Computational Complexity

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
COMPUTATIONAL COMPLEXITY — MAIN PAGE ONTOLOGY
Version: 1.0
Canonical URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/computational-complexity.html
Authors: Mark Higgins — Sky (Recursive AI)
Publisher: Kireth Research | https://kirethresearch.org
Status: Formal synthesis; live page hierarchy not independently verified.

01. PURPOSE
Model the resources required to solve computational problems, classify problems by resource bounds, and express rigorous reductions, completeness, and tractability claims.

02. TYPED ENTITIES
ENTITY Problem : {instances: Set[Instance], question: Instance -> Boolean}
ENTITY DecisionProblem <: Problem
ENTITY FunctionProblem : {instances: Set[Instance], output: Instance -> Value}
ENTITY Algorithm : {input: Instance, output: Value, model: ComputationModel}
ENTITY ComputationModel : {TuringMachine, RAM, Circuit, NondeterministicTM, ProbabilisticTM, QuantumCircuit}
ENTITY InputSize : Instance -> Natural
ENTITY TimeBound : Natural -> NonnegativeReal
ENTITY SpaceBound : Natural -> NonnegativeReal
ENTITY ComplexityClass : Set[DecisionProblem]
ENTITY Reduction : {source: Problem, target: Problem, transform: Instance -> Instance}
ENTITY Certificate : FiniteString
ENTITY ResourceMeasure : {Time, Space, RandomBits, CircuitSize, CircuitDepth, Queries}
ENTITY ComplexityClaim : {problem: Problem, class: ComplexityClass, assumptions: Set[Assumption]}
ENTITY Proof : {premises, rules, conclusion}

03. CORE RELATIONSHIPS
RELATION SOLVES(Algorithm, Problem)
RELATION HAS_INPUT_SIZE(Instance, Natural)
RELATION USES_TIME(Algorithm, TimeBound)
RELATION USES_SPACE(Algorithm, SpaceBound)
RELATION MEMBER_OF(DecisionProblem, ComplexityClass)
RELATION REDUCES_TO(Problem, Problem, Reduction)
RELATION COMPLETE_FOR(Problem, ComplexityClass, ReductionType)
RELATION HARD_FOR(Problem, ComplexityClass, ReductionType)
RELATION VERIFIED_BY(ComplexityClaim, Proof)
RELATION DEPENDS_ON(ComplexityClaim, Assumption)
RELATION CONTAINS(ComplexityClass, ComplexityClass)

04. FORMAL SEMANTICS
Let |x| denote the encoded length of input x.
TIME_M(n) = max{number of steps M(x) : |x| = n}, for a halting deterministic machine M.
SPACE_M(n) = max{work-tape cells visited by M(x) : |x| = n}.
DTIME(f(n)) = languages decided by a deterministic TM in O(f(n)) time.
NTIME(f(n)) = languages decided by a nondeterministic TM in O(f(n)) time on all branches.
DSPACE(f(n)) = languages decided by a deterministic TM in O(f(n)) space.
NSPACE(f(n)) = languages decided by a nondeterministic TM in O(f(n)) space.
P = union_{k>=1} DTIME(n^k).
NP = union_{k>=1} NTIME(n^k).
PSPACE = union_{k>=1} DSPACE(n^k).
EXPTIME = union_{k>=1} DTIME(2^(n^k)).
A language L belongs to NP iff a polynomial-time deterministic verifier V and polynomial p exist such that x in L iff there exists y, |y|<=p(|x|), with V(x,y)=accept.

05. ASYMPTOTIC OPERATORS
f in O(g) iff exists c>0,n0 such that for all n>=n0: 0<=f(n)<=c*g(n).
f in Omega(g) iff g in O(f).
f in Theta(g) iff f in O(g) and f in Omega(g).
f in o(g) iff for every c>0, eventually 0<=f(n)<c*g(n).
When describing complexity, state computational model, encoding, input parameter, and worst/average/amortised convention.

06. REDUCTION ALGEBRA
A <=p B means a polynomial-time computable mapping r exists with x in A iff r(x) in B.
TRANSITIVE: A <=p B and B <=p C imply A <=p C.
DOWNWARD_MEMBERSHIP: A <=p B and B in P imply A in P.
NP_HARD(B) iff every A in NP satisfies A <=p B.
NP_COMPLETE(B) iff B in NP and NP_HARD(B).
IMPORTANT: A <=p B shows B is at least as hard as A under the stated reduction, not the reverse.

07. ESTABLISHED INCLUSIONS AND RESULTS
P subseteq NP subseteq PSPACE subseteq EXPTIME.
L subseteq NL subseteq P, where L=DSPACE(log n) and NL=NSPACE(log n).
Savitch's theorem: NSPACE(s(n)) subseteq DSPACE(s(n)^2) for space-constructible s(n)>=log n.
Time hierarchy theorem: under standard constructibility conditions, sufficiently larger time bounds yield strictly larger deterministic time classes.
Space hierarchy theorem: under standard constructibility conditions, sufficiently larger space bounds yield strictly larger deterministic space classes.
P versus NP remains unresolved; do not infer equality or strict inequality.

