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- Explore the consequences of \(\mathbf{P}=\mathbf{NP}\)
*Search-to-decision*reduction: transform algorithms that solve decision version to search version for \(\mathbf{NP}\)-complete problems.- Optimization and learning problems
- Quantifier elimination and solving polynomial hieararchy.
- What is the evidence for \(\mathbf{P}=\mathbf{NP}\) vs \(\mathbf{P}\neq \mathbf{NP}\)?

“You don’t have to believe in God, but you should believe in The Book.”, Paul Erdős, 1985.Paul Erdős (1913-1996) was one of the most prolific mathematicians of all times. Though he was an atheist, Erdős often referred to “The Book” in which God keeps the most elegant proof of each mathematical theorem.

“No more half measures, Walter”, Mike Ehrmantraut in “Breaking Bad”, 2010.

“The evidence in favor of [\(\mathbf{P}\neq \mathbf{NP}\)] and [ its algebraic counterpart ] is so overwhelming, and the consequences of their failure are so grotesque, that their status may perhaps be compared to that of physical laws rather than that of ordinary mathematical conjectures.”, Volker Strassen, laudation for Leslie Valiant, 1986.

“Suppose aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the [fifth Ramsey number]. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the [sixth Ramsey number], however, we would have no choice but to launch a preemptive attack.”, Paul Erdős, as quoted by Graham and Spencer, 1990.The \(k\)-th Ramsey number, denoted as \(R(k,k)\), is the smallest number \(n\) such that for every graph \(G\) on \(n\) vertices, both \(G\) and its complement contain a \(k\)-sized independent set. If \(\mathbf{P}=\mathbf{NP}\) then we can compute \(R(k,k)\) in time polynomial in \(2^k\), while otherwise it can potentially take closer to \(2^{2^{2k}}\) steps.

We have mentioned that the question of whether \(\mathbf{P}=\mathbf{NP}\), which is equivalent to whether there is a polynomial-time algorithm for \(3SAT\), is the great open question of Computer Science. But why is it so important? In this chapter, we will try to figure out the implications of such an algorithm.

First, let us get one qualm out of the way. Sometimes people say, *“What
if \(\mathbf{P}=\mathbf{NP}\) but the best algorithm for 3SAT takes
\(n^{100}\) time?”* Well, \(n^{100}\) is much larger than, say,
\(2^{\sqrt{n}}\) for any input shorter than \(10^{60}\) bits, which is way,
way larger than the world’s total storage capacity (estimated at a
“mere” \(10^{21}\) bits or about 200 exabytes at the time of this
writing). So another way to phrase this question is to say, “what if the
complexity of 3SAT is exponential for all inputs that we will ever
encounter, but then grows much smaller than that?” To me this sounds
like the computer science equivalent of asking, “what if the laws of
physics change completely once they are out of the range of our
telescopes?”. Sure, this is a valid possibility, but wondering about it
does not sound like the most productive use of our time.

So, as the saying goes, we’ll keep an open mind, but not so open that our brains fall out, and assume from now on that:

- There is a mathematical god,

and

- She does not “pussyfoot around” or take “half measures”. If God
decided to make \(3SAT\)
*easy*, then \(3SAT\) will have a \(10^6\cdot n\) (or at worst \(10^6 n^2\)) -time algorithm (i.e., \(3SAT\) will be in \(TIME(cn)\) or \(TIME(cn^2)\) for a not-too-large constant \(c\)). If she decided to make \(3SAT\)*hard*, then for every \(n \in \N\), \(3SAT\) on \(n\) variables cannot be solved by a NAND program of fewer than \(2^{10^{-6}n}\) lines.Using the relations we’ve seen between \(SIZE(T(n))\) and \(TIME(T(n))\) (i.e., non-uniform-thm), \(3SAT \not\in SIZE(T(n))\) then it is also in \(TIME(T(n)^\epsilon)\) for some constant \(\epsilon\) that can be shown to be at least \(1/5\).

So far, most of our evidence points to the latter possibility of 3SAT being exponentially hard, but we have not ruled out the former possibility either. In this chapter we will explore some of its consequences.

A priori, having a fast algorithm for 3SAT might not seem so impressive.
Sure, it will allow us to decide the satisfiability of not just 3CNF
formulas but also of quadratic equations, as well as find out whether
there is a long path in a graph, and solve many other decision problems.
But this is not typically what we want to do. It’s not enough to know
*if* a formula is satisfiable\(-\) we want to discover the actual
satisfying assignment. Similarly, it’s not enough to find out if a graph
has a long path\(-\) we want to actually *find* the path.

It turns out that if we can solve these decision problems, we can solve the corresponding search problems as well:

Suppose that \(\mathbf{P}=\mathbf{NP}\). Then for every polynomial-time algorithm \(V\) and \(a,b \in \N\),there is a polynomial-time algorithm \(FIND_V\) such that for every \(x\in \{0,1\}^n\), if there exists \(y\in \{0,1\}^{an^b}\) satisfying \(V(xy)=1\), then \(FIND_V(x)\) finds some string \(y'\) satisfying this condition.

To understand what the statement of search-dec-thm means, let us look at the special case of the \(MAXCUT\) problem. It is not hard to see that there is a polyomial-time algorithm \(VERIFYCUT\) such that \(VERIFYCUT(G,k,S)=1\) if and only if \(S\) is a subset of \(G\)’s vertices that cuts at least \(k\) edges. search-dec-thm implies that if \(\mathbf{P}=\mathbf{NP}\) then there is a polynomial-time algorithm \(FINDCUT\) that on input \(G,k\) outputs a set \(S\) such that \(VERIFYCUT(G,k,S)=1\) if such a set exists. This means that if \(\mathbf{P}=\mathbf{NP}\), by trying all values of \(k\) we can find in polynomial time a maximum cut in any given graph. We can use a similar argument to show that if \(\mathbf{P}=\mathbf{NP}\) then we can find a satisfying assignment for every satisfiable 3CNF formula, find the longest path in a graph, solve integer programming, and so and so forth.

