From the Signal to the Symbol: Structure and Process in Artificial Intelligence

From the Signal to the Symbol:

Structure and Process in Artificial Intelligence

Marko A. RodriguezT-5, Center for Nonlinear StudiesLos Alamos National Laboratory

http://markorodriguez.com

November 13, 2008

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

2

General Introduction

• We receive signals that are noisy, never identical, and yet we have astable 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 withmore specificity, to a particular dog you know.).

• In higher-level thinking (i.e. at the level of “conscious awareness”), wereason 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.


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 processsignal features and relations feature distance and activation

symbol objects and relations rules of inference

1Ontological commitment means the assumptions about the world/environment that the system assumesto be true.


4

Introduction to our Experimental Subjects andNotation 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

2These images were found on the web many moons ago and apologies to the fine people who createdthem and will only get this meager credit.


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


6

Outline


? The HMAX Model? Self-Organizing Maps



? Evidential Logics




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 cortexresearch.

Riesenhuber, M., Poggio, T., “Hierarchical models of object recognition in cortex”, Nature Neuroscience, volume 2, pages

1019-1025, 1999.[6]

3Depends on the learning/training procedure used as well as the choice of the low-level features codedinto the system.


8

The HMAX Model - The Structure

• The HMAX network can be defined as G = (V,E) where V is a set ofvertices (i.e. neurons, feature selectors), E ⊆ (V × V ), and there existno cycles.

• There are two types of vertices: simple and complex, where V = S ∪Cand S ∩ C = ∅. Cells are “tuned” to respond to a particular inputfeature.

...

S1

C1

...S2

C2 ...


9

The HMAX Model - The Process

• Each vertex i ∈ S is tuned to a particular feature wi ∈ Rn and performsthe function.4

si : Rn → [0, 1] : si(x) 7→ exp

(−||wi − x||2

2σ2

)

• Each vertex i ∈ C has the same excitation value as its most excitedsimple, child vertex.

ci : Rm → [0, 1] : ci(x) 7→ max(x)

4The w features at S1 are the ontological commitments of the model and are usually simple line types.


10

The HMAX Model - Example


11


S11

3

2

41 2 3 4 1 2 3 4

1

3

2

4


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S1

C1

1

3

2

41 2 3 4

... ......

1 2 3 4

... ......

1

3

2

4


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S1

C1

1

3

2

41 2 3 4

S2

... ......

1 2 3 4

... ......

1

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2

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C2

1

3

2

4

1 2

3 4

1 2

3 4

1

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2

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...

...


14

The HMAX Model - Drawbacks

• What is captured at the highest point in the hierarchy is a large list offeatures, 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.

5Complex cells can be seen as “grandmother cells”. The further up the hierarchy, the more agnostic thecell is to its object representation’s under various transformations.


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]


16

Self-Organizing Maps - The Structure

• A self-organizing map is defined as G = (V,E, ω), where V is the setof vertices, E ⊆ {V ×V } is a set of edges, and ω : E → [0, 1] defines thestrength of coupling between vertices. If (i, j) /∈ E, then ω(i, j) 7→ 0.Finally, (i, i) ∈ E and ω(i, i) 7→ 1.

• Every vertex i ∈ V has an n-dimensional feature vector wi ∈ Rn. Initiallyall vertex features are randomly generated.6

• The environment is defined by an n-dimensional space. A sample fromthat space is denoted x ∈ Rn.

6Coupling strength between vertices (i.e. edge weight) can be determined by their relative distance toone another in Rn.


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.


18

Self-Organizing Maps - Example

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marko

fluffy

mammal

amount of fur

num

ber o

f leg

s


22

Outline


? The HMAX Model



? Description Logics? Evidential Logics




23

From Categorization to Reasoning on Categories

• With signal-based systems, the “grounded” entities are very primitiveconstructs (e.g. simple line types) and from these primitive constructs itis possible generate abstract representations of patterns that are invariantto various transformations (e.g. Fluffy regardless of his location in space).

• With symbol-based systems, the “grounded” entities are generally veryabstract (e.g. Fluffy) and from these concepts its possible to reasonabstract relationships (e.g. Fluffy must be a canine because he has fur.).


24

Knowledge Representation and Reasoning

• Knowledge representation: a model of a domain of discourserepresented in some medium – structure.

• Reasoning: the algorithm by which implicit knowledge in the model ismade explicit – process.

Reasoner

read/write

Knowledge Representation

f(x)


25

Description Logics - Introduction

• The purpose of description logics is to infer subsumption relationshipsin a knowledge structure.

• Given a set of individuals (i.e. real-world instances), determine whichconcept descriptions subsume the individuals. For example, is marko atype 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]


26

Description Logics - The Structure

• A multi-relational network (aka. semantic network, directed labeledgraph) 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, whereany 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 atriple (i, n, j) (e.g. (marko, bestFriend, fluffy)).

bestFriendmarko fluffy


27


• Individual: a unique identifier denoting some “real-world thing” thatexists. For example: marko.

