The Logic of IntelligenceThe Logic of Intelligence

Pei WangDepartment of Computer and Information Sciences

Temple University

Artificial General Intelligence

Main­stream­AI­treats­“Intelligence”­as­a­collection­of­problem-specific­and­domain-specific­parts

Artificial­General­Intelligence­(AGI)­takes­“Intelligence”­as­a­general-­purpose­capability­that­should­be­treated­as­a­whole

AGI­research­still­includes­different­research­objectives­and­strategies

Artificial Intelligence and Logic

“Intelligence”­can­be­understood­as­“rationality”­and­“validity”­---­“do­the­right­thing”

In­general,­“logic”­is­the­study­of­valid reasoning,­or­the­regularity­in­thinking

Therefore,­an­AI­system­may­be­built­according­to­a­logic,­by­converting­various­thinking­processes­into­reasoning­processes

Reasoning System

A­reasoning­system­typically­consists­of­the­following­major­components:

a­formal­language a­semantic­theory a­set­of­inference­rules a­memory­structure a­control­mechanismThe­first­three­are­usually­called­a­“logic”

Traditional Theories

Language­and­inference­rules:­first-order­predicate­calculus

Semantics:­model­theory Memory:­relational­or­object-oriented­data­structures­and­database

Inference­control:­theory­of­computation­(algorithm,­computability,­and­computational­complexity)

Problems of Traditional Theories Uncertainty:­fuzzy concepts, changing meanings

and truth values, plausible results, conflicting evidence, nondeterministic inference process, …

Semantic­justification­of­non-deductive­inference:­induction, abduction, analogy,­…

Counter-intuitive­results:­sorites paradox, implication paradox, confirmation paradox, Wason’s selection task,­…

Computability­and­complexity:­termination problem,­combinatorial explosion,­…

Proposed Solutions non-monotonic­logic paraconsistent­logic relevance­logic probabilistic­logic fuzzy­logic inductive­logic temporal­logic modal­logic situation­calculus possible­world­theory

mental­logic mental­model case-based­reasoning Bayesian­network neural­network genetic­algorithm heuristic­algorithm learning­algorithm anytime­algorithm­­­­…­…

Common Root of the Problems

The­traditional­theories­were­developed­in­the­study­of­the­foundation­of­mathematics,­while­the­problems­appear­outside­math

The­logic­of­mathematics­may­be­different­from­the­logic­of­cognition

In­mathematical­reasoning,­the­knowledge­and­resources­are­assumed­to­be­sufficient­(with­respect­to­the­tasks)

Different Types of Systems “Pure-axiomatic­system”:­the­system’s­knowledge­and­resources­are­assumed­to­be­sufficient

“Semi-axiomatic­system”:­certain­aspects­(but­not­all)­of­the­knowledge­and­resources­are­assumed­to­be­sufficient

“Non-axiomatic­system”:­the­knowledge­and­resources­of­the­system­are­assumed­to­be­generally­insufficient

NARS (Non-Axiomatic Reasoning System)

NARS­uses­a­formal­logic­(language,­semantics,­inference­rules)­and­is­implemented­in­a­computer­system

NARS­is­fully­based­on­the­assumption­of­insufficient­knowledge­and­resources,­in­the­sense­of­being­a­finite,­real time,­open,­and­adaptive­system

NARS­is­different­from­traditional­theories­in­all­major­components

Inheritance Based Representation

S P : there­is­an­inheritance relation­from­term­S­to­term­P

S is­a specialization­of P­P is­a­generalization­of S

Inheritance­is­reflexive­and­transitive

bird animal

Extension and Intension

For­a­given­term­T,its extension TE­=­{x­|­x­­T}its intension TI­=­{x­|­T ­x}

TTE TI

Theorem:­(S ­P)­ ­­(SE PE)­ (PI­­SI)

Therefore,­“Inheritance”­means­“inheritance­of­extension/intension”

Evidence

Positive­evidence­of­S ­P­:{x­|­x ­(SE

­PE)­­(PI ­SI)}

Negative­evidence­of­S ­P­:{x | x ­(SE

–­PE)­­(PI –­SI)}

S P

Amount­of­evidence:­­­­positive:­ w+­=­|­SE

­PE |­+­|­PI

­SI |

­­­negative:­ w–­=­|­SE –­PE|­+­|­PI

–­SI|

­­­total:­ w = w+­+­w–­=­|­SE |­+­|­PI

|

Truth Value

­­­In­NARS,­the­truth­value­of­a­statement­is­a­pair­of­numbers,­and­measures­the­evidential­support­to­the­statement.

S P[f, c]S P [f,­c] f:­frequency,­w+/w c:­confidence,­w­/­(w +1)­

Experience-Grounded Semantics

The­truth­value­of­a­statement­is­defined­according­to­certain­“idealized­experience”,­consisting­of­a­set­of­binary­inheritance­statements

The­meaning­of­a­term­is­defined­by­its­extension­and­intension,­according­to­certain­“idealized­experience”

So­meaning­and­truth-value­changes­according­to­the­system’s­experience

Syllogistic Inference Rules

A­typical­syllogistic­inference­rule­takes­a­pair­of­premises­with­a­common­term,­and­produces­a­conclusion

The­truth­value­of­the­conclusion­is­calculated­by­a­truth-value­function­

Different­combinations­of­premises­trigger­different­rules­(with­different­truth-value­functions)

