to apply Boolean algebra to the design of switching circuits" (M. Gardner, p. 116 n), and even speculated on the nature of abstract interpreters and other "Quasi-Minds" (Peirce, CP 4.536, 4.550 ff).

A symbolic calculus is needed to assist our reasoning and computation in the realm of propositions.  With an eye toward efficiency of computing and ease of human use, while preserving both functional and declarative properties of propositions, I have implemented an interpreter and assorted utilities for one such calculus.  The original form of this particular calculus goes back to the logician C.S. Peirce, who is my personal favorite candidate for the grand-uncle of AI.  Among other things, Peirce discovered the logical importance of NAND/NNOR operators (CP 4.12 ff, 4.264 f), (NE 4, ch. 5), inspired early ideas about logic machines (Peirce, 1883), is credited with "the first known effort to apply Boolean algebra to the design of switching circuits" (M. Gardner, p. 116 n), and even speculated on the nature of abstract interpreters and other "Quasi-Minds" (Peirce, CP 4.536, 4.550 ff).

All of these issues that occupied Peirce would be encountered again later in the 20th century when computer scientists, linguists, communication engineers, media theorists, and others would be forced to deal with them in their daily practice and would perforce discover many workable answers.  These are the topics that have come to be recognized as the reality of information and uncertainty, the physicality of symbol systems, the independent dimension of syntax, the complexity of semantics and evaluation, the pragmatic metes and bounds of interactive communication and interpretive control.  All in all, as acutely discovered in AI systems engineering, these factors sum up to the general resistance of matter to being impressed with our minds.

All of these issues that occupied Peirce would be encountered again later in the 20th century when computer scientists, linguists, communication engineers, media theorists, and others would be forced to deal with them in their daily practice and would perforce discover many workable answers.  These are the topics that have come to be recognized as the reality of information and uncertainty, the physicality of symbol systems, the independent dimension of syntax, the complexity of semantics and evaluation, the pragmatic metes and bounds of interactive communication and interpretive control.  All in all, as acutely discovered in AI systems engineering, these factors sum up to the general resistance of matter to being impressed with our minds.

3.1.1  Peirce's Existential Graphs

3.1.1. Peirce's Existential Graphs

Peirce devised a graphical notation for predicate calculus, or first order logic, that he called the system of "Existential Graphs" (EG).  In its emphasis on relations and its graphic depiction of their logic, EG anticipated many features of present-day semantic networks and conceptual graphs.  Not only does it remain logically more exact than most of these later formulations, but EG had transformation rules that rendered it a literal calculus, with a manifest power for inferring latent facts.  An explicit use of Peirce's EG for knowledge base representation appears in (Sowa, 1984).  A software package that uses EG to teach basic logic is documented in (Ketner, 1990).  The calculus presented below is related in its form and interpretation to the propositional part of Peirce's EG.  A similar calculus, but favoring an alternate interpretation, was developed in (Spencer-Brown, 1969).

Peirce devised a graphical notation for predicate calculus, or first order logic, that he called the system of "Existential Graphs" (EG).  In its emphasis on relations and its graphic depiction of their logic, EG anticipated many features of present-day semantic networks and conceptual graphs.  Not only does it remain logically more exact than most of these later formulations, but EG had transformation rules that rendered it a literal calculus, with a manifest power for inferring latent facts.  An explicit use of Peirce's EG for knowledge base representation appears in (Sowa, 1984).  A software package that uses EG to teach basic logic is documented in (Ketner, 1990).  The calculus presented below is related in its form and interpretation to the propositional part of Peirce's EG.  A similar calculus, but favoring an alternate interpretation, was developed in (Spencer-Brown, 1969).

3.1.1.1  Blank and Bound Connectives

3.1.1.1. Blank and Bound Connectives

Given an alphabet A = {a1,...,an} and a universe U = <A>, we write expressions for the propositions p: U -> B upon the following basis.  The ai: U -> B are interpreted as coordinate functions.  For each natural number k we have two k-ary operations, called the blank or unmarked connective and the bound or marked connective.

Given an alphabet A = {a1,...,an} and a universe U = <A>, we write expressions for the propositions p: U -> B upon the following basis.  The ai: U -> B are interpreted as coordinate functions.  For each natural number k we have two k-ary operations, called the blank or unmarked connective and the bound or marked connective.

As a consistent downward extension, the nullary (or 0-ary) connectives can be identified with logical constants.  That is, blank expressions " " are taken for the value "true" (silence assents), and empty bounds "()" are taken for the value "false".  By composing operations, negation and binary conjunction are enough in themselves to obtain all the other boolean functions, but the use of these k-ary connectives lends itself to a flexible and powerful representation as graph-theoretical data-structures in the computer.

As a consistent downward extension, the nullary (or 0-ary) connectives can be identified with logical constants.  That is, blank expressions " " are taken for the value "true" (silence assents), and empty bounds "()" are taken for the value "false".  By composing operations, negation and binary conjunction are enough in themselves to obtain all the other boolean functions, but the use of these k-ary connectives lends itself to a flexible and powerful representation as graph-theoretical data-structures in the computer.

3.1.2  Implementation Details

3.1.2. Implementation Details

The interpreter that has been implemented for EG employs advanced data-structures for the reprsentation of both lexical terms and logical expressions.

The interpreter that has been implemented for EG employs advanced data-structures for the reprsentation of both lexical terms and logical expressions.

Mathematically, all this verbiage is just a way of talking about two topics:  (1) functions and relations from structured objects to sets of features, and (2) equivalence relations (for example, orbits under symmetry group actions) on these structured objects.  But the visual metaphors seem to assist thought, most of the time, and are in any case a part of the popular iconography.