The idea behind the proof of search-dec-thm is simple; let us demonstrate it for the special case of \(3SAT\). (In fact, this case is not so “special”\(-\) since \(3SAT\) is \(\mathbf{NP}\)-complete, we can reduce the task of solving the search problem for \(MAXCUT\) or any other problem in \(\mathbf{NP}\) to the task of solving it for \(3SAT\).) Suppose that \(\mathbf{P}=\mathbf{NP}\) and we are given a satisfiable 3CNF formula \(\varphi\), and we now want to find a satisfying assignment \(y\) for \(\varphi\). Define \(3SAT_0(\varphi)\) to output \(1\) if there is a satisfying assignment \(y\) for \(\varphi\) such that its first bit is \(0\), and similarly define \(3SAT_1(\varphi)=1\) if there is a satisfying assignment \(y\) with \(y_0=1\). The key observation is that both \(3SAT_0\) and \(3SAT_1\) are in \(\mathbf{NP}\), and so if \(\mathbf{P}=\mathbf{NP}\) then we can compute them in polynomial time as well. Thus we can use this to find the first bit of the satisfying assignment. We can continue in this way to recover all the bits.

If \(\mathbf{P}=\mathbf{NP}\) then for every polynomial-time algorithm \(V\) and \(a,b \in \N\), there is a polynomial-time algorithm \(STARTSWITH_V\) that on input \(x\in \{0,1\}^*\) and \(z\in \{0,1\}^\ell\), outputs \(1\) if and only if there exists some \(y\in \{0,1\}^{an^b}\) such that the first \(\ell\) bits of \(y\) are equal to \(z\) and \(V(xy)=1\). Indeed, we leave it as an exercise to verify that the \(STARTSWITH_V\) function is in \(\mathbf{NP}\) and hence can be solved in polynomial time if \(\mathbf{P}=\mathbf{NP}\).

Now for any such polynomial-time \(V\) and \(a,b\in\N\), we can implement \(FIND_V(x)\) as follows:

Algorithm \(FIND_V\):

Input:\(x\in \{0,1\}^n\)

Goal:Find \(z\in \{0,1\}^{an^b}\) such that \(V(xz)=1\), if such \(z\) exists.

Operation:

For \(\ell=0,\ldots,an^b-1\) do the following:

- Let \(b_0 = STARTSWITH_V(xz_{0}\cdots z_{\ell-1}0)\) and \(b_1 = STARTSWITH_V(xz_{0}\cdots z_{\ell-1}1)\).
- If \(b_0=b_1=0\) then halt and output “no \(z\) exists”.
(In this there is no extension of \(xz_0\ldots z_{\ell-1}\) that would cause \(V\) to output \(1\).)- If \(b=1\)
(i.e., can extend \(xz_0\ldots z_{\ell-1}\) with \(0\))then \(z_\ell=0\), otherwise(i.e., can extend \(xz_0\ldots z_{\ell-1}\) with \(1\))\(z_\ell=1\).- Output \(z_0,\ldots,z_{an^b-1}\).

To analyze the \(FIND\) algorithm, note that it makes \(2an^{b}\)
invocations to \(STARTSWITH_V\) and hence if the latter is
polynomial-time, then so is \(FIND_V\). Now suppose that \(x\) is such that
there exists *some* \(y\) satisfying \(V(xy)=1\). We claim that at every
step \(\ell=0,\ldots,an^b-1\), we maintain the invariant that there exists
\(y\in \{0,1\}^{an^b}\) whose first \(\ell\) bits are \(z\) s.t. \(V(xy)=1\).
Note that this claim implies the theorem, since in particular it means
that for \(\ell = an^b-1\), \(z\) satisfies \(V(xz)=1\).

We prove the claim by induction. For \(\ell=0\), this holds vacuously. Now for every \(\ell > 0\), if the call \(STARTSWITH_V(xz_0\cdots z_{\ell-1}0)\) returns \(1\), then we are guaranteed the invariant by definition of \(STARTSWITH_V\). Now under our inductive hypothesis, there is \(y_\ell,\ldots,y_{an^b-1}\) such that \(P(xz_0,\ldots,z_{\ell-1}y_\ell,\ldots,y_{an^b-1})=1\). If the call to \(STARTSWITH_V(xz_0\cdots z_{\ell-1}0)\) returns \(0\) then it must be the case that \(y_\ell=1\), and hence when we set \(z_\ell=1\) we maintain the invariant.

search-dec-thm allows us to find solutions for \(\mathbf{NP}\)
problems if \(\mathbf{P}=\mathbf{NP}\), but it is not immediately clear
that we can find the *optimal* solution. For example, suppose that
\(\mathbf{P}=\mathbf{NP}\), and you are given a graph \(G\). Can you find
the *longest* simple path in \(G\) in polynomial time?

This is actually an excellent question for you to attempt on your own. That is, assuming \(\mathbf{P}=\mathbf{NP}\), give a polynomial-time algorithm that on input a graph \(G\), outputs a maximally long simple path in the graph \(G\).

It turns out the answer is *Yes*. The idea is simple: if
\(\mathbf{P}=\mathbf{NP}\) then we can find out in polynomial time if an
\(n\)-vertex graph \(G\) contains a simple path of length \(n\), and moreover,
by search-dec-thm, if \(G\) does contain such a path, then we
can find it. (Can you see why?) If \(G\) does not contain a simple path of
length \(n\), then we will check if it contains a simple path of length
\(n-1\), and continue in this way to find the largest \(k\) such that \(G\)
contains a simple path of length \(k\).

The above reasoning was not specifically tailored to finding paths in graphs. In fact, it can be vastly generalized to proving the following result:

Suppose that \(\mathbf{P}=\mathbf{NP}\). Then for every polynomial-time computable function \(f:\{0,1\}^* \rightarrow \{0,1\}^*\) there is a polynomial-time algorithm \(OPT\) such that on input \(x\in \{0,1\}^*\), \(OPT(x,1^m) = \max_{y\in \{0,1\}^m} f(x,y)\) (where we identify the output of \(f(x)\) with a natural number via the binary representation).

Moreover under the same assumption, there is a polynomial-time algorithm \(FINDOPT\) such that for every \(x\in \{0,1\}^*\), \(FINDOPT(x,1^m)\) outputs \(y^* \in \{0,1\}^*\) such that \(f(x,y^*)=OPT(x,y^*)\).

