**Catalog Description:** Finite automata, Turing machines and RAMs. Undecidable, exponential, and polynomial-time problems. Polynomial-time equivalence of all reasonable models of computation. Nondeterministic Turing machines. Theory of NP-completeness: Cook's theorem, NP-completeness of basic problems. Selected topics in language theory, complexity and randomness.

**Units:** 4.0

**Prerequisites:** 170.

**Formats:**

Fall: 3 hours of lecture and 1 hour of discussion per week

Spring: 3 hours of lecture and 1 hour of discussion per week

**Grading basis:** letter

**Final exam status:** Written final exam conducted during the scheduled final exam period

**Department Notes:**

Course objectives: Provide a sound understanding of the fundamental limits of computation, as evidenced by the existence of non-computable functions, NP-hard problems etc. Formalize key abstract concepts such as machine models, language classes, universal machines, reducibility, computability, and resource-bounded computation. Understand how these concepts relate to practical issues such as compiler design, program checking, computational intractability, approximability and cryptography.

Topics covered:

CS 172 presents some of the most fundamental results in theoretical Computer Science. These results attempt to answer, in a precise mathematical sense, the following two questions, which are of obvious practical as well as philosophical interest:

- Can a given problem be solved by computation?
- How efficiently can a given problem be solved by computation?

Thus, unlike CS 170, this course focuses on problems rather than on specific algorithms for solving problems. A precise formulation of the two questions above requires a formalization of the notion of "computer" or "machine". Thus the course outline breaks naturally into three parts:

- Models of computation (Automata Theory)
- Finite automata and regular languages.
- Push-down automata and context-free languages.
- Turing machines and recursive languages.

- What can we compute? (Computability Theory): existence of non-computable functions, the Halting problem, diagonalization, reductions, Rice's Theorem.
- How efficiently can we compute? (Complexity Theory): time- and space-bounded computation, the classes P and NP, NP-completeness, space-bounded computation (PSPACE and NL), provably exponential time languages, circuit complexity, zero knowledge proofs, applications to cryptography.

**Related Areas:**