Mathematically, all this verbiage is just a way of talking about two topics:  (1) functions and relations from structured objects to sets of features, and (2) equivalence relations (for example, orbits under symmetry group actions) on these structured objects.  But the visual metaphors seem to assist thought, most of the time, and are in any case a part of the popular iconography.

3.1.2.1  Painted Cacti

3.1.2.1. Painted Cacti

Viewing a propositional expression in EG as a "cactus", the bound connectives ( , , , ) constitute its "lobes" (edges or lines) and the positive literals ai are tantamount to "colors" (paints or tints) on its "points" (vertices or nodes).  One of the chief tasks of processing logical expressions is their systematic clarification.  This involves transforming arbitrary expressions into logically equivalent expressions whose latent meaning is manifest, their "canonical" or "normal" forms.  The normalization process implemented for EG, in the graphical language just given, takes an arbitrary tinted cactus and turns it into a special sort of painted cactus.

Viewing a propositional expression in EG as a "cactus", the bound connectives ( , , , ) constitute its "lobes" (edges or lines) and the positive literals ai are tantamount to "colors" (paints or tints) on its "points" (vertices or nodes).  One of the chief tasks of processing logical expressions is their systematic clarification.  This involves transforming arbitrary expressions into logically equivalent expressions whose latent meaning is manifest, their "canonical" or "normal" forms.  The normalization process implemented for EG, in the graphical language just given, takes an arbitrary tinted cactus and turns it into a special sort of painted cactus.

3.1.2.2  Concept and Purpose

3.1.2.2. Concept and Purpose

What good is this?  What conceivable purpose is there for these inductive and deductive capacities, that enable the personal computer to learn formal languages and to turn propositional calculi into painted cacti?  By developing these abilities for inductive learning and accurate inference, aided by a facility for integrating their alternate "takes" on the world, I hope that AI software will gain a new savvy, one that helps it be both friendly to people and faithful to truth, both politic and correct.  To do this demands a form of artificial intelligence that can do both, without the kinds of trade-off that make it a travesty to both.

What good is this?  What conceivable purpose is there for these inductive and deductive capacities, that enable the personal computer to learn formal languages and to turn propositional calculi into painted cacti?  By developing these abilities for inductive learning and accurate inference, aided by a facility for integrating their alternate "takes" on the world, I hope that AI software will gain a new savvy, one that helps it be both friendly to people and faithful to truth, both politic and correct.  To do this demands a form of artificial intelligence that can do both, without the kinds of trade-off that make it a travesty to both.

3.1.3  Applications

3.1.3. Applications

The current implementation of this calculus is efficient enough to have played a meaningful part in realistically complex investigations, both practical and theoretical.  For example, it has been used in qualitative research to represent observational protocols of event sequences as propositional data bases.  It has also been used to analyze the behavior of finite state machines and space-time limited Turing machines, exploiting a coding that is similar to but more succinct than the one used in Cook's theorem (on the NP-completeness of propositional calculus satisfiability).  See (Garey & Johnson, 1979) and (Wilf, 1986).

The current implementation of this calculus is efficient enough to have played a meaningful part in realistically complex investigations, both practical and theoretical.  For example, it has been used in qualitative research to represent observational protocols of event sequences as propositional data bases.  It has also been used to analyze the behavior of finite state machines and space-time limited Turing machines, exploiting a coding that is similar to but more succinct than the one used in Cook's theorem (on the NP-completeness of propositional calculus satisfiability).  See (Garey & Johnson, 1979) and (Wilf, 1986).

3.2  Differential Extensions of Propositional Calculi

3.2. Differential Extensions of Propositional Calculi

In order to define a differential extension of a propositional universe of discourse U, the alphabet A of U's defining features must be extended to include a set of symbols for differential features, or elementary "changes" in the universe of discourse.  Intuitively, these symbols may be construed as denoting primitive features of change, or propositions about how things or points in U change with respect to the features noted in the original alphabet A.  Hence, let dA = {da1,...,dan} and dU = <dA> = <da1,...,dan>.  As before, we may express dU concretely as a product of distinct factors:

In order to define a differential extension of a propositional universe of discourse U, the alphabet A of U's defining features must be extended to include a set of symbols for differential features, or elementary "changes" in the universe of discourse.  Intuitively, these symbols may be construed as denoting primitive features of change, or propositions about how things or points in U change with respect to the features noted in the original alphabet A.  Hence, let dA = {da1,...,dan} and dU = <dA> = <da1,...,dan>.  As before, we may express dU concretely as a product of distinct factors:

Up to this point my terminology, to the extent that it matters for the qualitative case, has been roughly consistent with the usage in standard accounts, e.g. (Chevalley, 1946) and (Doolin & Martin, 1990).  The treatment that follows is much more tentative.  I am less certain here about the best way to adapt the geometric concepts to the logical context.

Up to this point my terminology, to the extent that it matters for the qualitative case, has been roughly consistent with the usage in standard accounts, e.g. (Chevalley, 1946) and (Doolin & Martin, 1990).  The treatment that follows is much more tentative.  I am less certain here about the best way to adapt the geometric concepts to the logical context.

A word of preparation for what is to come:  much of the scaffolding we need to build will seem overly definitional and lacking in substance.  These te deums are not recited for their own sake merely, but are dictated by our desire for computational implementations, for which careful specifications are of course crucial, and yes, sometimes a bit excruciating.

A word of preparation for what is to come:  much of the scaffolding we need to build will seem overly definitional and lacking in substance.  These te deums are not recited for their own sake merely, but are dictated by our desire for computational implementations, for which careful specifications are of course crucial, and yes, sometimes a bit excruciating.