The Logic of Intelligence

1
The Logic of Intelligence

• Pei Wang
• Department of Computer and Information Sciences
• Temple University

2
Artificial General Intelligence

• Mainstream AI treats Intelligence as a
  collection of problem-specific and
  domain-specific parts
• Artificial General Intelligence (AGI) takes
  Intelligence as a general-purpose capability
  that should be treated as a whole
• AGI research still includes different research
  objectives and strategies

3
Artificial Intelligence and Logic

• Intelligence can be understood as rationality
  and validity --- do the right thing
• In general, logic is the study of valid
  reasoning, or the regularity in thinking
• Therefore, an AI system may be built according to
  a logic, by converting various thinking processes
  into reasoning processes

4
Reasoning System

• A reasoning system typically consists of the
  following major components
• a formal language
• a semantic theory
• a set of inference rules
• a memory structure
• a control mechanism
• The first three are usually called a logic

5
Traditional Theories

• Language and inference rules first-order
  predicate calculus
• Semantics model theory
• Memory relational or object-oriented data
  structures and database
• Inference control theory of computation
  (algorithm, computability, and computational
  complexity)

6
Problems of Traditional Theories

• Uncertainty fuzzy concepts, changing meanings
  and truth values, plausible results, conflicting
  evidence, nondeterministic inference process,
• Semantic justification of non-deductive
  inference induction, abduction, analogy,
• Counter-intuitive results sorites paradox,
  implication paradox, confirmation paradox,
  Wasons selection task,
• Computability and complexity termination
  problem, combinatorial explosion,

7
Proposed Solutions

• non-monotonic logic
• paraconsistent logic
• relevance logic
• probabilistic logic
• fuzzy logic
• inductive logic
• temporal logic
• modal logic
• situation calculus
• possible world theory

• mental logic
• mental model
• case-based reasoning
• Bayesian network
• neural network
• genetic algorithm
• heuristic algorithm
• learning algorithm
• anytime algorithm

8
Common Root of the Problems

• The traditional theories were developed in the
  study of the foundation of mathematics, while the
  problems appear outside math
• The logic of mathematics may be different from
  the logic of cognition
• In mathematical reasoning, the knowledge and
  resources are assumed to be sufficient (with
  respect to the tasks)

9
Different Types of Systems

• Pure-axiomatic system the systems knowledge
  and resources are assumed to be sufficient
• Semi-axiomatic system certain aspects (but not
  all) of the knowledge and resources are assumed
  to be sufficient
• Non-axiomatic system the knowledge and
  resources of the system are assumed to be
  generally insufficient

10
NARS (Non-Axiomatic Reasoning System)

• NARS uses a formal logic (language, semantics,
  inference rules) and is implemented in a computer
  system
• NARS is fully based on the assumption of
  insufficient knowledge and resources, in the
  sense of being a finite, real time, open, and
  adaptive system
• NARS is different from traditional theories in
  all major components

11
Inheritance Based Representation

• S ? P there is an inheritance relation from
  term S to term P
• S is a specialization of P
• P is a generalization of S

12
Extension and Intension

• For a given term T,
• its extension TE x x ? T
• its intension TI x T ? x

Theorem (S ? P) ? (SE ? PE) ? (PI ? SI)
Therefore, Inheritance means inheritance of
extension/intension
13
Evidence

• Positive evidence of S ? P
• x x ? (SE ? PE) ? (PI ? SI)
• Negative evidence of S ? P
• x x ? (SE PE) ? (PI SI)

Amount of evidence positive w SE ? PE
PI ? SI negative w SE PE
PI SI total w w w SE PI
14
Truth Value

• In NARS, the truth value of a statement is a
  pair of numbers, and measures the evidential
  support to the statement.

• S ?P f, c
• f frequency, w/w
• c confidence, w / (w 1)

15
Experience-Grounded Semantics

• The truth value of a statement is defined
  according to certain idealized experience,
  consisting of a set of binary inheritance
  statements
• The meaning of a term is defined by its extension
  and intension, according to certain idealized
  experience
• So meaning and truth-value changes according to
  the systems experience

16
Syllogistic Inference Rules

• A typical syllogistic inference rule takes a pair
  of premises with a common term, and produces a
  conclusion
• The truth value of the conclusion is calculated
  by a truth-value function
• Different combinations of premises trigger
  different rules (with different truth-value
  functions)

17
To Design a Truth-value Function

• 1. Treat all involved variables as Boolean
  (binary) variables
• 2. For each value combination in premises, decide
  the values in conclusion
• 3. Build Boolean functions among the variables
• 4. Extend the functions to real-number
• not(x) 1 x
• and(x, y) x y
• or(x, y) 1 (1 x) (1 y)