• Simple Concepts: a unique identifier denoting a “ground” concept. Forexample: Mammal.

• Simple Roles (aka properties): a unique identifier denoting a binaryrelationship. For example: numberOfLegs, hasFur, bestFriend.

• Compound Concept: a concept that is defined in terms of anotherconcept. For example: a Canine is a thing that has 4 legs and is furry.7

7There are many description logic languages. Distinctions between these languages are made explicit bydefining their “expressivity” (i.e. the possible forms a compound concept description can take).


28


• Terminological Box (T-Box): a collection of descriptions. Also knownas an ontology.

? Human ≡ (= 2 numberOfLegs) u (= false hasFur) u ∃bestFriend.Canine? Canine ≡ (= 4 numberOfLegs) u (= true hasFur)? Human v Mammal? Canine v Mammal

• Assertion Box (A-Box): a collection of individuals and their relationshipsto one another.

? numberOfLegs(marko, 2), hasFur(marko, false), bestFriend(marko, fluffy),

numberOfLegs(fluffy, 4), hasFur(fluffy, true).


29

Description Logics - The Process

• Inference rules (Reasoner): a collection of pattern descriptions are usedto 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)

? . . .


30

Description Logics - Example

Mammal

Canine

subClassOf

Canine ! MammalHuman ! Mammal

subClassOf

Restriction_D

onProperty

numberOfLegs

hasValue

4

subClassOf

Restriction_E

onProperty

hasFur

hasValue

true

Human

subClassOf

Restriction_B

onProperty

numberOfLegs

hasValue

2

subClassOf

Restriction_C

onProperty

hasFur

hasValue

false

subClassOf

Restriction_A

onProperty

bestFriend

someValuesFrom

subClassOf

Canine ! (= 4 numberOfLegs) " (= true hasFur)Human ! (= 2 numberOfLegs) " (= false hasFur) " #bestFriend.Canine

T-Box


31


marko fluffybestFriend

2 false 4 true

numberOfLegs hasFur numberOfLegs hasFur

numberOfLegs(fuffy, 4)

bestFriend(marko, fluffy)

numberOfLegs(marko, 2)

hasFur(marko, false) hasFur(fluffy, true) A-Box


32


marko fluffybestFriend

2 false 4 true

numberOfLegs hasFur numberOfLegs hasFur

Mammal

Human Canine

type

subClassOf

type

subClassOf

T-Box

A-Box

* The T-Box includes other description information, but for diagram clarity, this was left out.

type type

inferred

Yes — marko is a type of Mammal.


33

Description Logics - Drawbacks

• With “nested” descriptions and complex quantifiers, you can run intoexponential running times.

• Requires that all assertions in the A-Box are “true”. For example, ifthe T-Box declares that a country can have only one president and youassert that barack is the president of the United States and that markois the president of the United States, then it is inferred that barack andmarko are the same person. And this can have rippling effects such astheir 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 predicatescome from? Where do all the inference rules come from?


34

Evidential Logics - Introduction

Evidential logics are multi-valued logics founded on AIKIR (Assumption ofInsufficient Knowledge and Insufficient Resources) and are:

• non-bivalent: there is no inherent truth in a statement, only differingdegrees of support or negation.

• non-monotonic: the evaluation of the “truth” of a statement is notimmutable, but can change as new experiences occur. In other words, asnew evidence is accumulated.

Wang, P., “Cognitive Logic versus Mathematical Logic”, Proceedings of the Third International Seminar on Logic and Cognition,

May 2004.[8]


35

Evidential Logics - The Structure

• An evidence network is defined as G = (V,E, ω), where V is the setof vertex (i.e. symbols), E ⊆ (V × V ) is a set of directed edges, andω : 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. “Howmuch positive and negative evidence is there for marko inheriting theproperties of Human”?

8Every evidence tuple 〈w+, w−〉 has a mapping to 〈f, c〉 ∈ 〈[0, 1], [0, 1]〉 that is perhaps more

“natural” to work with. f = w+

w++w−denotes frequency of positive evidence and c = w++w−

w++w−+kdenotes

confidence in stability of the frequency, where k ∈ N+ is a user-defined constant.


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 fluffy9It is helpful to think of the copula as “inherits the properties of” instead of “is a”.

10Exemplification is a much less used syllogism in evidential reasoning.


37

Evidential Logics - Example

Assume that the past experience of the evidential system has providedthese 〈w+, w−〉 evidential tuples for the following relationships.11

Human Canine

Mammal

<1,0> <1,0>

4-legs2-legs fur

<1,0><1,0> <0,1> <1,0>

11The example to follow is not completely faithful to NAL-* (Non-Axiomatic Logic). Please refer to moreexpressive NAL constructs for a better representation of the ideas presented in this example.


