There is a divide in the domain of artificial intelligence. On the one end of this divide are the various sub-symbolic, or signal-based systems that are able to distill stable representations from a potentially noisy signal. Pattern recognition and classification are typical uses of such signal-based systems. On the other side of the divide are various symbol-based systems. In these systems, the lowest-level of representation is that of the a priori determined symbol, which can denote something as high-level as a person, place, or thing. Such symbolic systems are used to model and reason over some domain of discourse given prescribed rules of inference. An example of the unification of this divide is the human. The human perceptual system performs signal processing to yield the rich symbolic models that form the majority of our interpretation of and reasoning about the world. This presentation will provide an introduction to different signal and symbol systems and discuss the unification of this divide.

1. From the Signal to the Symbol:
Structure and Process in Artificial Intelligence
Marko A. Rodriguez
T-5, Center for Nonlinear Studies
Los Alamos National Laboratory
http://markorodriguez.com
November 13, 2008

2. 1
Abstract
There is a divide in the domain of artificial intelligence. On the one end of this divide
are the various sub-symbolic, or signal-based systems that are able to distill stable
representations from a potentially noisy signal. Pattern recognition and classification
are typical uses of such signal-based systems. On the other side of the divide are
various symbol-based systems. In these systems, the lowest-level of representation
is that of the a priori determined symbol, which can denote something as high-level
as a person, place, or thing. Such symbolic systems are used to model and reason
over some domain of discourse given prescribed rules of inference. An example of
the unification of this divide is the human. The human perceptual system performs
signal processing to yield the rich symbolic models that form the majority of our
interpretation of and reasoning about the world. This presentation will provide an
introduction to different signal and symbol systems and discuss the unification of
this divide.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

3. 2
General Introduction
• We receive signals that are noisy, never identical, and yet we have a
stable representation of “reality”.
• Signals from different modalities can map to the same abstract concepts
(e.g. hearing a dog bark and seeing a dog, both map to dog. Or with
more specificity, to a particular dog you know.).
• In higher-level thinking (i.e. at the level of “conscious awareness”), we
reason in terms of these abstract concepts, not in terms of the signals
(e.g. “This dog has no owner, it must be a stray.”).
• Both signal and symbol processing occur in the same neural substrate.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

4. 3
General Introduction
• A distinction between signal and symbol systems:
signal: the information processed by the system is very “low-level” (e.g. simple
geometric patterns) and makes few ontological commitments.1
symbol: the information processed by the system is very “high-level” (e.g. people)
and makes many ontological commitments.
• A distinction between the structure and process of systems:
structure: the types of objects that compose the system.
process: the types of mappings that evolve the system.
structure process
signal features and relations feature distance and activation
symbol objects and relations rules of inference
1
Ontological commitment means the assumptions about the world/environment that the system assumes
to be true.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

5. 4
Introduction to our Experimental Subjects and
Notation Conventions
Our example subjects are Marko and Fluffy:2
All formalisms are going to be presented in graph notation and using the same variable
names as best as possible.
• G graph, V vertices, E edges, E family of edge sets
• i, j ∈ V , (i, j) ∈ E , (i, n, j) a statement or triple, w+, w− evidence tuple
• x ∈ Rn input vector, w ∈ Rm feature vector
2
These images were found on the web many moons ago and apologies to the fine people who created
them and will only get this meager credit.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

6. 5
Outline
• Signal Representation
The HMAX Model
Self-Organizing Maps
• Symbol Representation
Description Logics
Evidential Logics
• Unifying Signals and Symbols
• A Distributed Graph in an Infinite Space
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

7. 6
Outline
• Signal Representation
The HMAX Model
Self-Organizing Maps
• Symbol Representation
Description Logics
Evidential Logics
• Unifying Signals and Symbols
• A Distributed Graph in an Infinite Space
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

8. 7
The HMAX Model - Introduction
• Object recognition/classification through low-level feature analysis.
• Can support scale, translation, and rotation invariance.3
• Anatomically realistic with respect to the Hubel and Wiesel visual cortex
research.
Riesenhuber, M., Poggio, T., “Hierarchical models of object recognition in cortex”, Nature Neuroscience, volume 2, pages
1019-1025, 1999.[6]
3
Depends on the learning/training procedure used as well as the choice of the low-level features coded
into the system.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