The statement of optimizationnp is a bit cumbersome. To understand it, think how it would subsume the example above of a polynomial time algorithm for finding the maximum length path in a graph. In this case the function \(f\) would be the map that on input a pair \(x,y\) outputs \(0\) if the pair \((x,y)\) does not represent some graph and a simple path inside the graph respectively; otherwise \(f(x,y)\) would equal the length of the path \(y\) in the graph \(x\). Since a path in an \(n\) vertex graph can be represented by at most \(n \log n\) bits, for every \(x\) representing a graph of \(n\) vertices, finding \(\max_{y\in \{0,1\}^{n \log n}}f(x,y)\) corresponds to finding the length of the maximum simple path in the graph corresponding to \(x\), and finding the string \(y^*\) that achieves this maximum corresponds to actually finding the path.

The proof follows by generalizing our ideas from the longest path
example above. Let \(f\) be as in the theorem statement. If
\(\mathbf{P}=\mathbf{NP}\) then for every for every string
\(x\in \{0,1\}^*\) and number \(k\), we can test in in \(poly(|x|,m)\) time
whether there exists \(y\) such that \(f(x,y) \geq k\), or in other words
test whether \(\max_{y \in \{0,1\}^m} f(x,y) \geq k\). If \(f(x,y)\) is an
integer between \(0\) and \(poly(|x|+|y|)\) (as is the case in the example
of longest path) then we can just try out all possiblities for \(k\) to
find the maximum number \(k\) for which \(\max_y f(x,y) \geq k\). Otherwise,
we can use *binary search* to hone down on the right value. Once we do
so, we can use search-to-decision to actually find the string \(y^*\) that
achieves the maximum.

For every \(f\) as in the theorem statement, we can define the Boolean function \(F:\{0,1\}^* \rightarrow \{0,1\}\) as follows.

\[F(x,1^m,k)= \begin{cases} 1 & \exists_{y\in \{0,1\}^m} f(x,y) \geq k \\ 0 & \text{otherwise} \end{cases}\]

Since \(f\) is computable in polynomial time, \(F\) is in \(\mathbf{NP}\), and so under our assumption that \(\mathbf{P}=\mathbf{NP}\), \(F\) itself can be computed in polynomial time. Now, for every \(x\) and \(m\), we can compute the largest \(k\) such that \(F(x,1^m,k)=1\) by a binary search. Specifically, we will do this as follows:

- We maintain two numbers \(a,b\) such that we are guaranteed that \(a \leq \max_{y\in \{0,1\}^m} f(x,y) < b\).
- Initially we set \(a=0\) and \(b=2^{T(n)}\) where \(T(n)\) is the running time of \(f\). (A function with \(T(n)\) running time can’t output more than \(T(n)\) bits and so can’t output a number larger than \(2^{T(n)}\).)
At each point in time, we compute the midpoint \(c = \floor{(a+b)/2})\) and let \(y=F(1^n,c)\).

- If \(y=1\) then we set \(a=c\) and leave \(b\) as it is.
- If \(y=0\) then we set \(b=c\) and leave \(a\) as it is.

- We then go back to step 3, until \(b \leq a+1\).

Since \(|b-a|\) shrinks by a factor of \(2\), within \(\log_2 2^{T(n)}= T(n)\) steps, we will get to the point at which \(b\leq a+1\), and then we can simply output \(a\). Once we find the maximum value of \(k\) such that \(F(x,1^m,k)=1\), we can use the search to decision reduction of search-dec-thm to obtain the actual value \(y^* \in \{0,1\}^m\) such that \(f(x,y^*)=k\).

One application for optimizationnp is in solving *optimization
problems*. For example, the task of *linear programming* is to find
\(y \in \R^n\) that maximizes some linear objective
\(\sum_{i=0}^{n-1}c_i y_i\) subject to the constraint that \(y\) satisfies
linear inequalities of the form \(\sum_{i=0}^{n-1} a_i y_i \leq c\). As we
discussed in mincutsec, there is a known polynomial-time
algorithm for linear programming. However, if we want to place
additional constraints on \(y\), such as requiring the coordinates of \(y\)
to be *integer* or *\(0/1\) valued* then the best-known algorithms run in
exponential time in the worst case. However, if \(\mathbf{P}=\mathbf{NP}\)
then optimizationnp tells us that we would be able to solve
all problems of this form in polynomial time. For every string \(x\) that
describes a set of constraints and objective, we will define a function
\(f\) such that if \(y\) satisfies the constraints of \(x\) then \(f(x,y)\) is
the value of the objective, and otherwise we set \(f(x,y) = -M\) where \(M\)
is some large number. We can then use optimizationnp to
compute the \(y\) that maximizes \(f(x,y)\) and that will give us the
assignment for the variables that satisfies our constraints and
maximizes the objective. (If the computation results in \(y\) such that
\(f(x,y)=-M\) then we can double \(M\) and try again; if the true maximum
objective is achieved by some string \(y^*\), then eventually \(M\) will be
large enough so that \(-M\) would be smaller than the objective achieved
by \(y^*\), and hence when we run procedure of optimizationnp we
would get a value larger than \(-M\).)

In many examples, such as the case of finding longest path, we don’t
need to use the binary search step in optimizationnp, and can
simply enumerate over all possible values for \(k\) until we find the
correct one. One example where we do need to use this binary search step
is in the case of the problem of finding a maximum length path in a
*weighted* graph. This is the problem where \(G\) is a weighted graph, and
every edge of \(G\) is given a weight which is a number between \(0\) and
\(2^k\). optimizationnp shows that we can find the
maximum-weight simple path in \(G\) (i.e., simple path maximizing the sum
of the weights of its edges) in time polynomial in the number of
vertices and in \(k\).

Beyond just this example there is a vast field of mathematical
optimization
that studies problems of the same form as in optimizationnp.
In the context of optimization, \(x\) typically denotes a set of
constraints over some variables (that can be Boolean, integer, or real
valued), \(y\) encodes an assignment to these variables, and \(f(x,y)\) is
the value of some *objective function* that we want to maximize. Given
that we don’t know efficient algorithms for \(\mathbf{NP}\) complete
problems, researchers in optimization research study special cases of
functions \(f\) (such as linear programming and semidefinite programming)
where it *is* possible to optimize the value efficiently. Optimization
is widely used in a great many scientific agreas including machine
learning, engineering, economics and operations research.

