Skip to content

README.md

Latest commit

 

History

History
53 lines (34 loc) · 2.2 KB

README.md

File metadata and controls

53 lines (34 loc) · 2.2 KB
Raw

Complexitylib

A Lean 4 formalization of computational complexity theory, built on Mathlib.

Motivation

Complexity theory studies the resources required to solve computational problems — time, space, nondeterminism, randomness, and more. Despite its mathematical maturity, surprisingly little of the theory has been machine-checked. This library aims to build verified foundations for complexity classes, reductions, and the relationships between them.

Approach

The definitions follow Arora and Barak's Computational Complexity: A Modern Approach:

  • A concrete 4-symbol alphabet {0, 1, □, ▷}
  • Deterministic TMs with a single transition function (AB Definition 1.1)
  • Nondeterministic/probabilistic TMs with two transition functions δ₀, δ₁ (AB Definition 2.1 / Section 7.1) — same machine structure, different acceptance semantics

Tapes are custom one-sided infinite (cell 0 is leftmost, head cannot move left of it), matching AB exactly. Configurations use named tapes (read-only input, read-write work tapes, read-write output), making the read-only/read-write distinction structural.

Contents

Module Description
Complexitylib.Models.TuringMachine Alphabet, directions, TM, NTM, configurations, step/trace, acceptance, deciding, PTM counting, language type

Building

Requires elan (the Lean version manager).

lake build

The first build will download and compile Mathlib, which takes a while. Subsequent builds are incremental.

Roadmap

  • Deterministic multi-tape Turing machine (AB Definition 1.1)
  • Nondeterministic TM with two transition functions (AB Definition 2.1)
  • Acceptance and deciding predicates (Accepts, DecidesInTime)
  • PTM counting primitives (acceptCount, acceptProb)
  • DTM-to-NTM embedding (TM.toNTM)
  • Complexity classes (P, NP, BPP, PSPACE, ...)
  • Reductions and completeness
  • Simulation theorems
  • Hierarchy theorems

Contributing

See CONTRIBUTING.md for commit format, code style, and guidelines.

License

Licensed under the Apache License, Version 2.0.