9. 8
The HMAX Model - The Structure
• The HMAX network can be defined as G = (V, E) where V is a set of
vertices (i.e. neurons, feature selectors), E ⊆ (V × V ), and there exist
no cycles.
• There are two types of vertices: simple and complex, where V = S ∪ C
and S ∩ C = ∅. Cells are “tuned” to respond to a particular input
feature.
C2 ...
S2 ...
C1 ...
S1
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

10. 9
The HMAX Model - The Process
• Each vertex i ∈ S is tuned to a particular feature wi ∈ Rn and performs
the function.4
n ||wi − x||2
si : R → [0, 1] : si(x) → exp −
2σ 2
• Each vertex i ∈ C has the same excitation value as its most excited
simple, child vertex.
ci : Rm → [0, 1] : ci(x) → max(x)
4
The w features at S1 are the ontological commitments of the model and are usually simple line types.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

11. 10
The HMAX Model - Example
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

12. 11
The HMAX Model - Example
1 2 1 2
S1
3 4 3 4
1 2 3 4 1 2 3 4
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

13. 12
The HMAX Model - Example
C1
... ... ... ... ... ...
1 2 1 2
S1
3 4 3 4
1 2 3 4 1 2 3 4
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

14. 13
The HMAX Model - Example
C2 ...
1 2 1 2 1 2 1 2
S2 3 4 3 4 ... 3 4 3 4
C1
... ... ... ... ... ...
1 2 1 2
S1
3 4 3 4
1 2 3 4 1 2 3 4
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

15. 14
The HMAX Model - Drawbacks
• What is captured at the highest point in the hierarchy is a large list of
features, not their relative positions to each other. With high resolution,
the list of features turns into a unique identifier for an object (hopefully).5
• There is a distinction between learning/training and categorizing/perceiving.
5
Complex cells can be seen as “grandmother cells”. The further up the hierarchy, the more agnostic the
cell is to its object representation’s under various transformations.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

16. 15
Self-Organizing Maps - Introduction
• Self-organizing maps (aka. Kohonen networks) can be used to generate a map
(i.e. model) of an input space (i.e. environment) in an unsupervised manner.
• Each vertex in the map specializes on representing a particular region of the input
space (i.e. each vertex specializes on particular features of the environment). Denser
regions of the input space receive more vertices for their representation.
• There is no separation between learning/training and categorzing/perceiving. Every
input adjusts the feature tunings of the vertices. The more “learned” the system is to
the environment, the smaller the adjustments.
Kohonen, T., “Self-organized formation of topologically correct feature maps”, Biological Cybernetics, volume 42, pages 59-69,
1982.[5]
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

17. 16
Self-Organizing Maps - The Structure
• A self-organizing map is defined as G = (V, E, ω), where V is the set
of vertices, E ⊆ {V × V } is a set of edges, and ω : E → [0, 1] defines the
strength of coupling between vertices. If (i, j) ∈ E, then ω(i, j) → 0.
/
Finally, (i, i) ∈ E and ω(i, i) → 1.
• Every vertex i ∈ V has an n-dimensional feature vector wi ∈ Rn. Initially
all vertex features are randomly generated.6
• The environment is defined by an n-dimensional space. A sample from
that space is denoted x ∈ Rn.
6
Coupling strength between vertices (i.e. edge weight) can be determined by their relative distance to
one another in Rn .
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

18. 17
Self-Organizing Maps - The Process
The SOM algorithm proceeds according to the following looping rules:
1. Generate a sample x ∈ Rn from the environment.
2. Determine which vertex in V is closest to x via some distance function
(e.g. ||x − wi||2). Denote that vertex i.
3. For each vertex j ∈ V ,
wj ← wj + ω(j, i)(wj − x)η,
where η ∈ [0, 1] is some learning parameter.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