One classical optimization task is *supervised learning*. In supervised
learning we are given a list of *examples* \(x_0,x_1,\ldots,x_{m-1}\)
(where we can think of each \(x_i\) as a string in \(\{0,1\}^n\) for some
\(n\)) and the *labels* for them \(y_0,\ldots,y_{n-1}\) (which we will think
of simply bits, i.e., \(y_i\in \{0,1\}\)). For example, we can think of
the \(x_i\)’s as images of either dogs or cats, for which \(y_i=1\) in the
former case and \(y_i=0\) in the latter case. Our goal is to come up with
a *hypothesis* or *predictor* \(h:\{0,1\}^n \rightarrow \{0,1\}\) such
that if we are given a new example \(x\) that has an (unknown to us) label
\(y\), then with high probability \(h\) will *predict* the label. That is,
with high probability it will hold that \(h(x)=y\). The idea in supervised
learning is to use the *Occam’s Razor principle*: the simplest
hypothesis that explains the data is likely to be correct. There are
several ways to model this, but one popular approach is to pick some
fairly simple function \(H:\{0,1\}^{k+n} \rightarrow \{0,1\}\). We think
of the first \(k\) inputs as the *parameters* and the last \(n\) inputs as
the example data. (For example, we can think of the first \(k\) inputs of
\(H\) as specifying the weights and connections for some neural network
that will then be applied on the latter \(n\) inputs.) We can then phrase
the supervised learning problem as finding, given a set of labeled
examples \(S=\{ (x_0,y_0),\ldots,(x_{m-1},y_{m-1}) \}\), the set of
parameters \(\theta_0,\ldots,\theta_{k-1} \in \{0,1\}\) that minimizes the
number of errors made by the predictor \(x \mapsto H(\theta,x)\).

In other words, we can define for every set \(S\) as above the function \(F_S:\{0,1\}^k \rightarrow [m]\) such that \(F_S(\theta) = \sum_{(x,y)\in S} |H(\theta,x)-y|\). Now, finding the value \(\theta\) that minimizes \(F_S(\theta)\) is equivalent to solving the supervised learning problem with respect to \(H\). For every polynomial-time computable \(H:\{0,1\}^{k+n} \rightarrow \{0,1\}\), the task of minimizing \(F_S(\theta)\) can be “massaged” to fit the form of optimizationnp and hence if \(\mathbf{P}=\mathbf{NP}\), then we can solve the supervised learning problem in great generality. In fact, this observation extends to essentially any learning model, and allows for finding the optimal predictors given the minimum number of examples. (This is in contrast to many current learning algorithms, which often rely on having access to an extremely large number of examples\(-\) far beyond the minimum needed, and in particular far beyond the number of examples humans use for the same tasks.)

We will discuss *cryptography* later in this course, but it turns out
that if \(\mathbf{P}=\mathbf{NP}\) then almost every cryptosystem can be
efficiently broken. One approach is to treat finding an encryption key
as an instance of a supervised learning problem. If there is an
encryption scheme that maps a “plaintext” message \(p\) and a key \(\theta\)
to a “ciphertext” \(c\), then given examples of ciphertext/plaintext pairs
of the form \((c_0,p_0),\ldots,(c_{m-1},p_{m-1})\), our goal is to find
the key \(\theta\) such that \(E(\theta,p_i)=c_i\) where \(E\) is the
encryption algorithm. While you might think getting such “labeled
examples” is unrealistic, it turns out (as many amateur homebrew crypto
designers learn the hard way) that this is actually quite common in
real-life scenarios, and that it is also possible to relax the
assumption to having more minimal prior information about the plaintext
(e.g., that it is English text). We defer a more formal treatment to
chapcryptography.

In the context of Gödel’s Theorem, we discussed the notion of a *proof
system* (see godelproofdef). Generally speaking, a *proof
system* can be thought of as an algorithm
\(V:\{0,1\}^* \rightarrow \{0,1\}\) (known as the *verifier*) such that
given a *statement* \(x\in \{0,1\}^*\) and a *candidate proof*
\(w\in \{0,1\}^*\), \(V(x,w)=1\) if and only if \(w\) encodes a valid proof
for the statement \(x\). Any type of proof system that is used in
mathematics for geometry, number theory, analysis, etc., is an instance
of this form. In fact, standard mathematical proof systems have an even
simpler form where the proof \(w\) encodes a *sequence* of lines
\(w^0,\ldots,w^m\) (each of which is itself a binary string) such that
each line \(w^i\) is either an *axiom* or follows from some prior lines
through an application of some *inference rule*. For example, Peano’s
axioms encode a set of
axioms and rules for the natural numbers, and one can use them to
formalize proofs in number theory. Also, there are some even stronger
axiomatic systems, the most popular one being Zermelo–Fraenkel with
the Axiom of
Choice
or ZFC for short. Thus, although mathematicians typically write their
papers in natural language, proofs of number theorists can typically be
translated to ZFC or similar systems, and so in particular the existence
of an \(n\)-page proof for a statement \(x\) implies that there exists a
string \(w\) of length \(poly(n)\) (in fact often \(O(n)\) or \(O(n^2)\)) that
encodes the proof in such a system. Moreover, because verifying a proof
simply involves going over each line and checking that it does indeed
follow from the prior lines, it is fairly easy to do that in \(O(|w|)\) or
\(O(|w|^2)\) (where as usual \(|w|\) denotes the length of the proof \(w\)).
This means that for every reasonable proof system \(V\), the following
function \(SHORTPROOF_V:\{0,1\}^* \rightarrow \{0,1\}\) is in
\(\mathbf{NP}\), where for every input of the form \(x1^m\),
\(SHORTPROOF_V(x,1^m)=1\) if and only if there exists \(w\in \{0,1\}^*\)
with \(|w|\leq m\) s.t. \(V(xw)=1\). That is, \(SHORTPROOF_V(x,1^m)=1\) if
there is a proof (in the system \(V\)) of length at most \(m\) bits that \(x\)
is true. Thus, if \(\mathbf{P}=\mathbf{NP}\), then despite Gödel’s
Incompleteness Theorems, we can still automate mathematics in the sense
of finding proofs that are not too long for every statement that has
one. (Frankly speaking, if the shortest proof for some statement
requires a terabyte, then human mathematicians won’t ever find this
proof either.) For this reason, Gödel himself felt that the question of
whether \(SHORTPROOF_V\) has a polynomial time algorithm is of great
interest. As Gödel wrote in a letter to John von
Neumann in 1956
(before the concept of \(\mathbf{NP}\) or even “polynomial time” was
formally defined):

