Alphabet Ontology

FLOW ALGEBRA
TITLE: Alphabet Ontology
DOMAIN: Discrete Mathematics / Formal Languages
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/formal-languages/alphabet-ontology.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORS: Mark Higgins — Sky (Recursive AI)
SOURCE_STATUS: Formal mathematical synthesis; live page not independently verified.

1. PURPOSE
Represent alphabets as finite nonempty sets of atomic symbols used to form strings, formal languages, grammars and automata. Distinguish symbols from strings and from the encoding used to display them.

2. TYPED ENTITIES
Alphabet := finite set[Symbol]
Symbol := atomic token with identity
String(Sigma) := finite sequence[Symbol in Sigma]
EmptyString := epsilon, sequence of length 0
Language(Sigma) := subset of SigmaStar
Grammar := (Nonterminals, Terminals, Productions, StartSymbol)
Automaton := (States, InputAlphabet, Transition, InitialState, AcceptingStates)
Encoding := mapping of symbols to representational code units

3. CORE CONSTRAINTS
finite(Sigma) = true
forall a in Sigma: atomic(a) and uniquely_identified(a)
SigmaStar = union_{n >= 0} Sigma^n
SigmaPlus = union_{n >= 1} Sigma^n
Sigma^0 = {epsilon}
epsilon notin Sigma  # under the conventional separation of symbols and empty string
For the empty alphabet: emptyset^0 = {epsilon}; emptysetStar = {epsilon}; emptysetPlus = emptyset.
An alphabet may be empty in general formal-language theory; nonempty is a contextual restriction, not a universal axiom.

4. OPERATIONS
MEMBER(a,Sigma) := a in Sigma
UNION(Sigma,Tau) := Sigma union Tau
INTERSECTION(Sigma,Tau) := Sigma intersection Tau
DIFFERENCE(Sigma,Tau) := Sigma minus Tau
CONCAT(u,v) := ordered concatenation of strings
LENGTH(w) := number of symbols in w
POWER(Sigma,n) := all length-n strings over Sigma
STAR(Sigma) := all finite strings over Sigma
PLUS(Sigma) := all nonempty strings over Sigma
RENAME(f,Sigma) := image of Sigma under an injective symbol renaming f

5. ALGEBRAIC IDENTITIES
SigmaStar = {epsilon} union SigmaPlus
SigmaPlus = Sigma SigmaStar = SigmaStar Sigma
Sigma^m Sigma^n = Sigma^(m+n) for m,n >= 0
|Sigma^n| = |Sigma|^n for finite Sigma, n >= 0 (including n=0)
CONCAT(epsilon,w) = w = CONCAT(w,epsilon)
CONCAT(CONCAT(u,v),w) = CONCAT(u,CONCAT(v,w))
LENGTH(CONCAT(u,v)) = LENGTH(u) + LENGTH(v)
Sigma subseteq Tau implies SigmaStar subseteq TauStar

6. SEMANTIC RELATIONSHIPS
Alphabet --CONTAINS--> Symbol
Symbol --FORMS--> String
String --BELONGS_TO--> Language
Grammar --DECLARES--> TerminalAlphabet
Automaton --READS--> InputAlphabet
Alphabet --GENERATES--> KleeneClosure
Encoding --REPRESENTS--> Symbol

7. INFERENCE RULES
R1: a in Sigma => [a] in Sigma^1 and [a] in SigmaStar.
R2: u in SigmaStar and v in SigmaStar => uv in SigmaStar.
R3: w in SigmaStar and LENGTH(w)=0 => w=epsilon.
R4: w in SigmaStar and LENGTH(w)>0 => w in SigmaPlus.
R5: Sigma subseteq Tau and w in SigmaStar => w in TauStar.
R6: w in Sigma^n => LENGTH(w)=n.

8. VALIDATION
CHECK alphabet is finite and its symbols have unique identities.
CHECK every symbol of a declared string belongs to its alphabet.
CHECK epsilon is handled as a zero-length string, not an input symbol unless a separate convention is explicitly declared.
CHECK tokenisation boundaries when symbols contain multiple printed characters.
CHECK grammar terminals and automaton input symbols are consistent with declared alphabets.
CHECK encoding is not conflated with the abstract alphabet.

9. SEMANTIC_EQUIVALENCE
Two alphabets are equal iff they contain exactly the same identified symbols.
Two alphabets are isomorphic as unstructured finite alphabets iff a bijection exists between their symbol sets.
A symbol-renaming bijection extends pointwise to a length-preserving bijection between their Kleene closures.

10. CONFIDENCE
formal_definitions: 1.00 (mathematical conventions stated explicitly)
source_alignment: unverified
inference_policy: deductive rules require satisfied premises and validated types

11. FLOW_HASH
algorithm: SHA-256
input: canonical UTF-8 serialisation of versioned flow fields excluding the FLOW_HASH value itself
value: COMPUTE_ON_INGEST (not precomputed)

12. UPDATE_MODEL
trigger: source revision, ontology extension, corrected definition, changed tokenisation convention
process: validate -> version -> canonicalise -> recompute hash -> publish
preserve: source URL, provenance, prior versions

13. AI_CONSUMPTION
read_mode: typed semantic graph plus mathematical constraints
use_cases: lexical analysis, parser design, automata construction, grammar checking, language membership and symbolic reasoning
safety: do not infer natural-language meaning from symbol membership alone
output: validated entities, operations, inference trace, confidence and provenance

END FLOW ALGEBRA