19. 18
Self-Organizing Maps - Example
iteration 0 iteration 1
1.0
1.0
q q q
q
q q
q
q q q
0.8
0.8
q q
q q q q
q q q q
q q q
q q q q q
q q
q qq q
qq q q qq q
qq q
q q q q q q
q qq qq q qq qq
0.6
q qq
0.6
q
q qq q qq qq
q qq q
q q
q q q
q q q q
q q q
qq qq q
qq qq
qq q q qq q q
q q
q q
q q
0.4
0.4
q
q
q q q
q q
q
q q q q
q q
q q
q
0.2
0.2
q q
q
q q
q q
q
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

20. 19
Self-Organizing Maps - Example
iteration 10 iteration 25
1.0
1.0
q q
q q
0.8
0.8
q q
q q
q q q q
q q
q q q q q q
q q
q q
q q q
q q
q qq q
qq q q q qq q
q q
qq q
q q q q q q q
q qq q q qq q
q q q
0.6
q qq
0.6
q qq
q q qq q q qqq qq q qq
q
q q q qq q q q q
qq q
q
qq qq
qq q q qq
qq q
q qq qq q
q
q
q q q
qq q q q
q
0.4
0.4
q q
q q
q q q q q
q
q q
q q
q
q q q
0.2
0.2
q q
q q
q q
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

21. 20
Self-Organizing Maps - Example
iteration 50 iteration 75
1.0
1.0
q q
q q
0.8
0.8
q q
q q q
q q
q q q q q
q q q
q q q q qq
q q
q q qq q q
q
qq qq
q q
qq
q qq q
q q q q q qq q
q qq
qq qq q
q
q q q qq
0.6
q qq
0.6
q
q qqq q q q
q qq q
q q q q q
qqq q q q
qq
qq q qq
q
q
q
q qq
q q q
qq q qq
q qq q
q
q
q q q
q q q
q q
0.4
0.4
q
q
q
q
q
q q q
q
q
q q q q q
q q
0.2
0.2
q q
q q
q q
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

22. 21
Self-Organizing Maps - Example
iteration 75
1.0
●
●
0.8
●
●
● ●
● ●
● ● ●●
● ●
● ●● ●● ● fluffy
●●
●● ●● ●● ●
number of legs
● ●● ●
● ●●
0.6
● ●
● ● ●● ●
●●● ● ● ●
●
●●
● ●●
●● ● ●
● ●
●
● ●
0.4
●
●
● mammal ●
●
● ● ●
●
0.2
●
●
●
marko
0.0
0.0 0.2 0.4 0.6 0.8 1.0
amount of fur
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

23. 22
Outline
• Signal Representation
The HMAX Model
Self-Organizing Maps
• Symbol Representation
Description Logics
Evidential Logics
• Unifying Signals and Symbols
• A Distributed Graph in an Infinite Space
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

24. 23
From Categorization to Reasoning on Categories
• With signal-based systems, the “grounded” entities are very primitive
constructs (e.g. simple line types) and from these primitive constructs it
is possible generate abstract representations of patterns that are invariant
to various transformations (e.g. Fluffy regardless of his location in space).
• With symbol-based systems, the “grounded” entities are generally very
abstract (e.g. Fluffy) and from these concepts its possible to reason
abstract relationships (e.g. Fluffy must be a canine because he has fur.).
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

25. 24
Knowledge Representation and Reasoning
• Knowledge representation: a model of a domain of discourse
represented in some medium – structure.
• Reasoning: the algorithm by which implicit knowledge in the model is
made explicit – process.
f (x) Reasoner
read/write
Knowledge Representation
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

26. 25
Description Logics - Introduction
• The purpose of description logics is to infer subsumption relationships
in a knowledge structure.
• Given a set of individuals (i.e. real-world instances), determine which
concept descriptions subsume the individuals. For example, is marko a
type of Mammal?
F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi, P. F. Patel-Schneider: The Description Logic Handbook: Theory,
Implementation, Applications. Cambridge University Press, Cambridge, UK, 2003.[1]
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

