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The Logic of Intelligence

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### The Logic of Intelligence

Pei Wang

Department of Computer and Information Sciences

Temple University

Logic, Cognition, and AI

In general, “logic” is the study of valid reasoning, and reasoning plays an important role in cognition.

The study of artificial intelligence aims at computer systems with the cognitive capacity of the human mind.

“Intelligence” can be understood as “rationality” and “validity” --- “to do the right thing”.

Reasoning System

A computerized 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”, and the last two use the logic in a computer system.

Traditional Theories

- language and inference rules: first-order predicate calculus,
- semantics: model theory,
- memory: relational or object-oriented database,
- inference control: theory of computation (algorithm, computability, and computational complexity).

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, Wason’s selection task, …
- computability and complexity: termination problem, combinatorial explosion, …

paraconsistent logic

relevance logic

probabilistic logic

fuzzy logic

inductive logic

temporal logic

modal logic

situation calculus

possible world theory

Proposed Solutions- mental logic
- mental model
- case-based reasoning
- Bayesian network
- neural network
- genetic algorithm
- heuristic algorithm
- learning algorithm
- anytime algorithm
… …

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).

Different Types of Systems

“pure-axiomatic system”: the system’s 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.

Non-Axiomatic Reasoning System

NARS is a reasoning system. It 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.

animal

Inheritance is reflexive and transitive.

Inheritance Based RepresentationS P : there is an inheritance relation from term S to term P.

S is aspecialization of P;

P is a generalization of S.

TI

TE

T

Extension and IntensionFor a given term T,

its extension TE = {x | xT}

its intension TI = {x | T x}

Theorem:

(S P) (SE PE) (PI SI)

Therefore, “Inheritance” means

“inheritance of extension/intension”.

Å

S

P

Å

Æ

EvidencePositive 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|

P

[f, c]

Truth ValueIn 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)

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”.

Syllogistic Inference Rules

A typical syllogistic inference rule takes a pair of premises, and produces a conclusion. The truth value of the conclusion is calculated by a function according to the semantics of NARS.

Different combinations of premises trigger different rules (with different truth-value functions).

To Design Truth-value Functions

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).

S

P

Deductionbird animal [1.00, 0.90]

robin bird [1.00, 0.90]

¾¾¾¾¾¾¾¾¾¾

robin animal [1.00, 0.81]

M P [f1, c1]

S M [f2, c2]

¾¾¾¾¾¾¾

S P [f, c]

f = f1 * f2

c = c1 * c2 * f1 * f2

S

P

Inductionswan bird [1.00, 0.90]

swan swimmer [1.00, 0.90]

¾¾¾¾¾¾¾¾¾¾¾

bird swimmer [1.00, 0.45]

M P [f1, c1]

M S [f2, c2]

¾¾¾¾¾¾¾

S P [f, c]

f = f1

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

S

P

Abductionseabird swimmer [1.00, 0.90]

gull swimmer [1.00, 0.90]

¾¾¾¾¾¾¾¾¾¾¾¾¾

gull seabird [1.00, 0.45]

P M [f1, c1]

S M [f2, c2]

¾¾¾¾¾¾¾

S P [f, c]

f = f2

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

S

Revisionbird swimmer [1.00, 0.62]

bird swimmer [0.00, 0.45]

¾¾¾¾¾¾¾¾¾¾¾

bird swimmer [0.67, 0.71]

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 =

Other Inference Rules

M P [f1, c1]

S M [f2, c2]

¾¾¾¾¾¾¾

S P [f, c]

analogy

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]

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)
“ais such an xthat satisfies R(x, b, c).”

- b(/, R,a, _, c)
“bis such an xthat satisfies R(a, x, c).”

- c(/, R,a, b, _)
“cis such an xthat satisfies R(a, b, x).”

swimmer

robin

[1.00, 0.90]

[1.00, 0.90]

[1.00, 0.90]

feathered_creature

[0.00, 0.90]

[1.00, 0.90]

[1.00, 0.90]

[1.00, 0.90]

[1.00, 0.90]

crow

bird

swan

Memory as a Belief NetworkThe knowledge of the system is a network of beliefs

among terms. A term with all of its beliefs is a concept.

Cbird

Inference Tasks

The current version of NARS accepts two types of inference tasks:

- questions to be answered,
- new knowledge to be absorbed.
For both types of tasks, the system let them interact with available beliefs.

For questions, the interactions produce answers or derived questions. For new knowledge, the interactions add or modify the relevant beliefs.

This process usually generates new tasks recursively.

A task is also stored in the corresponding concepts.

Inference Process

When running, NARS repeats the following cycle:

- Choose a concept within the memory;
- Choose a task within the concept;
- Choose a belief within the concept;
- Use applicable rules to produce new tasks;
- Return the used items to memory;
- Add the new (input and derived) tasks into the memory. Provide an answer if available.

Control Strategy

NARS maintains priority distributions among concepts, tasks, and beliefs, 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.

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.

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 cognitive sciences.

There is a logic of intelligence, though it is fundamentally different from the logic of mathematics.

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