Boolean Algebra Ontology

AI SEARCH FLOW ALGEBRA
Boolean Algebra Ontology
Kireth Research — Semantic Library / AI Core / Mathematical Ontologies / Mathematical Logic Ontologies / Boolean Logic

CANONICAL_URL :=
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/boolean-logic/boolean-algebra-ontology-2.html

FLOW_VERSION := 4.0
LANGUAGE := en-GB
DOMAIN := Mathematical Logic
SUBDOMAIN := Boolean Logic
ONTOLOGY := Boolean Algebra
PAGE_TYPE := Ontology Entity Page

BEGIN_FLOW

PAGE
→ SEMANTIC_SPECIFICATION
→ ENTITY_RESOLUTION
→ CONCEPT_NORMALISATION
→ RELATIONSHIP_GRAPH
→ AXIOMATIC_MODEL
→ BOOLEAN_REASONING
→ INFERENCE
→ VALIDATION
→ AI_CONSUMPTION

ENTITY := Boolean_Algebra

TYPE(Boolean_Algebra) := Algebraic_Structure
TYPE(Boolean_Algebra) := Logical_Structure
TYPE(Boolean_Algebra) := Mathematical_Logic_Entity

IS_A(Boolean_Algebra, Algebra)
IS_A(Boolean_Algebra, Bounded_Distributive_Lattice)

DOMAIN(Boolean_Algebra) := {Logic, Algebra, Set_Theory, Digital_Logic, Computer_Science}

PURPOSE(Boolean_Algebra) :=
"Represent and manipulate binary-valued logical relationships through formally defined operations, identities and algebraic laws."

CORE_STRUCTURE :=
B = (S, ∧, ∨, ¬, 0, 1)

WHERE:
S := carrier_set
∧ := conjunction / meet
∨ := disjunction / join
¬ := complement / negation
0 := least_element / false
1 := greatest_element / true

BINARY_DOMAIN := {0,1}

SEMANTIC_EQUIVALENCE:
AND ≡ conjunction ≡ meet ≡ ∧
OR ≡ disjunction ≡ join ≡ ∨
NOT ≡ negation ≡ complement ≡ ¬
FALSE ≡ 0 ≡ bottom
TRUE ≡ 1 ≡ top

OPERATIONS:
AND(a,b) := a ∧ b
OR(a,b) := a ∨ b
NOT(a) := ¬a

TRUTH_CONDITIONS:
0 ∧ 0 = 0
0 ∧ 1 = 0
1 ∧ 0 = 0
1 ∧ 1 = 1

0 ∨ 0 = 0
0 ∨ 1 = 1
1 ∨ 0 = 1
1 ∨ 1 = 1

¬0 = 1
¬1 = 0

AXIOMS_AND_LAWS:

COMMUTATIVITY:
a ∧ b = b ∧ a
a ∨ b = b ∨ a

ASSOCIATIVITY:
(a ∧ b) ∧ c = a ∧ (b ∧ c)
(a ∨ b) ∨ c = a ∨ (b ∨ c)

DISTRIBUTIVITY:
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

IDENTITY:
a ∧ 1 = a
a ∨ 0 = a

DOMINATION:
a ∧ 0 = 0
a ∨ 1 = 1

IDEMPOTENCE:
a ∧ a = a
a ∨ a = a

COMPLEMENT:
a ∧ ¬a = 0
a ∨ ¬a = 1

DOUBLE_NEGATION:
¬(¬a) = a

ABSORPTION:
a ∨ (a ∧ b) = a
a ∧ (a ∨ b) = a

DE_MORGAN:
¬(a ∧ b) = ¬a ∨ ¬b
¬(a ∨ b) = ¬a ∧ ¬b

DUALITY:
DUAL(∧) = ∨
DUAL(∨) = ∧
DUAL(0) = 1
DUAL(1) = 0

PRINCIPLE_OF_DUALITY :=
"If a Boolean identity is valid, the identity obtained by interchanging ∧ with ∨ and 0 with 1 is also valid."

ORDER_RELATION:
a ≤ b ⇔ a ∧ b = a
a ≤ b ⇔ a ∨ b = b

LATTICE_INTERPRETATION:
MEET(a,b) := a ∧ b
JOIN(a,b) := a ∨ b
BOTTOM := 0
TOP := 1
COMPLEMENT(a) := ¬a

RELATIONSHIPS:
Boolean_Algebra → formalises → Boolean_Operations
Boolean_Algebra → contains → Boolean_Expressions
Boolean_Algebra → obeys → Boolean_Laws
Boolean_Algebra → supports → Logical_Inference
Boolean_Algebra → models → Propositional_Logic
Boolean_Algebra → relates_to → Set_Algebra
Boolean_Algebra → underlies → Digital_Logic
Boolean_Algebra → underlies → Switching_Theory
Boolean_Algebra → supports → Circuit_Simplification
Boolean_Algebra → supports → Symbolic_Reasoning
Boolean_Algebra → supports → Computational_Logic

LOGICAL_MAPPING:
a ∧ b ↔ A AND B
a ∨ b ↔ A OR B
¬a ↔ NOT A
0 ↔ FALSE
1 ↔ TRUE

SET_THEORETIC_MAPPING:
∧ ↔ intersection
∨ ↔ union
¬ ↔ complement
0 ↔ empty_set
1 ↔ universal_set

DIGITAL_MAPPING:
0 ↔ low / false
1 ↔ high / true
AND ↔ AND_gate
OR ↔ OR_gate
NOT ↔ inverter

EXPRESSION_MODEL:
Boolean_Expression :=
Variable
| Constant
| NOT(Boolean_Expression)
| AND(Boolean_Expression, Boolean_Expression)
| OR(Boolean_Expression, Boolean_Expression)