27. 26
Description Logics - The Structure
• A multi-relational network (aka. semantic network, directed labeled
graph) is defined as G = (V, E), where V is the set of vertices
(i.e. symbols), E = {E1, E2, . . . , En} is a family of edge sets, where
any En ⊆ (V × V ). Each edge set has a categorical or nominal meaning
(e.g. bestFriend, hasFur, numberOfLegs, etc.).
• An edge (i, j) ∈ En is called a “statement” and is usually denoted as a
triple (i, n, j) (e.g. (marko, bestFriend, fluffy)).
marko bestFriend fluffy
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

28. 27
Description Logics - The Structure
• Individual: a unique identifier denoting some “real-world thing” that
exists. For example: marko.
• Simple Concepts: a unique identifier denoting a “ground” concept. For
example: Mammal.
• Simple Roles (aka properties): a unique identifier denoting a binary
relationship. For example: numberOfLegs, hasFur, bestFriend.
• Compound Concept: a concept that is defined in terms of another
concept. For example: a Canine is a thing that has 4 legs and is furry.7
7
There are many description logic languages. Distinctions between these languages are made explicit by
defining their “expressivity” (i.e. the possible forms a compound concept description can take).
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

29. 28
Description Logics - The Structure
• Terminological Box (T-Box): a collection of descriptions. Also known
as an ontology.
Human ≡ (= 2 numberOfLegs) (= false hasFur) ∃bestFriend.Canine
Canine ≡ (= 4 numberOfLegs) (= true hasFur)
Human Mammal
Canine Mammal
• Assertion Box (A-Box): a collection of individuals and their relationships
to one another.
numberOfLegs(marko, 2), hasFur(marko, false), bestFriend(marko, fluffy),
numberOfLegs(fluffy, 4), hasFur(fluffy, true).
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

30. 29
Description Logics - The Process
• Inference rules (Reasoner): a collection of pattern descriptions are used
to assert new statements:
(?x, subClassOf, ?y) ∧ (?y, subClassOf, ?z) ⇒ (?x, subClassOf, ?z)
(?x, subClassOf, ?y) ∧ (?y, subClassOf, ?x) ⇒ (?x, equivalentClass, ?y)
(?x, subPropertyOf, ?y) ∧ (?y, subPropertyOf, ?z) ⇒ (?x, subPropertyOf, ?z)
(?x, type, ?y) ∧ (?y, subClassOf, ?z) ⇒ (?x, type, ?z)
(?x, onProperty, ?y) ∧ (?x, hasValue, ?z) ∧ (?a, subClassOf, ?x) ⇒ (?a, ?y, ?z)
(?x, onProperty, ?y) ∧ (?x, hasValue, ?z) ∧ (?a, ?y, ?z) ⇒ (?a, type, ?x)
. . .
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

31. 30
Description Logics - Example
Human ≡ (= 2 numberOfLegs) (= false hasFur) ∃bestFriend.Canine Canine ≡ (= 4 numberOfLegs) (= true hasFur)
bestFriend numberOfLegs 2 false hasFur numberOfLegs 4 true hasFur
onProperty hasValue onProperty
onProperty hasValue onProperty
hasValue onProperty hasValue
Restriction_A Restriction_B Restriction_C Restriction_D Restriction_E
subClassOf subClassOf
subClassOf Mammal
subClassOf subClassOf
someValuesFrom subClassOf subClassOf
Human Human Mammal Canine Mammal Canine
T-Box
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

32. 31
Description Logics - Example
bestFriend(marko, fluffy)
marko bestFriend fluffy
numberOfLegs hasFur numberOfLegs hasFur
2 false 4 true
numberOfLegs(marko, 2) numberOfLegs(fuffy, 4)
hasFur(marko, false) hasFur(fluffy, true) A-Box
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

33. 32
Description Logics - Example
inferred Mammal
subClassOf subClassOf
Human Canine
type type
T-Box
A-Box
type type
marko bestFriend fluffy
numberOfLegs hasFur numberOfLegs hasFur
2 false 4 true
* The T-Box includes other description information, but for diagram clarity, this was left out.
Yes — marko is a type of Mammal.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