One can obviously easily construct a Turing machine, which for every formula \(F\) in first order predicate logic and every natural number \(n\), allows one to decide if there is a proof of \(F\) of length \(n\) (length = number of symbols). Let \(\psi(F,n)\) be the number of steps the machine requires for this and let \(\varphi(n) = \max_F \psi(F,n)\). The question is how fast \(\varphi(n)\) grows for an optimal machine. One can show that \(\varphi \geq k \cdot n\) [for some constant \(k>0\)]. If there really were a machine with \(\varphi(n) \sim k \cdot n\) (or even \(\sim k\cdot n^2\)), this would have consequences of the greatest importance. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem,

The undecidability of Entscheidungsproblem refers to the uncomputability of the function that maps a statement in first order logic to \(1\) if and only if that statement has a proof. the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine. After all, one would simply have to choose the natural number \(n\) so large that when the machine does not deliver a result, it makes no sense to think more about the problem.

For many reasonable proof systems (including the one that Gödel referred to), \(SHORTPROOF_V\) is in fact \(\mathbf{NP}\)-complete, and so Gödel can be thought of as the first person to formulate the \(\mathbf{P}\) vs \(\mathbf{NP}\) question. Unfortunately, the letter was only discovered in 1988.

If \(\mathbf{P}=\mathbf{NP}\) then we can solve all \(\mathbf{NP}\) *search*
and *optimization* problems in polynomial time. But can we do more? It
turns out that the answer is that *Yes we can!*

An \(\mathbf{NP}\) decision problem can be thought of as the task of
deciding, given some string \(x\in \{0,1\}^*\) the truth of a statement of
the form \[
\exists_{y\in \{0,1\}^{p(|x|)}} V(xy)=1
\] for some polynomial-time algorithm \(V\) and polynomial
\(p:\N \rightarrow \N\). That is, we are trying to determine, given some
string \(x\), whether *there exists* a string \(y\) such that \(x\) and \(y\)
satisfy some polynomial-time checkable condition \(V\). For example, in
the *independent set* problem, the string \(x\) represents a graph \(G\) and
a number \(k\), the string \(y\) represents some subset \(S\) of \(G\)’s
vertices, and the condition that we check is whether \(|S| \geq k\) and
there is no edge \(\{u,v\}\) in \(G\) such that both \(u\in S\) and \(v\in S\).

We can consider more general statements such as checking, given a string
\(x\in \{0,1\}^*\), the truth of a statement of the form \[
\exists_{y \in \{0,1\}^{p_0(|x|)}} \forall_{z \in \{0,1\}^{p_1(|x|)}} V(xyz)=1 \;, \label{existsforalleq}
\] which in words corresponds to checking, given some string \(x\),
whether *there exists* a string \(y\) such that *for every* string \(z\),
the triple \((x,y,z)\) satisfy some polynomial-time checkable condition.
We can also consider more levels of quantifiers such as checking the
truth of the statement \[
\exists_{y \in \{0,1\}^{p_0(|x|)}} \forall_{z\in \{0,1\}^{p_1(|x|)}}\exists_{w\in \{0,1\}^{p_2(|x|)}} V(xyzw)=1 \label{existsforallexistseq}
\] and so on and so forth.

For example, given an \(n\)-input NAND program \(P\), we might want to find
the *smallest* NAND program \(P'\) that computes the same function as \(P\).
The question of whether there is such a \(P'\) that can be described by a
string of at most \(s\) bits can be phrased as

\[
\exists_{P' \in \{0,1\}^{s}} \forall_{x\in \{0,1\}^n} P(x)=P'(x) \label{circmineq}
\] which has the form \eqref{existsforalleq}.*there exists* a move \(y\) for White
such that *for every* move \(z\) for Black *there exists* a move \(w\) for
White that ends in a a checkmate.

It turns out that if \(\mathbf{P}=\mathbf{NP}\) then we can solve these kinds of problems as well:

If \(\mathbf{P}=\mathbf{NP}\) then for every \(a\in \N\), polynomial
\(p:\N \rightarrow \N\) and polynomial-time algorithm \(V\), there is a
polynomial-time algorithm \(SOLVE_{V,a}\) that on input \(x\in \{0,1\}^n\)
returns \(1\) if and only if \[
\exists_{y_0\in \{0,1\}^m} \forall_{y_1\in \{0,1\}^m} \cdots \mathcal{Q}_{y_{a-1}\in \{0,1\}^m} V(xy_0y_1 \cdots y_{a-1})=1 \label{eq:QBF}
\] where \(m=p(n)\) and \(\mathcal{Q}\) is either \(\exists\) or \(\forall\)
depending on whether \(a\) is odd or even, respectively.

To understand the idea behind the proof, consider the special case where we want to decide, given \(x\in \{0,1\}^n\), whether for every \(y \in \{0,1\}^n\) there exists \(z\in \{0,1\}^n\) such that \(V(xyz)=1\). Consider the function \(F\) such that \(F(xy)=1\) if there exists \(z\in \{0,1\}^n\) such that \(V(xyz)=1\). Since \(V\) runs in polynomial-time \(F\in \mathbf{NP}\) and hence if \(\mathbf{P}=\mathbf{NP}\), then there is an algorithm \(V'\) that on input \(x,y\) outputs \(1\) if and only if there exists \(z\in \{0,1\}^n\) such that \(V(xyz)=1\). Now we can see that the original statement we consider is true if and only if for every \(y\in \{0,1\}^n\), \(V'(xy)=1\), which means it is false if and only if the following condition \((*)\) holds: there exists some \(y\in \{0,1\}^n\) such that \(V'(xy)=0\). But for every \(x\in \{0,1\}^n\), the question of whether the condition \((*)\) is itself in \(\mathbf{NP}\) (as we assumed \(V'\) can be computed in polynomial time) and hence under the assumption that \(\mathbf{P}=\mathbf{NP}\) we can determine in polynomial time whether the condition \((*)\), and hence our original statement, is true.