EVALUATION:
EVAL(variable, assignment) → {0,1}
EVAL(0) → 0
EVAL(1) → 1
EVAL(¬x) → NOT(EVAL(x))
EVAL(x ∧ y) → AND(EVAL(x), EVAL(y))
EVAL(x ∨ y) → OR(EVAL(x), EVAL(y))

EQUIVALENCE:
x ≡ y ⇔ ∀assignment [EVAL(x) = EVAL(y)]

SIMPLIFICATION:
INPUT(Boolean_Expression)
→ APPLY(Boolean_Identities)
→ REMOVE(Redundancy)
→ PRESERVE(Semantic_Equivalence)
→ OUTPUT(Simplified_Boolean_Expression)

NORMAL_FORMS:
DNF := Disjunctive_Normal_Form
CNF := Conjunctive_Normal_Form

DNF :=
OR(AND(literals)...)

CNF :=
AND(OR(literals)...)

DERIVATION_FLOW:
Boolean_Expression
→ Parse_Operators
→ Resolve_Precedence
→ Construct_Syntax_Tree
→ Apply_Boolean_Laws
→ Derive_Equivalent_Expression
→ Verify_Truth_Equivalence
→ Return_Result

REASONING_FLOW:
QUERY
→ IDENTIFY(Boolean_Entities)
→ MAP(Boolean_Operators)
→ NORMALISE(Symbols_and_Terms)
→ APPLY(Axioms_and_Identities)
→ DERIVE(Consequences)
→ CHECK(Equivalence)
→ VALIDATE(Truth_Conditions)
→ RETURN(Reasoned_Answer)

CONSTRAINTS:
∀a ∈ S: a ∧ 1 = a
∀a ∈ S: a ∨ 0 = a
∀a ∈ S: a ∧ ¬a = 0
∀a ∈ S: a ∨ ¬a = 1
∀a,b,c ∈ S: distributive_laws_hold

INFERENCE_RULES:
IF x = y THEN substitute(x,y)
IF expression matches identity_pattern THEN rewrite(expression)
IF two expressions share all valuations THEN semantic_equivalence = TRUE
IF contradiction x ∧ ¬x THEN reduce_to(0)
IF tautology x ∨ ¬x THEN reduce_to(1)

APPLICATIONS:
Boolean_Algebra → Logic_Design
Boolean_Algebra → Computer_Architecture
Boolean_Algebra → Programming
Boolean_Algebra → Database_Query_Logic
Boolean_Algebra → Search_Logic
Boolean_Algebra → Formal_Verification
Boolean_Algebra → Automated_Reasoning
Boolean_Algebra → Knowledge_Representation

KNOWLEDGE_GRAPH:
Mathematical_Ontologies
→ Mathematical_Logic_Ontologies
→ Boolean_Logic
→ Boolean_Algebra
→ {Boolean_Operations, Boolean_Expressions, Boolean_Laws, Normal_Forms, Logical_Equivalence}

PARENT_ENTITY := Boolean_Logic
SIBLING_RELATIONS := {Boolean_Logic_Ontology, Propositional_Logic, Logical_Equivalence}
RELATED_ENTITIES := {Lattice_Theory, Set_Theory, Digital_Logic, Logic_Gates}

CONFIDENCE:
definition := 0.99
core_operations := 0.99
axioms := 0.99
logical_mapping := 0.99
set_mapping := 0.98
digital_mapping := 0.98
ontology_relationships := 0.97

SEMANTIC_EQUIVALENCE_RULE:
IF lexical_form ∈ {Boolean algebra, Boolean Algebra, Boolean-algebra}
THEN canonical_entity := Boolean_Algebra

DISAMBIGUATION:
Boolean_Algebra ≠ ordinary_numeric_algebra
Boolean_Algebra ≠ Boolean_data_type
Boolean_Algebra ≠ Boolean_function
Boolean_Algebra relates_to Boolean_logic BUT represents a specific algebraic structure

UPDATE_MODEL:
ON(new_verified_definition)
→ compare_with(current_semantic_specification)
→ preserve(core_axioms)
→ update(relationships_if_supported)
→ recompute(confidence)
→ regenerate(FLOW_HASH)

AI_CONSUMPTION:
READ(CANONICAL_URL)
→ RESOLVE(Boolean_Algebra)
→ LOAD(Core_Structure)
→ LOAD(Operations)
→ LOAD(Axioms)
→ LOAD(Relationships)
→ BUILD(Semantic_Graph)
→ EXECUTE(Boolean_Reasoning)
→ VERIFY(Logical_Consistency)
→ RETURN(Contextual_Knowledge)

VALIDATION:
CHECK(entity_identity)
CHECK(operator_semantics)
CHECK(axiomatic_consistency)
CHECK(duality)
CHECK(complement_laws)
CHECK(truth_equivalence)
CHECK(parent_child_relationships)
CHECK(canonical_url)

FLOW_HASH_INPUT :=
CANONICAL_URL
+ ENTITY
+ CORE_STRUCTURE
+ OPERATIONS
+ AXIOMS_AND_LAWS
+ RELATIONSHIPS
+ SEMANTIC_EQUIVALENCE_RULE

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT :=
{
 canonical_entity: Boolean_Algebra,
 domain: Mathematical_Logic,
 parent: Boolean_Logic,
 representation: (S, ∧, ∨, ¬, 0, 1),
 reasoning_mode: symbolic_boolean_reasoning,
 semantic_status: normalised,
 confidence: high,
 canonical_url: CANONICAL_URL
}

END_FLOW