34. 33
Description Logics - Drawbacks
• With “nested” descriptions and complex quantifiers, you can run into
exponential running times.
• Requires that all assertions in the A-Box are “true”. For example, if
the T-Box declares that a country can have only one president and you
assert that barack is the president of the United States and that marko
is the president of the United States, then it is inferred that barack and
marko are the same person. And this can have rippling effects such as
their mothers and fathers must be the same people, etc.
• Not very “organic” as concepts descriptions are driven, not by the system,
but by a human designer. Where do all the meta-language predicates
come from? Where do all the inference rules come from?
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

35. 34
Evidential Logics - Introduction
Evidential logics are multi-valued logics founded on AIKIR (Assumption of
Insufficient Knowledge and Insufficient Resources) and are:
• non-bivalent: there is no inherent truth in a statement, only differing
degrees of support or negation.
• non-monotonic: the evaluation of the “truth” of a statement is not
immutable, but can change as new experiences occur. In other words, as
new evidence is accumulated.
Wang, P., “Cognitive Logic versus Mathematical Logic”, Proceedings of the Third International Seminar on Logic and Cognition,
May 2004.[8]
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

36. 35
Evidential Logics - The Structure
• An evidence network is defined as G = (V, E, ω), where V is the set
of vertex (i.e. symbols), E ⊆ (V × V ) is a set of directed edges, and
ω : E → R+, R+ maps each edge to its evidence tuple.8
• Edge (i, j) can be thought of as stating “i inherits from j”, “i is a j”,
“i has properties of j”, etc.
• Every edge has two values: total amount of positive (w+) and negative
(w−) evidence supporting or negating the inheritance statement. “How
much positive and negative evidence is there for marko inheriting the
properties of Human”?
8
Every evidence tuple w+ , w− has a mapping to f, c ∈ [0, 1], [0, 1] that is perhaps more
w+ + −
“natural” to work with. f = denotes frequency of positive evidence and c = w +w denotes
w+ +w− w+ +w− +k
+
confidence in stability of the frequency, where k ∈ N is a user-defined constant.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

37. 36
Evidential Logics - The Process
Evidential reasoning is done using various syllogisms:9
• deduction: (?x, ?y) ∧ (?y, ?z) ⇒ (?x, ?z)
fluffy is a canine, canine is a mammal ⇒ fluffy is a mammal
• induction: (?x, ?y) ∧ (?z, ?y) ⇒ (?x, ?z)
fluffy is a canine, fifi is a canine ⇒ fluffy is a fifi
• abduction: (?x, ?y) ∧ (?x, ?z) ⇒ (?y, ?z)
fluffy is a canine, fluffy is a dog ⇒ canine is a dog
• exemplification: (?x, ?y) ∧ (?y, ?z) ⇒ (?z, ?x)10
fluffy is a canine, canine is a mammal ⇒ mammal is a fluffy
9
It is helpful to think of the copula as “inherits the properties of” instead of “is a”.
10
Exemplification is a much less used syllogism in evidential reasoning.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

38. 37
Evidential Logics - Example
Assume that the past experience of the evidential system has provided
these w+, w− evidential tuples for the following relationships.11
Mammal
<1,0> <1,0>
Human Canine
<1,0> <0,1> <1,0> <1,0>
2-legs fur 4-legs
11
The example to follow is not completely faithful to NAL-* (Non-Axiomatic Logic). Please refer to more
expressive NAL constructs for a better representation of the ideas presented in this example.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

39. 38
Evidential Logics - Example
experienced Mammal
<1,0> <1,0>
Human Canine
<1,0> <0,1> <1,0> <1,0>
2-legs fur 4-legs
<1,0> <0,1> <1,0> <1,0>
marko fluffy
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

40. 39
Evidential Logics - Example
inferred Mammal
<1,0> <1,0>
Human Canine
<1,0> <0,1> <1,0> <1,0>
<1,0> D <2,0> D
2-legs fur 4-legs
<1,0> <0,1> <1,0> <1,0>
D deduction
marko I induction fluffy
A abduction
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

41. 40
Evidential Logics - Example
inferred Mammal
<1,0> <1,0>
Human Canine
<1,0> <0,1> <1,0> <1,0>
<1,0> <2,0>
2-legs <0,1> fur <1,0> 4-legs
I A
<1,0> <0,1> <1,0> <1,0>
D deduction
marko I induction fluffy
A abduction
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

