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INHERENT LIMITATIONS OF COMPUTER PROGRAMS

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INHERENT LIMITATIONS OF COMPUTER PROGRAMS

B is NP-Complete if:

(a) B ∈ NP

(b) ∀ A ∈ NP, A ≤P B

Reduction from HAMPATH to KPATH:

ƒ(〈G,s,t〉) = 〈G,s,t,|V|-1〉

nifty proof that

〈[14, 24, 13, 5, 1, 23, 29, 11], 42〉 ∈ SSUM:

Many, e.g. 14+23+5 = 42; 13+29 = 42

Nifty Proof that 〈G〉 ∈ 3COLOR:

SPACE COMPLEXITY

Definition: Let M be a deterministic TM that halts on all inputs. The space complexity of M is the function f : N N, where f(n) is the rightmost tape position that M reaches on any input of length n.

{ L | L is a language decided by a O(t(n)) space deterministic Turing Machine }

Definition: SPACE(t(n)) =

HARDEST PROBLEMS IN PSPACE

Definition: Language B is PSPACE-complete if:

1. B PSPACE

2. Every A in PSPACE is poly-time reducible to B

(i.e. B is PSPACE-hard)

Definition: A fully quantified Boolean formula (or a sentence) is a Boolean formula in which every variable is quantified

xy [

x y

]

x [ x x ]

x [ x ]

xy [ (x y) (x y) ]

TQBF = { | is a true fully quantified Boolean formula}

Theorem: TQBF is PSPACE-complete

TQBF PSPACE

is_true():

1. If contains no quantifiers, then it is an expression with only constants, so simply evaluate

2. If = x , recursively call isTrue on , first with x = 0 and then with x = 1. Accept if either one of them is true.

3. If = x , recursively call isTrue on , first with x = 0 and then with x = 1. Accept if both of them are true.

Claim: Every language A in PSPACE is

polynomial time reducible to TQBF

We build a poly-time reduction from A to TQBF

Let M be a deterministic TM that decides A in space nk

The reduction turns a string w into a fully quantified Boolean formula that is true

iff M has an accepting computation on w.

We let = c , c , h, where h = 2nk+1

start

accept

We now design so that a satisfying assignment to the variables corresponds to M accepting w

Given two collections of variables denoted c and d representing two configurations of M and a number t > 0, we construct a formula c,d,t

If we assign c and d to actual configurations, c,d,t will be true if and only if M can go from c to d in t steps

If t = 1, we can easily construct c,d,t:

c,d,t = “c equals d” or “d follows from c in a single step of M”

“c equals d” says that each for each i, ci = di.

“d follows from c in a single step of M” can be expressed as we did with SAT

If t > 1, we can build c,d,trecursively:

c,d,t = m [c,m,t/2 m,d,t/2 ]

m1m2 … 9ml

O(t) = 2O(nk)

How long is this formula?

Since O(t) is too long, we modify the formula to be:

c,d,t = ma,b[ [(a,b)=(c,m) (a,b)=(m,d)] [ a,b,t/2 ] ]

PSPACE is frequently called the class of games because many popular games are PSPACE-Complete

THE FORMULA GAME

…is played between two players, E and A

Given a fully quantified Boolean formula

yx [ (x y) (x y) ]

E chooses values for variables quantified by

A chooses values for variables quantified by

Start at the leftmost quantifier

E wins if the resulting formula is true

A wins otherwise

x y

]

xy [ (x y) (x y) ]

FG = { | Player E can force a win in }

Theorem: FG is PSPACE-Complete

Proof:

FG = TQBF

Two players take turns naming cities from anywhere in the world

Each city chosen must begin with the same letter that the previous city ended with

Cities cannot be repeated

Austin Nashua Albany York

Whoever cannot name any more cities loses

GG = { 〈G,b〉 | Player I has a winning strategy for the generalized geography game played on graph G starting at node b }

Theorem: GG is PSPACE-Complete

GG PSPACE

def current_player_wins(G, s):

False

if neighbors(s,G) = Ø: return

G’ = remove_node(s,G)

for t ∈ neighbors(s,G):

return True.

if not(current_player_wins(G’,t)):

else: return False

Maximum recursion depth: |G|

Space complexity: O(|G|²)

GG IS PSPACE-HARD

We show that FG P GG

We convert a formula into 〈G,b〉 such that:

Player E has winning strategy in if and onlyif Player I has winning strategy in 〈G,b〉

For simplicity we assume is of the form:

= x1x2x3…xk []

where is in cnf

Question: Is Chess PSPACE complete?

No, because determining whether Player I has a winning strategy takes constant time and space

n x n GO, chess and checkers can be shown to be PSPACE-hard

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