To Design a Truth-value Function

1.­Treat­all­involved­variables­as­Boolean­(binary)­variables

2.­For­each­value­combination­in­premises,­decide­the­values­in­conclusion

3.­Build­Boolean­functions­among­the­variables4.­Extend­the­functions­to­real-number:

not(x)­=­1­–­xand(x,­y)­=­x­*­yor(x,­y)­=­1­–­(1­–­x)­*­(1­–­y)

Deduction

­­bird­­animal­[1.00,­0.90]­robin­bird­­­­­[1.00,­0.90]

­robin­­animal­[1.00,­0.81]

M P [f1, c1] S M [f2, c2] S P [f, c]

f = f1 * f2

c = c1 * c2 * f1 * f2

M

S P

Induction

­swan­bird­­­­­­­­­[1.00,­0.90]­swan­­swimmer­[1.00,­0.90]

­­­bird­­swimmer­[1.00,­0.45]

M P [f1, c1] M S [f2, c2] S P [f, c]

f = f1

c = f2 * c1 * c2 / (f2 * c1 * c2 + 1)

S

M

P

Abduction

seabird­swimmer­[1.00,­0.90]­­­­­gull­­swimmer­[1.00,­0.90]­­­­­gull­­seabird­­­[1.00,­0.45]

P M [f1, c1] S M [f2, c2] S P [f, c]

f = f2

c = f1 * c1 * c2 / (f1 * c1 * c2 + 1)

S

M

P

Revision

­bird­­swimmer­[1.00,­0.62]­bird­­swimmer­[0.00,­0.45]

­bird­­swimmer­[0.67,­0.71]

S P [f1, c1] S P [f2, c2] S P [f, c]

f =f1 * c1 * (1 - c2) + f2 * c2 * (1 - c1)

c1 * (1 - c2) + c2 * (1 - c1)

c1 * (1 - c2) + c2 * (1 - c1)

c1 * (1 - c2) + c2 * (1 - c1) + (1 - c2) * (1 - c1)

c =

S P

Other Inference Rules

M­­P­[f1,­c1]S­­M­[f2,­c2]­­S­­P­[f,­c]

analogy

union

implication P M [f1, c1] S M [f2, c2](S P) M [f, c]

B C [f1, c1]A B [f2, c2] A C [f, c]

Other Relations and InheritanceAn­arbitrary­statement­R(a, b, c)­can­be­rewritten­as­inheritance­relations­with­compound­terms:

(*,­a, b, c)­­R“The­relation­among­a, b, c­is­a­kind­of­R.”

a­­(/,­R,­_,­b, c)“a is­such­an­x that­satisfies­R(x, b, c).”

b­­(/,­R,­a,­_,­c)“b is­such­an­x that­satisfies­R(a, x, c).”

c­­(/,­R,­a, b,­_)“c is­such­an­x that­satisfies­R(a, b, x).”

Memory as a Belief Network

bird

gull

swan

robinswimmer

crow

feathered_creature

[1.00, 0.90] [1.00, 0.90]

[0.0

0, 0

.90]

[1.00, 0.90]

[1.00, 0.90] [1.00, 0.90]

[1.00, 0.90]

[1.00, 0.90]The­knowledge­of­the­system­is­a­network­of­beliefs­among­terms.­A­term­with­all­of­its­beliefs­is­a­concept

Cbird

Inference TasksNARS­accepts­several­types­of­inference­tasks: Knowledge­to­be­absorbed Questions­to­be­answered Goals­to­be­achievedA­task­is­stored­in­the­corresponding­conceptsTo­process­each­task­means­letting­it­interacts­

with­the­available­beliefs­in­the­conceptThis­process­usually­generates­new­tasks,­

beliefs,­and­concepts,­recursively

Inference Process

NARS­runs­by­repeating­the­following­cycle:1. Choose­a­concept­within­the­memory2. Choose­a­task­within­the­concept3. Choose­a­belief­within­the­concept4. Use­inference­rules­to­produce­new­tasks5. Return­the­used­items­to­memory6. Add­the­new­tasks­into­the­memory­and­

provide­an­answer­if­available

Control Strategy

NARS­maintains priority distributions­among­data­items,­uses­them­to­make­choice,­and­adjusts­them­after­each­step

Factors­influence­priority: quality­of­the­item usefulness­of­the­item­in­history relevance­of­the­item­to­the­current­context

Architecture and Working Cycle

Design and ImplementationThe­conceptual­design­of­NARS­has­been­

described­in­a­series­of­publicationsMost­parts­of­the­design­have­been­

implemented­in­several­prototypes,­and­the­current­version­is­open­source­in­Java

Working­examples­exist­as­proof­of­concept,­and­only­cover­single-step­inference­or­short­inference­processes

The­project­is­on-going,­though­has­produced­novel­and­interesting­results­

Unified Solutions The­truth­value­uniformly­represents­various­kinds­of­uncertainty

The­truth­value­depends­on­both­positive­and­negative­evidence

The­non-deductive­inference­rules­is­justified­according­to­the­semantics

The­meaning­of­a­term­is­determined­by­its­experienced­relations­with­other­terms

With­syllogistic­rules,­the­premises­and­conclusions­must­be­semantically­related

The­inference­processes­in­NARS­does­not­follow­predetermined­algorithms

ConclusionsIt­is­possible­to­build­a­reasoning­system­that­adapts­to­its­environment,­and­works­with­insufficient­knowledge­and­resources

Such­a­system­provides­a­unified­solution­to­many­problems­in­A(G)I

There­is­a logic of intelligence,­­though­it­is­fundamentally­different­from­the logic of mathematics