42. 41
Evidential Logics - Example
<1,0> Mammal
inferred D
<1,0> <1,0>
Human <1,0> Canine
<1,0> <0,1> <1,0> <1,0>
<1,0> <2,0>
2-legs <0,1> fur <1,0> 4-legs
<1,0> <0,1> <1,0> <1,0>
D deduction
marko I induction fluffy
A abduction
Yes — currently, marko is believed to be a type of Mammal.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

43. 42
Evidential Logics - Versions
What was presented was an evidential logic known as NAL-1 or
Non-Axomatic Logic 1. There exist more expressive forms that are based
on the NAL-1 core formalisms:
• NAL-0: binary inheritance – (marko, Human)
• NAL-1: inference rules – (?x, ?y) ∧ (?y, ?z) ⇒ (?x, ?z)
• NAL-2: sets and variants of inheritance – (fluffy, [fur]), ({marko}, Human)
• NAL-3: intersections and differences
• NAL-4: products, images, and ordinary relations – ((marko × fluffy), bestFriend)
• NAL-5: statement reification – ((marko × (fluffy, Canine)), knows)
• NAL-6: variables – (?x, Human) ∧ (?y, Canine) ⇒ ((?x×?y), bestFriend)
• NAL-7: temporal statements
• NAL-8: procedural statements – can model FOPL and thus, utilize an axiomatic
“subsystem”
Pei, W., “Rigid Flexibility”, Springer, 2006.[9]
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008

