{"query": "Computability and complexity — Turing to P vs NP", "count": 9, "results": [{"id": "card_cs_computability", "title": "Computability and complexity — Turing to P vs NP", "shelf": "codex", "surface": "secular", "snippet": "A Turing machine defines what is computable; the universal machine runs any program (the engine, the laptop, the brain's computation). Some problems are undecidable (the halting problem); among the de", "authority_tier": "engine_derived", "source": "Narrow Highway — computer science", "readable": false, "generated": false}, {"id": "card_theory_church_turing_thesis_computability", "title": "Church–Turing thesis / computability", "shelf": "theories", "surface": "secular", "snippet": "Church–Turing thesis / computability — an engine domain that can touch it: computer_science. Calibration: map-only — a thesis, not a theorem. Everything effectively computable is computable by a Turin", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_sys_oneway_function", "title": "The one-way function — the computational rectifier", "shelf": "systems", "surface": "secular", "snippet": "Some computations are cheap one way and infeasible the reverse. A one-way function is easy to evaluate (polynomial time) but computationally infeasible to invert; a TRAPDOOR one-way function adds a se", "authority_tier": "reference", "source": "The recurring form — the system analogies (standard engineering) + the design they witness to", "readable": false, "generated": false}, {"id": "card_bridge_theory_kolmogorov_complexity__church_turing_thesis_computability", "title": "Bridge: Algorithmic information (Kolmogorov complexity)  ↔  Church–Turing thesis / computability", "shelf": "bridges", "surface": "secular", "snippet": "Algorithmic information (Kolmogorov complexity) and Church–Turing thesis / computability are the same form in different domains. its uncomputability is a Turing-halting result in disguise", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_bridge_theory_structural_generative_linguistics__church_turing_thesis_computability", "title": "Bridge: Structural & generative linguistics (Saussure, Chomsky)  ↔  Church–Turing thesis / computability", "shelf": "bridges", "surface": "secular", "snippet": "Structural & generative linguistics (Saussure, Chomsky) and Church–Turing thesis / computability are the same form in different domains. the Chomsky hierarchy IS a computability hierarchy: regular, co", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_floor_p_vs_np", "title": "The P versus NP chain - from the machine and the circuit to the three barriers", "shelf": "codex", "surface": "secular", "snippet": "Two trees. Computability: Turing's machine (1936), time as a resource (Hartmanis-Stearns 1965), feasible as polynomial in the input's length (Cobham 1965, Edmonds 1965). Circuits: Shannon's count of g", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_diophantine_rh", "title": "The Riemann hypothesis is one Diophantine equation with no solutions", "shelf": "codex", "surface": "secular", "snippet": "Davis, Matiyasevich and Robinson (1976): the Riemann hypothesis is equivalent to a specific polynomial Diophantine equation having no solutions in the integers. The hypothesis is a Pi-1 statement; its", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_floor_computer_science", "title": "Computer science — bits, logic, and what can be computed", "shelf": "codex", "surface": "secular", "snippet": "Information is bits (a byte is 2^8 = 256 values); logic gates build from the 16 Boolean functions of two inputs; good algorithms beat bad ones (merge sort's n log n against n^2); and computability and", "authority_tier": "engine_derived", "source": "Narrow Highway — computer science", "readable": false, "generated": false}, {"id": "card_joint_manin_algorithm", "title": "If Sha is finite, the rank is computable", "shelf": "codex", "surface": "secular", "snippet": "Manin (1971): if the Tate-Shafarevich group is finite - part of what BSD asserts - then the rank of an elliptic curve over Q is effectively computable by descent. Without it, no algorithm is known. A ", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_cs_computability"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}