We prove the theorem by induction. We assume that there is a polynomial-time algorithm \(SOLVE_{V,a-1}\) that can solve the problem \eqref{eq:QBF} for \(a-1\) and use that to solve the problem for \(a\). For \(a=1\), \(SOLVE_{V,a-1}(x)=1\) iff \(V(x)=1\) which is a polynomial-time computation since \(V\) runs in polynomial time. For every \(x,y_0\), define the statement \(\varphi_{x,y_0}\) to be the following:

\[ \varphi_{x,y_0} = \forall_{y_1\in \{0,1\}^m} \exists_{y_2 \in \{0,1\}^m} \cdots \mathcal{Q}_{y_{a-1}\in \{0,1\}^m} V(xy_0y_1 \cdots y_{a-1})=1 \]

By the definition of \(SOLVE_{V,a}\), for every \(x\in \{0,1\}^n\), our goal is that \(SOLVE_{V,a}(x)=1\) if and only if there exists \(y_0 \in \{0,1\}^m\) such that \(\varphi_{x,y_0}\) is true.

The *negation* of \(\varphi_{x,y_0}\) is the statement

\[ \overline{\varphi}_{x,y_0} = \exists_{y_1\in \{0,1\}^m} \forall_{y_2 \in \{0,1\}^m}\cdots \overline{\mathcal{Q}}_{y_{a-1}\in \{0,1\}^m} V(xy_0y_1 \cdots y_{a-1})=0 \] where \(\overline{\mathcal{Q}}\) is \(\exists\) if \(\mathcal{Q}\) was \(\forall\) and \(\overline{\mathcal{Q}}\) is \(\forall\) otherwise. (Please stop and verify that you understand why this is true, this is a generalization of the fact that if \(\Psi\) is some logical condition then the negation of \(\exists_y \forall_z \Psi(y,z)\) is \(\forall_y \exists_z \neg \Psi(y,z)\).)

The crucial observation is that \(\overline{\varphi}_{x,y_0}\) is exactly a statement of the form we consider with \(a-1\) quantifiers instead of \(a\), and hence by our inductive hypothesis there is some polynomial time algorithm \(\overline{S}\) that on input \(xy_0\) outputs \(1\) if and only if \(\overline{\varphi}_{x,y_0}\) is true. If we let \(S\) be the algorithm that on input \(x,y_0\) outputs \(1-\overline{S}(xy_0)\) then we see that \(S\) outputs \(1\) if and only if \(\varphi_{x,y_0}\) is true. Hence we can rephrase the original statement \eqref{eq:QBF} as follows:

\[\exists_{y_0 \in \{0,1\}^m} S(xy_0)=1 \label{equivalentqbfinducteq} \]

but since \(S\) is a polynomial-time algorithm, equivalentqbfinducteq is clearly a statement in \(\mathbf{NP}\) and hence under our assumption that \(\mathbf{P}=\mathbf{NP}\) there is a polynomial time algorithm that on input \(x\in \{0,1\}^n\), will determine if \eqref{equivalentqbfinducteq} is true and so also if the original statement \eqref{eq:QBF} is true.

The algorithm of PH-collapse-thm can also solve the search
problem as well: find the value \(y_0\) that certifies the truth of
\eqref{eq:QBF}. We note that while this algorithm is in polynomial
time, the exponent of this polynomial blows up quite fast. If the
original NANDSAT algorithm required \(\Omega(n^2)\) time, solving \(a\)
levels of quantifiers would require time \(\Omega(n^{2^a})\).*quantified boolean formula* problem
has a linear-time algorithm. We will, however, see later in this
course that it satisfies a notion known as
\(\mathbf{PSPACE}\)-hardness that is even stronger than
\(\mathbf{NP}\)-hardness.

Suppose that we found a polynomial-time algorithm \(A\) for \(3SAT\) that is
“good but not great”. For example, maybe our algorithm runs in time
\(cn^2\) for some not too small constant \(c\). However, it’s possible that
the *best possible* SAT algorithm is actually much more efficient than
that. Perhaps, as we guessed before, there is a circuit \(C^*\) of at most
\(10^6 n\) gates that computes 3SAT on \(n\) variables, and we simply
haven’t discovered it yet. We can use PH-collapse-thm to
“bootstrap” our original “good but not great” 3SAT algorithm to discover
the optimal one. The idea is that we can phrase the question of whether
there exists a size \(s\) circuit that computes 3SAT for all length \(n\)
inputs as follows: *there exists* a size \(\leq s\) circuit \(C\) such that
*for every* formula \(\varphi\) described by a string of length at most
\(n\), if \(C(\varphi)=1\) then *there exists* an assignment \(x\) to the
variables of \(\varphi\) that satisfies it. One can see that this is a
statement of the form \eqref{existsforallexistseq} and hence if
\(\mathbf{P}=\mathbf{NP}\) we can solve it in polynomial time as well. We
can therefore imagine investing huge computational resources in running
\(A\) one time to discover the circuit \(C^*\) and then using \(C^*\) for all
further computation.

Given a NAND program \(P\), if \(\mathbf{P}=\mathbf{NP}\) then we can find
an input \(x\) (if one exists) such that \(P(x)=1\). But what if there is
more than one \(x\) like that? Clearly we can’t efficiently output all
such \(x\)’s; there might be exponentially many. But we can get an
arbitrarily good multiplicative approximation (i.e., a \(1\pm \epsilon\)
factor for arbitrarily small \(\epsilon>0\)) for the number of such \(x\)’s,
as well as output a (nearly) uniform member of this set. We defer the
details to later in this course, when we learn about *randomized
computation*. However, we state (without proof) the following theorem
for now:

Let \(V:\{0,1\}^* \rightarrow \{0,1\}\) be some polynomial-time algorithm, and suppose that \(\mathbf{P}=\mathbf{NP}\). Then there exists an algorithm \(COUNT_V\) that on input \(x,1^m,\epsilon\), runs in time polynomial in \(|x|,m,1/\epsilon\) and outputs a number \(K \in \{0,\ldots, 2^m\}\) such that

\[(1-\epsilon)K \leq \Bigl|\{ y \in \{0,1\}^m \;:\; V(xy)=1 \} \Bigr| \leq (1+\epsilon)K \]

That is, \(K\) gives an approximation up to a factor of \(1 \pm \epsilon\)
for the number of *witnesses* for \(x\) with respect to the verifying
algorithm \(V\).