44. 43
Evidential Logics - Drawbacks
• The model does not provide a mechanism for how evidence is “perceived”.
All communication with the system is by means of statement-based
assertions (marko, Human) and queries (marko, ?x).
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Outline
• Signal Representation
The HMAX Model
Self-Organizing Maps
• Symbol Representation
Description Logics
Evidential Logics
• Unifying Signals and Symbols
• A Distributed Graph in an Infinite Space
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Unification of Symbol and Signal - Introduction
• Signal to symbol: receive input signals and map them to transformation
invariant symbols.12 – categorization
• Explicit relations between symbols: from similarities in input signals,
make explicit inheritance relations between symbols. – relations
• Implicit relations between symbols: utilize various rules of inference
to generate new relations that might not be based on external signal
alone. – reasoning
12
Symbols need not be labeled, just unique. In other words, some vertex must denote Fluffy, yet need not
be labeled fluffy.
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Signals to Symbols
• Signal-based systems are able to provide a (fuzzy) unique identifier for a concept. For
example, if ci(x) ≈ 1, then marko was perceived. Another way to think of it is that
ci denotes markoness. With ci : Rn → [0, 1], ci is a fuzzy classifier of the concept
“marko” (aka. “grandmother cell”).
marko Human Mammal
ci Cm
arm
... ...
C1
S1
Symbols need not exist. They are provided for diagram clarity
e.g. marko's c vertex is just some unique identifier (e.g. abcd1234)
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Explicit Relations Between Symbols
• Symbols (i.e. derived abstract concepts) can be related to one another according to
inheritance relationships. Simply, this can be based on the intersection of their features.
• For example, how much are the features that make up marko are part of the features
that make up Human? Likewise, for Human and Mammal?
marko <1,0> Human <1,0> Mammal
<1,0>
ci Cm
arm
... ...
C1
S1
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Implicit Relations Between Symbols
• Once experience has dictated the relationship between various concepts, utilize rules of
inference to “predict” or “assume” other relationships in the world.
• Validate these inferences with more experiential data.
<1,0>
marko <1,0> Human <1,0> Mammal
<1,0>
ci Cm
arm
... ...
C1
S1
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Outline
• Signal Representation
The HMAX Model
Self-Organizing Maps
• Symbol Representation
Description Logics
Evidential Logics
• Unifying Signals and Symbols
• A Distributed Graph in an Infinite Space
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A Distributed Graph in an Infinite Space - Introduction
• The Uniform Resource Identifier (URI) provides an infinite, global
address space for denoting “resources” (i.e. discrete entities, symbols,
vertices). An example URI is http://www.lanl.gov#marko.13 14
• The Resource Description Framework (RDF) is a means of graphing
URIs in a standardized, machine processable representation.
• The URI and RDF form the foundation standards of the Semantic Web.
At its most general-level, the Semantic Web is a distributed directed
labeled graph. The Semantic Web is for data what the World Wide Web
is for documents.
13
Namespace prefixes are denoted for brevity, where http://www.lanl.gov#marko is expressed as
lanl:marko.
14
Universally Unique Identifiers (UUIDs) are 232 bit identifiers that can be used as globally unique
identifiers (e.g. lanl:fb5d2990-b111-11dd-ad8b-0800200c9a66).
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A Distributed Graph in an Infinite Space - Example
127.0.0.2
127.0.0.1
lanl:marko lanl:bestFriend vub:fluffy
lanl:hasFur lanl:hasFur
lanl:numberOfLegs lanl:numberOfLegs
"2"^^xsd:integer "false"^^xsd:boolean "4"^^xsd:integer "true"^^xsd:boolean
• The concept of lanl:marko and the properties lanl:numberOfLegs, lanl:hasFur,
and lanl:bestFriend is maintained by LANL.
• The concept of vub:fluffy is maintained by VUB.
• The data types of xsd:integer and xsd:boolean are maintained by XML standards
organization.
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A Distributed Graph in an Infinite Space - Example
127.0.0.3 127.0.0.4
ad8a ad8b ad8c ad8d n:region00
n:super n:super n:super
... ... n:super
ad8e ad8f ad81
n:super http://www.images.com/marko.jpg
n:super n:super
ad82
owl:sameAs 127.0.0.2
127.0.0.1
lanl:marko lanl:bestFriend vub:fluffy
lanl:hasFur lanl:hasFur
lanl:numberOfLegs lanl:numberOfLegs
"2"^^xsd:integer "false"^^xsd:boolean "4"^^xsd:integer "true"^^xsd:boolean
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Related Interesting Work
Healy, M.J., Caudell, T.P., “Ontologies and Worlds in Category Theory: Implications for
Neural Systems”, Axiomathes, volume 16, pages 165-214, 2006.[2]
Jackendoff, R., “Languages of the Mind”, MIT Press, September 1992.[4]
Serre, T., Oliva, A., Poggio, T., “A feedforward architecture accounts for rapid
categorization”, Proceedings of the National Academy of Science, volume 104, number
15, pages 6424-6429, April 2007.[7]
Heylighen, F., “Collective Intelligence and its Implementation on the Web: Algorithms to
Develop a Collective Mental Map”, Computational & Mathematical Organization Theory,
volume 5, number 3, pages 253-280, 1999.[3]
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References
[1] Franz Baader, Diego Calvanese, Deborah L. Mcguinness, Daniele Nardi,
and Peter F. Patel-Schneider, editors. The Description Logic Handbook:
Theory, Implementation and Applications. Cambridge University Press,
January 2003.
[2] Michael John Healy and Thomas Preston Caudell. Ontologies and
worlds in category theory: Implications for neural systems. Axiomathes,
16:165–214, 2006.
[3] Francis Heylighen. Collective intelligence and its implementation on the
web: Algorithms to develop a collective mental map. Computational &
Mathematical Organization Theory, 5(3):253–280, 1999.
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56. 55
[4] Ray S. Jackendoff. Languages of the Mind. MIT Press, 1992.
[5] Teuvo Kohonen. Self-organized formation of topologically correct
feature maps. Biological Cybernetics, 43:59–69, 1982.
[6] M. Riesenhuber and T. Poggio. Hierarchical models of ob ject
recognition in cortex. Nature Neuroscience, 2:1019–1025, 1999.
[7] Thomas Serre, Aude Oliva, and Tomaso Poggio. A feedforward
architecture accounts for rapid categorization. Proceedings of the
National Academy of Science, 104(15):6424–6429, April 2007.
[8] Pei Wang. Cognitive logic versus mathematical logic. In Proceedings of
the Third International Seminar on Logic and Cognition, May 2004.
[9] Pei Wang. Rigid Flexibility. Springer, 2006.
Center for Non-Linear Studies – Los Alamos, New Mexico – November 13, 2008