today
	time complexity
		defn
		examples
		robustness
admin
	ps1 out tonight, due 01-31
	sign up for piazza
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defn
	TM
	picture
prop: TM M with L(M)={0^n1^n}
pf
Q. how many steps does this take?
	eg, # of iterations of transition function
	n=input length
	n^2
Q. can we do better?
	not just (n^2)/10, but asymptotically better?
Fact: cannot run in time O(n)
	Sipser #7.47
Prop: TM M with L(M)={0^n1^n} in O(n\lg n) steps
idea
	check for 0*1*
	count
		check equality of \lg n bits each
	\lg n sweeps of entire tape
		n steps each
		count parity of 0's vs parity of 1's
		cross off half of 0's and half of 1's
Fact: cannot run in time o(n\lg n)
	Sipser 7.47
	so this is tight
Prop: two-tape TM M with with L(M)={0^n1^n} in O(n) steps
Pf
	defn of two-tape TM
		by picture
	algo
		check for 0*1*
		copy 0's to second tape
		compare 0's to 1's in real-time using tapes in parallel
rmk
	time complexity changed with model
		but not by much
defn
	TIME
	picture of heirarchy theorem
Prop: TIME_{2-tape TM}(t(n))\subseteq TIME((t(n)+n)^2)
pf
	simulate all updates in sweeps
	use larger tape alphabet to encode head position
rmk
	two-tape and one-tape models are polynomially related in time complexity
	   \-> also applies to all known realistic deterministic models of computational
	Extended Church Turing thesis says this extends to all known realistic models
		potentially falsified by quantum computing
defn of P
rmk:
	model invariant
	worst case notion
	roughly corresponds to theoretical efficiency
		big Oh
			due to model invariance
		closed under subroutines
		often efficient in practice
			many natural algo run in ~n^3
			n^1000 is not efficient
			problems outside P are usually not efficiently solvable
Q. What is in P?
ex
	graph connectivity
		do algo
	Q. encoding issues?
	PRIMES is in P in unary
		is in P in binary, but this is harder
	HAMPATH
		not known to be in P
---
today:
	time complexity
	robustness of P
	defn of P
next time:
	non-determinism
	NP