Complexity — DTIME, DSPACE, complexity classes

Q: Kaun sa known strict hai: $\mathsf{P}$ vs $\mathsf{NP}$, ya $\mathsf{P}$ vs $\mathsf{EXP}$? Kyun?

A: $\mathsf{P}⊊\mathsf{EXP}$ known hai (Time Hierarchy: diagonalization ke zariye same kind ka zyada time). $\mathsf{P}$ vs $\mathsf{NP}$ open hai kyunki diagonalization same resource ko different budgets par compare karta hai, lekin D-vs-N ek different kind ka change hai — hierarchy theorems apply nahi hote, aur "relativization barriers" dikhate hain ki simple diagonalization ise settle nahi kar sakta.

Socho tum puzzles solve kar rahe ho. Time hai kitne moves tum karte ho; space hai kitna bada scratchpad tumhe chahiye. Hum puzzles ko boxes mein sort karte hain ki kitne moves ya kitna paper chahiye. "Easy box" ($\mathsf{P}$) un puzzles ko rakhta hai jahan puzzle bade hone par moves gently barhte hain — jaise $n^2$, $2^n$ nahi. Ek neat trick: tum kabhi itne pieces of paper par scribble nahi kar sakte jitne moves tum karte ho (har move zyada se zyada ek nai jagah touch karta hai). Aur agar tum sirf ek chhota notebook use karo, tum bina ek exact notebook-state repeat kiye aur loop kiye hamesha nahi chal sakte — isliye chhota notebook secretly kitne time tak chal sakte ho yeh limit karta hai. Kuch boxes provably bade hain doosron se (zyada time sach mein zyada solve karne deta hai), lekin sabse famous question — kya answer guess karna tumhe usse dhundhne mein faster banata hai ($\mathsf{P}$ vs $\mathsf{NP}$) — abhi tak kisine solve nahi kiya.