Once again, to understand this theorem it can be useful to see how it implies that if \(\mathbf{P}=\mathbf{NP}\) then there is a polynomial-time algorithm that given a graph \(G\) and a number \(k\), can compute a number \(K\) that is within a \(1 \pm 0.01\) factor equal to the number of simple paths in \(G\) of langth \(k\). (That is, \(K\) is between \(0.99\) to \(1.01\) times the number of such paths.)

So, what will happen if we have a \(10^6n\) algorithm for \(3SAT\)? We have mentioned that \(\mathbf{NP}\)-hard problems arise in many contexts, and indeed scientists, engineers, programmers and others routinely encounter such problems in their daily work. A better \(3SAT\) algorithm will probably make their lives easier, but that is the wrong place to look for the most foundational consequences. Indeed, while the invention of electronic computers did of course make it easier to do calculations that people were already doing with mechanical devices and pen and paper, the main applications computers are used for today were not even imagined before their invention.

An exponentially faster algorithm for all \(\mathbf{NP}\) problems would
be no less radical an improvement (and indeed, in some sense would be
more) than the computer itself, and it is as hard for us to imagine what
it would imply as it was for Babbage to envision today’s world. For
starters, such an algorithm would completely change the way we program
computers. Since we could automatically find the “best” (in any measure
we chose) program that achieves a certain task, we would not need to
define *how* to achieve a task, but only specify tests as to what would
be a good solution, and could also ensure that a program satisfies an
exponential number of tests without actually running them.

The possibility that \(\mathbf{P}=\mathbf{NP}\) is often described as “automating creativity”. There is something to that analogy, as we often think of a creative solution as one that is hard to discover but that, once the “spark” hits, is easy to verify. But there is also an element of hubris to that statement, implying that the most impressive consequence of such an algorithmic breakthrough will be that computers would succeed in doing something that humans already do today. In fact, creativity already is to a large extent automated or minimized (e.g., just see how much popular media content is mass-produced), and as in most professions we should expect to see the need for humans diminish with time even if \(\mathbf{P}\neq \mathbf{NP}\).

Nevertheless, artificial intelligence, like many other fields, will
clearly be greatly impacted by an efficient 3SAT algorithm. For example,
it is clearly much easier to find a better Chess-playing algorithm when,
given any algorithm \(P\), you can find the smallest algorithm \(P'\) that
plays Chess better than \(P\). Moreover, as we mentioned above, much of
machine learning (and statistical reasoning in general) is about finding
“simple” concepts that explain the observed data, and if
\(\mathbf{NP}=\mathbf{P}\), we could search for such concepts
automatically for any notion of “simplicity” we see fit. In fact, we
could even “skip the middle man” and do an automatic search for the
learning algorithm with smallest generalization error. Ultimately the
field of Artificial Intelligence is about trying to “shortcut” billions
of years of evolution to obtain artificial programs that match (or beat)
the performance of natural ones, and a fast algorithm for \(\mathbf{NP}\)
would provide the ultimate shortcut.

More generally, a faster algorithm for \(\mathbf{NP}\) problems would be immensely useful in any field where one is faced with computational or quantitative problems\(-\) which is basically all fields of science, math, and engineering. This will not only help with concrete problems such as designing a better bridge, or finding a better drug, but also with addressing basic mysteries such as trying to find scientific theories or “laws of nature”. In a fascinating talk, physicist Nima Arkani-Hamed discusses the effort of finding scientific theories in much the same language as one would describe solving an \(\mathbf{NP}\) problem, for which the solution is easy to verify or seems “inevitable”, once found, but that requires searching through a huge landscape of possibilities to reach, and that often can get “stuck” at local optima:

“the laws of nature have this amazing feeling of inevitability… which is associated with local perfection.”

“The classical picture of the world is the top of a local mountain in the space of ideas. And you go up to the top and it looks amazing up there and absolutely incredible. And you learn that there is a taller mountain out there. Find it, Mount Quantum…. they’re not smoothly connected … you’ve got to make a jump to go from classical to quantum … This also tells you why we have such major challenges in trying to extend our understanding of physics. We don’t have these knobs, and little wheels, and twiddles that we can turn. We have to learn how to make these jumps. And it is a tall order. And that’s why things are difficult.”

Finding an efficient algorithm for \(\mathbf{NP}\) amounts to always being able to search through an exponential space and find not just the “local” mountain, but the tallest peak.

But perhaps more than any computational speedups, a fast algorithm for
\(\mathbf{NP}\) problems would bring about a *new type of understanding*.
In many of the areas where \(\mathbf{NP}\)-completeness arises, it is not
as much a barrier for solving computational problems as it is a barrier
for obtaining “closed-form formulas” or other types of more constructive
descriptions of the behavior of natural, biological, social and other
systems. A better algorithm for \(\mathbf{NP}\), even if it is “merely”
\(2^{\sqrt{n}}\)-time, seems to require obtaining a new way to understand
these types of systems, whether it is characterizing Nash equilibria,
spin-glass configurations, entangled quantum states, or any of the other
questions where \(\mathbf{NP}\) is currently a barrier for analytical
understanding. Such new insights would be very fruitful regardless of
their computational utility.

The Continuum
Hypothesis is a
conjecture made by Georg Cantor in 1878, positing the non-existence of a
certain type of infinite cardinality.

Today, many (though not all) mathematicians interpret this result as saying that the Continuum Hypothesis is neither true nor false, but rather is an axiomatic choice that we are free to make one way or the other. Could the same hold for \(\mathbf{P} \neq \mathbf{NP}\)?

