Cellular Automata Generalized to an Inferential System：元胞自动机广义推理系统共67页文档
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cellular automata).
Some Basic Boolean Questions
1.How are Boolean functions defined? 2.How are syntactically correct Boolean
formulas produced? 3.What is a good canonical expression for
Saratoga Springs, New York, 11 July 2019
David J. Blower Cogon Systems Pensacola FL
Motivation
Why is it impossible to predict the behaviour of a cellular automaton?
Disjunctive Normal Form
For example, here is the expansion of any Boolean
function
f(x,y) with n = 2 arguments.
The carrier set
Special Elements
Characterized by the quintuple
Binary operators
Function definition: A mapping from the set of ordered pairs of the carrier to an element in the carrier set.
Boolean Algebra
An example of a carrier set with four elements
All 16 ordered pairs from the carrier set
A mapping from an element of B x B into an element of B
Cellular Automata are a stand-in for any sufficiently detailed complicated ontological explanation for how the world works.
Issues in Order Addressed
Boolean Algebra Logic Functions Cellular Automata
Cellular Automata Generalized To An Inferential System
27th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering,
Example of Boolean Function
Function Table Boolean Formula
Substitute specific arguments
Boole’s Expansion Theorem
Any Boolean function can be expanded in the following manner. Applying this theorem in a recursive manner yields the disjunctive normal form (DNF).
Why Start with Boolean Algebra?
The following few slides on Boolean Algebra are solely to set the stage for analogous operations with classical logic functions (and
Formal Rules, Boolean Algebra, and Probability Theory
Boolean Algebra on a finite carrier set is a “closed” system, that is, there is always an answer. Moreover, neither numbers nor arithmetic operations are required to find that answer.
Jaynes used probability theory to generalize classical logic functions.
Treat CA from an inferential and informational point of view.
Why Examine Cellular Automata?
Perfect choice for discussing formal manipulation rules of probability theory where the actual numerical assignments are not the issue.
Boolean Algebra
The Motivating Question
But isn’t the quintessential feature of probability theory and inferential systems the ability to predict future events?
Proposed Solution
Boolean functions?
Why Is It Helpful?
1.Useful for both logic functions and cellular automata
2.Axioms oFra Baidu bibliotek Boolean Algebra used in Bayes’s Theorem
3.Canonical expressions substituted for complicated logic and CA rules
Some Basic Boolean Questions
1.How are Boolean functions defined? 2.How are syntactically correct Boolean
formulas produced? 3.What is a good canonical expression for
Saratoga Springs, New York, 11 July 2019
David J. Blower Cogon Systems Pensacola FL
Motivation
Why is it impossible to predict the behaviour of a cellular automaton?
Disjunctive Normal Form
For example, here is the expansion of any Boolean
function
f(x,y) with n = 2 arguments.
The carrier set
Special Elements
Characterized by the quintuple
Binary operators
Function definition: A mapping from the set of ordered pairs of the carrier to an element in the carrier set.
Boolean Algebra
An example of a carrier set with four elements
All 16 ordered pairs from the carrier set
A mapping from an element of B x B into an element of B
Cellular Automata are a stand-in for any sufficiently detailed complicated ontological explanation for how the world works.
Issues in Order Addressed
Boolean Algebra Logic Functions Cellular Automata
Cellular Automata Generalized To An Inferential System
27th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering,
Example of Boolean Function
Function Table Boolean Formula
Substitute specific arguments
Boole’s Expansion Theorem
Any Boolean function can be expanded in the following manner. Applying this theorem in a recursive manner yields the disjunctive normal form (DNF).
Why Start with Boolean Algebra?
The following few slides on Boolean Algebra are solely to set the stage for analogous operations with classical logic functions (and
Formal Rules, Boolean Algebra, and Probability Theory
Boolean Algebra on a finite carrier set is a “closed” system, that is, there is always an answer. Moreover, neither numbers nor arithmetic operations are required to find that answer.
Jaynes used probability theory to generalize classical logic functions.
Treat CA from an inferential and informational point of view.
Why Examine Cellular Automata?
Perfect choice for discussing formal manipulation rules of probability theory where the actual numerical assignments are not the issue.
Boolean Algebra
The Motivating Question
But isn’t the quintessential feature of probability theory and inferential systems the ability to predict future events?
Proposed Solution
Boolean functions?
Why Is It Helpful?
1.Useful for both logic functions and cellular automata
2.Axioms oFra Baidu bibliotek Boolean Algebra used in Bayes’s Theorem
3.Canonical expressions substituted for complicated logic and CA rules