18
Deduction
M ? P f1, c1 S ? M f2, c2 ¾¾¾¾¾¾¾ S ? P
f, c
f f1 f2 c c1 c2 f1 f2

• bird ? animal 1.00, 0.90
• robin ? bird 1.00, 0.90
• ¾¾¾¾¾¾¾¾¾¾
• robin ? animal 1.00, 0.81

19
Induction
M ? P f1, c1 M ? S f2, c2 ¾¾¾¾¾¾¾ S ? P
f, c
f f1 c f2 c1 c2 / (f2 c1 c2 1)

• swan ? bird 1.00, 0.90
• swan ? swimmer 1.00, 0.90
• ¾¾¾¾¾¾¾¾¾¾¾
• bird ? swimmer 1.00, 0.45

20
Abduction
P ? M f1, c1 S ? M f2, c2 ¾¾¾¾¾¾¾ S ? P
f, c
f f2 c f1 c1 c2 / (f1 c1 c2 1)

• seabird ? swimmer 1.00, 0.90
• gull ? swimmer 1.00, 0.90
• ¾¾¾¾¾¾¾¾¾¾¾¾¾
• gull ? seabird 1.00, 0.45

21
Revision
f1 c1 (1 - c2) f2 c2 (1 -
c1) ¾¾¾¾¾¾¾¾¾¾ c1 (1 - c2) c2 (1 -
c1) c1 (1 - c2) c2 (1 -
c1) ¾¾¾¾¾¾¾¾¾¾¾¾¾ c1 (1 - c2) c2 (1 - c1)
(1 - c2) (1 - c1)
S ? P f1, c1 S ? P f2, c2 ¾¾¾¾¾¾¾ S ? P
f, c
f
c

• bird ? swimmer 1.00, 0.62
• bird ? swimmer 0.00, 0.45
• ¾¾¾¾¾¾¾¾¾¾¾
• bird ? swimmer 0.67, 0.71

22
Other Inference Rules
analogy

• M ? P f1, c1
• S ? M f2, c2
• ¾¾¾¾¾¾¾
• S ? P f, c

union
P ? M f1, c1 S ? M f2,
c2 ¾¾¾¾¾¾¾¾¾ (S ? P) ? M f, c
implication
B ? C f1, c1 A ? B f2, c2 ¾¾¾¾¾¾¾ A ? C f,
c
23
Other Relations and Inheritance

• An arbitrary statement R(a, b, c) can be
  rewritten as inheritance relations with compound
  terms
• (, a, b, c) ? R
• The relation among a, b, c is a kind of R.
• a ? (/, R, _, b, c)
• a is such an x that satisfies R(x, b, c).
• b ? (/, R, a, _, c)
• b is such an x that satisfies R(a, x, c).
• c ? (/, R, a, b, _)
• c is such an x that satisfies R(a, b, x).

24
Memory as a Belief Network
The knowledge of the system is a network of
beliefs among terms. A term with all of its
beliefs is a concept
Cbird
25
Inference Tasks

• NARS accepts several types of inference tasks
• Knowledge to be absorbed
• Questions to be answered
• Goals to be achieved
• A task is stored in the corresponding concepts
• To process each task means letting it interacts
  with the available beliefs in the concept
• This process usually generates new tasks,
  beliefs, and concepts, recursively

26
Inference Process

• NARS runs by repeating the following cycle
• Choose a concept within the memory
• Choose a task within the concept
• Choose a belief within the concept
• Use inference rules to produce new tasks
• Return the used items to memory
• Add the new tasks into the memory and provide an
  answer if available

27
Control Strategy

• NARS maintains priority distributions among data
  items, uses them to make choice, and adjusts them
  after each step
• Factors influence priority
• quality of the item
• usefulness of the item in history
• relevance of the item to the current context

28
Architecture and Working Cycle
29
Design and Implementation

• The conceptual design of NARS has been described
  in a series of publications
• Most parts of the design have been implemented in
  several prototypes, and the current version is
  open source in Java
• Working examples exist as proof of concept, and
  only cover single-step inference or short
  inference processes
• The project is on-going, though has produced
  novel and interesting results

30
Unified Solutions

• The truth value uniformly represents various
  kinds of uncertainty
• The truth value depends on both positive and
  negative evidence
• The non-deductive inference rules is justified
  according to the semantics
• The meaning of a term is determined by its
  experienced relations with other terms
• With syllogistic rules, the premises and
  conclusions must be semantically related
• The inference processes in NARS does not follow
  predetermined algorithms

31
Conclusions

• It is possible to build a reasoning system that
  adapts to its environment, and works with
  insufficient knowledge and resources
• Such a system provides a unified solution to many
  problems in A(G)I
• There is a logic of intelligence, though it is
  fundamentally different from the logic of
  mathematics