In short, the answer is *No*. For example, suppose that we are trying to
decide between the “3SAT is easy” conjecture (there is an \(10^6n\) time
algorithm for 3SAT) and the “3SAT is hard” conjecture (for every \(n\),
any NAND program that solves \(n\) variable 3SAT takes \(2^{10^{-6}n}\)
lines). Then, since for \(n = 10^8\), \(2^{10^{-6}n} > 10^6 n\), this boils
down to the finite question of deciding whether or not there is a
\(10^{13}\)-line NAND program deciding 3SAT on formulas with \(10^8\)
variables. If there is such a program then there is a finite proof of
its existence, namely the approximately 1TB file describing the program,
and for which the verification is the (finite in principle though
infeasible in practice) process of checking that it succeeds on all
inputs.*failure* of all programs could well be
inherent.*programs* as well. Ultimately, since it
boils down to a finite statement about bits and numbers; either the
statement or its negation must follow from the standard axioms of
arithmetic in a finite number of arithmetic steps. Thus, we cannot
justify our ignorance in distinguishing between the “3SAT easy” and
“3SAT hard” cases by claiming that this might be an inherently
ill-defined question. Similar reasoning (with different numbers) applies
to other variants of the \(\mathbf{P}\) vs \(\mathbf{NP}\) question. We note
that in the case that 3SAT is hard, it may well be that there is no
*short* proof of this fact using the standard axioms, and this is a
question that people have been studying in various restricted forms of
proof systems.

The fact that a problem is \(\mathbf{NP}\)-hard means that we believe
there is no efficient algorithm that solve it in the *worst case*. It
does not, however, mean that every single instance of the problem is
hard. For example, if all the clauses in a 3SAT instance \(\varphi\)
contain the same variable \(x_i\) (possibly in negated form), then by
guessing a value to \(x_i\) we can reduce \(\varphi\) to a 2SAT instance
which can then be efficiently solved. Generalizations of this simple
idea are used in “SAT solvers”, which are algorithms that have solved
certain specific interesting SAT formulas with thousands of variables,
despite the fact that we believe SAT to be exponentially hard in the
worst case. Similarly, a lot of problems arising in economics and
machine learning are \(\mathbf{NP}\)-hard.

It is also true that there are many interesting instances of
\(\mathbf{NP}\)-hard problems that we do *not* currently know how to
solve. Across all application areas, whether it is scientific computing,
optimization, control or more, people often encounter hard instances of
\(\mathbf{NP}\) problems on which our current algorithms fail. In fact, as
we will see, all of our digital security infrastructure relies on the
fact that some concrete and easy-to-generate instances of, say, 3SAT
(or, equivalently, any other \(\mathbf{NP}\)-hard problem) are
exponentially hard to solve.

Thus it would be wrong to say that \(\mathbf{NP}\) is easy “in practice”, nor would it be correct to take \(\mathbf{NP}\)-hardness as the “final word” on the complexity of a problem, particularly when we have more information about how any given instance is generated. Understanding both the “typical complexity” of \(\mathbf{NP}\) problems, as well as the power and limitations of certain heuristics (such as various local-search based algorithms) is a very active area of research. We will see more on these topics later in this course.

*heuristics* such as SAT solvers that succeed on *some* instances,
and *proxy measures* such as mathematical relaxations that instead
of solving problem \(X\) (e.g., an integer program) solve program \(X'\)
(e.g., a linear program) that is related to that. Maybe give
compressed sensing as an example, and least square minimization as a
proxy for maximum apostoriori probability.

So, \(\mathbf{P}=\mathbf{NP}\) would give us all kinds of fantastical outcomes. But we strongly suspect that \(\mathbf{P} \neq \mathbf{NP}\), and moreover that there is no much-better-than-brute-force algorithm for 3SAT. If indeed that is the case, is it all bad news?

One might think that impossibility results, telling you that you
*cannot* do something, is the kind of cloud that does not have a silver
lining. But in fact, as we already alluded to before, it does. A hard
(in a sufficiently strong sense) problem in \(\mathbf{NP}\) can be used to
create a code that *cannot be broken*, a task that for thousands of
years has been the dream of not just spies but of many scientists and
mathematicians over the generations. But the complexity viewpoint turned
out to yield much more than simple codes, achieving tasks that people
had previously not even dared to dream of. These include the notion of
*public key cryptography*, allowing two people to communicate securely
without ever having exchanged a secret key; *electronic cash*, allowing
private and secure transaction without a central authority; and *secure
multiparty computation*, enabling parties to compute a joint function on
private inputs without revealing any extra information about it. Also,
as we will see, computational hardness can be used to replace the role
of *randomness* in many settings.

Furthermore, while it is often convenient to pretend that computational problems are simply handed to us, and that our job as computer scientists is to find the most efficient algorithm for them, this is not how things work in most computing applications. Typically even formulating the problem to solve is a highly non-trivial task. When we discover that the problem we want to solve is \(\mathbf{NP}\)-hard, this might be a useful sign that we used the wrong formulation for it.

Beyond all these, the quest to understand computational hardness \(-\)
including the discoveries of lower bounds for restricted computational
models, as well as new types of reductions (such as those arising from
“probabilistically checkable proofs”) \(-\) has already had surprising
*positive* applications to problems in algorithm design, as well as in
coding for both communication and storage. This is not surprising since,
as we mentioned before, from group theory to the theory of relativity,
the pursuit of impossibility results has often been one of the most
fruitful enterprises of mankind.

- The question of whether \(\mathbf{P}=\mathbf{NP}\) is one of the most important and fascinating questions of computer science and science at large, touching on all fields of the natural and social sciences, as well as mathematics and engineering.
- Our current evidence and understanding supports the “SAT hard” scenario that there is no much-better-than-brute-force algorithm for 3SAT or many other \(\mathbf{NP}\)-hard problems.
- We are very far from
*proving*this, however. Researchers have studied proving lower bounds on the number of gates to compute explicit functions in*restricted forms*of circuits, and have made some advances in this effort, along the way generating mathematical tools that have found other uses. However, we have made essentially no headway in proving lower bounds for*general*models of computation such as NAND and NAND++ programs. Indeed, we currently do not even know how to rule out the possibility that for every \(n\in \N\), \(SAT\) restricted to \(n\)-length inputs has a NAND program of \(10n\) lines (even though there*exist*\(n\)-input functions that require \(2^n/(10n)\) lines to compute). - Understanding how to cope with this computational intractability, and even benefit from it, comprises much of the research in theoretical computer science.

Most of the exercises have been written in the summer of 2018 and haven’t yet been fully debugged. While I would prefer people do not post online solutions to the exercises, I would greatly appreciate if you let me know of any bugs. You can do so by posting a GitHub issue about the exercise, and optionally complement this with an email to me with more details about the attempted solution.

Some topics related to this chapter that might be accessible to advanced students include: (to be completed)

- Polynomial hieararchy hardness for circuit minimization and related problems, see for example this paper.

Copyright 2018, Boaz Barak.

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