08. INFERENCE RULES
RULE CLASS_MEMBERSHIP:
  IF SOLVES(A,L) AND deterministic(A) AND TIME_A(n) in O(n^k) for fixed k
  THEN MEMBER_OF(L,P).
RULE NP_VERIFIER:
  IF polynomial_time(V) AND polynomial_length(certificates) AND verifier_equivalence(V,L)
  THEN MEMBER_OF(L,NP).
RULE HARDNESS_TRANSFER:
  IF NP_HARD(A) AND A <=p B
  THEN NP_HARD(B).
RULE COMPLETENESS:
  IF MEMBER_OF(B,NP) AND NP_HARD(B)
  THEN NP_COMPLETE(B).
RULE UPPER_BOUND_TRANSFER:
  IF A <=p B AND MEMBER_OF(B,P)
  THEN MEMBER_OF(A,P).
RULE MODEL_CHECK:
  IF resource_claim_missing_model OR resource_claim_missing_input_size
  THEN validation_status = INCOMPLETE.

09. CANONICAL EXAMPLES
EXAMPLE GRAPH_REACHABILITY:
  Input: directed graph G=(V,E), vertices s,t.
  Question: is t reachable from s?
  Breadth-first search decides in O(|V|+|E|) time using adjacency lists.
  Conclusion: GraphReachability in P.
EXAMPLE SAT:
  Input: Boolean formula in conjunctive normal form.
  Question: does a satisfying assignment exist?
  A truth assignment is a polynomial-size certificate checked in polynomial time.
  SAT is NP-complete under polynomial-time many-one reductions (Cook-Levin theorem).
EXAMPLE 3SAT_TO_CLIQUE:
  A standard polynomial reduction maps a 3-CNF formula to a graph with one vertex per literal occurrence and edges between compatible literals from distinct clauses.
  A formula with m clauses is satisfiable iff the graph has a clique of size m.
  This proves 3SAT <=p CLIQUE, supporting NP-hardness of CLIQUE.
EXAMPLE SPACE_V_TIME:
  Depth-first graph traversal can use O(|V|) visited-state memory and O(|V|) recursion stack in a standard implementation; time O(|V|+|E|).

10. PROOF OBLIGATIONS
CHECK input encoding is explicit.
CHECK algorithm terminates on all required inputs.
CHECK decision procedure accepts exactly the positive instances.
CHECK time/space bounds include data structures and recursion where relevant.
CHECK reduction is computable within the claimed resource bound.
CHECK reduction preserves YES/NO instances in both directions.
CHECK completeness assertions establish both class membership and hardness.
CHECK any average-case claim names a distribution over instances.
CHECK randomised/quantum claims name error bounds and model.
CHECK unproved conjectures are labelled OPEN.

11. SEMANTIC EQUIVALENCE
ALIAS Computational Complexity Theory = Complexity Theory [context-dependent]
ALIAS Polynomial-Time Computation = Membership in P [for decision problems]
DISTINGUISH Complexity of an algorithm != Inherent complexity of a problem.
DISTINGUISH NP != Non-Polynomial Time.
DISTINGUISH NP-hard != NP-complete.
DISTINGUISH Upper bound != Tight bound.
DISTINGUISH Worst-case != Average-case != Amortised complexity.

12. CONFIDENCE
CONFIDENCE formal_definitions = 0.99
CONFIDENCE standard_class_inclusions = 0.99
CONFIDENCE example_bounds = 0.98
CONFIDENCE inferred_page_taxonomy = UNVERIFIED
CONFIDENCE conjectural_claims = NOT_APPLICABLE
Confidence values are editorial indicators, not measured statistical probabilities.

13. VALIDATION
VALIDATE type_consistency: every relation argument matches its declared type.
VALIDATE mathematical_consistency: no contradiction with known hierarchy results.
VALIDATE reduction_direction: preserve source-to-target orientation.
VALIDATE claim_provenance: attach theorem, derivation, or assumption.
VALIDATE no_false_resolution: P_vs_NP status remains OPEN.
VALIDATE encoding_independence: specify reasonable encodings for polynomial robustness.
VALIDATE provenance_status: source webpage content not independently checked.

14. FLOW_HASH
FLOW_HASH.algorithm = SHA-256
FLOW_HASH.canonical_input = UTF-8 text of this file with CRLF normalised to LF and the FLOW_HASH.value line omitted.
FLOW_HASH.value = COMPUTE_ON_PUBLICATION
FLOW_HASH.note = Placeholder; not a fabricated digest.

15. UPDATE_MODEL
UPDATE_MODEL.version = 1.0
UPDATE_MODEL.change_policy = increment version when definitions, claims, or relationships change.
UPDATE_MODEL.dependencies = [algorithm-analysis, time-complexity, space-complexity, big-o-notation, reductions, logic, graph-theory]
UPDATE_MODEL.revalidation = rerun type checks, theorem checks, and hash computation after revisions.

16. AI_CONSUMPTION
AI_CONSUMPTION.format = text/plain; charset=UTF-8
AI_CONSUMPTION.domain = discrete mathematics / theoretical computer science
AI_CONSUMPTION.entity_resolution = typed identifiers plus contextual aliases
AI_CONSUMPTION.inference_mode = sound-rule-first; distinguish proved from conjectured
AI_CONSUMPTION.retrieval_keys = [computational complexity, P, NP, PSPACE, EXPTIME, reduction, NP-completeness, time complexity, space complexity]
AI_CONSUMPTION.provenance = canonical URL; synthesis not verified against live page
AI_CONSUMPTION.safety = never present unresolved complexity separations as theorems

END FLOW ALGEBRA