38


marko

Human Canine

fluffy

Mammal

<1,0> <1,0>

4-legs2-legs fur

<1,0><1,0> <0,1> <1,0>

<0,1> <1,0><1,0><1,0>

experienced


39


marko

Human Canine

fluffy

Mammal

<1,0> <1,0>

4-legs2-legs fur

<1,0><1,0> <0,1> <1,0>

<0,1> <1,0><1,0>

<1,0> <2,0>

<1,0>

D D

A

deductioninductionabduction

DI

inferred


40


marko

Human Canine

fluffy

Mammal

<1,0> <1,0>

4-legs2-legs fur

<1,0><1,0> <0,1> <1,0>

<0,1> <1,0><1,0>

<1,0> <2,0>

<1,0>I

<1,0>A

A


DI

inferred

<0,1>


41


marko

Human Canine

fluffy

Mammal

<1,0> <1,0>

4-legs2-legs fur

<1,0><1,0> <0,1> <1,0>

<0,1> <1,0><1,0>

<1,0> <2,0>

<1,0>

<1,0>

<0,1> <1,0>

inferred<1,0>D

A


DI

Yes — currently, marko is believed to be a type of Mammal.


42

Evidential Logics - Versions

What was presented was an evidential logic known as NAL-1 orNon-Axomatic Logic 1. There exist more expressive forms that are basedon 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]


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-basedassertions (marko, Human) and queries (marko, ?x).


44

Outline


? The HMAX Model




? Evidential Logics




45

Unification of Symbol and Signal - Introduction

• Signal to symbol: receive input signals and map them to transformationinvariant 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 inferenceto generate new relations that might not be based on external signalalone. – reasoning

12Symbols need not be labeled, just unique. In other words, some vertex must denote Fluffy, yet need notbe labeled fluffy.


46

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”).

Mammal

S1

C1

Cm...

Human

...

marko

Symbols need not exist. They are provided for diagram claritye.g. marko's c vertex is just some unique identifier (e.g. abcd1234)

armci


47

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?

Mammal

S1

C1

Cm...

Human

...

marko

arm

<1,0>

<1,0>

<1,0>

ci


48

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.

Mammal

S1

C1

Cm...

Human

...

marko

arm

<1,0>

<1,0>

<1,0>

<1,0>

ci


49

Outline


? The HMAX Model




? Evidential Logics




50

A Distributed Graph in an Infinite Space - Introduction

• The Uniform Resource Identifier (URI) provides an infinite, globaladdress 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 graphingURIs 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 directedlabeled graph. The Semantic Web is for data what the World Wide Webis for documents.

13Namespace prefixes are denoted for brevity, where http://www.lanl.gov#marko is expressed aslanl:marko.

14Universally Unique Identifiers (UUIDs) are 232 bit identifiers that can be used as globally uniqueidentifiers (e.g. lanl:fb5d2990-b111-11dd-ad8b-0800200c9a66).


51

A Distributed Graph in an Infinite Space - Example

127.0.0.1127.0.0.2

lanl:marko vub:fluffylanl:bestFriend

"2"^^xsd:integer "false"^^xsd:boolean

lanl:numberOfLegslanl:hasFur


"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.


52

A Distributed Graph in an Infinite Space - Example

127.0.0.1127.0.0.2

lanl:marko vub:fluffylanl:bestFriend

"2"^^xsd:integer "false"^^xsd:boolean


"4"^^xsd:integer "true"^^xsd:boolean

ad8e ad8f ad81

ad8bad8a ad8c ad8d

ad82


owl:sameAs

n:super... ...

127.0.0.3

n:supern:super

n:super

n:supern:super

n:superhttp://www.images.com/marko.jpg

127.0.0.4

n:region00


53

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]


54

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 andworlds in category theory: Implications for neural systems. Axiomathes,16:165–214, 2006.

[3] Francis Heylighen. Collective intelligence and its implementation on theweb: Algorithms to develop a collective mental map. Computational &Mathematical Organization Theory, 5(3):253–280, 1999.


55

[4] Ray S. Jackendoff. Languages of the Mind. MIT Press, 1992.

[5] Teuvo Kohonen. Self-organized formation of topologically correctfeature maps. Biological Cybernetics, 43:59–69, 1982.

[6] M. Riesenhuber and T. Poggio. Hierarchical models of ob jectrecognition in cortex. Nature Neuroscience, 2:1019–1025, 1999.

[7] Thomas Serre, Aude Oliva, and Tomaso Poggio. A feedforwardarchitecture accounts for rapid categorization. Proceedings of theNational Academy of Science, 104(15):6424–6429, April 2007.

[8] Pei Wang. Cognitive logic versus mathematical logic. In Proceedings ofthe Third International Seminar on Logic and Cognition, May 2004.

[9] Pei Wang. Rigid Flexibility. Springer, 2006.


Technology

From the Signal to the Symbol: Structure and Process in Artificial Intelligence