Decidability

Decision problem P is language P ⊂ Σ^*

Decidable: P is recursive Semidecidable: P is recursively enumerable Undecidable: otherwise

Decidable problem:

• algorithm that always halts
• always returns yes or no

Semidecidable problem:

• algorithm halts eventually if the answer is yes (w ∈ P)
• may or may nto halt if answer is no
• but you don't know how long you'll have to wait for an answer

Decision problem is decidable if:

• P is semidecidable, and
• $\overline{P}$ is semidecidable

The halting problem: given deterministic Turing machine and a word, does the machine halt when started with the word as input? It's undecidable, as the language H = { (M, x) | M reaches halting state on input x } is not recursive.

Rice's theorem: every non-trivial property of recursively enumerable languages is undecidable. A property of class K is trivial if it holds for all or no k ∈ K.

Post correspondence problem

Given n pairs of words (w₁, v₁)…(w_n, v_n), are there indices i₁…i_k st $w_{i_{1}}…w_{i_{k}} = v_{i_{1}}…v_{i_{k}}$?

Modified: is there a way to make the first words the same?

The PCP is undecidable, as evidenced by Joerg spending 2 hours in a lecture trying to make us decide it.

Undecidable properties of context-free languages

Stuff that's undecidable:

• the question L₁ ∩ L₂ = ∅
• ambiguity of context-free grammars
• whether a context-free language contains a palindrome
• the question L = Σ^* (and hence equality of languages)

Semidecidability

A decision P ⊂ Σ^* is:

• decidable if P recursive
• semidecidable if P recursively enumerable

Examples of semidecidable: halting problem, PCP, non-empty intersection of context-free languages, ambiguity of context-free grammars

Examples of undecidable:

• derivability of formula φ in predicate logic
• Hilbert's 10th problem (algorithm to decide if system of Diophantine equations has solution in Z)

Complexity

Big O notation

Wtf is this definition. TODO: find a better definition.

The highest exponent is what matters. e.g. n³ is in O(n³) but also in O(n⁵) List of big Os:

• exponentials are in O(n^e), where e is the exponent
• polynomials are O(n^e), where e is the largest exponent

Time complexity

Let f,g : N → N. Nondeterministic Turing machine (M) runs in time f if for every input (w), every computation of M reaches a halting state after at most f(|w|) steps.

A Turing machine has time complexity O(g) if there exists f ∈ O(g) st the machine runs in time f.

Complexity classes P and NP

Nondeterministic Turing machine is in polynomial time if it has time complexity O(n^k) for some k.

NP are languages accepted by nondeterministic polynomial time Turing machines.

P is class of languages accepted by deterministic polynomial time Turing machines.

P ⊂ NP, but nobody knows if P = NP.

A problem is in NP if:

• every instance has finite set of possible solutions
• correctness of solution can be checked in polynomial time