Code as data, data as code

“The term code script is, of course, too narrow. The chromosomal structures are at the same time instrumental in bringing about the development they foreshadow. They are law-code and executive power - or, to use another simile, they are architect’s plan and builder’s craft - in one.” , Erwin Schrödinger, 1944.

“A mathematician would hardly call a correspondence between the set of 64 triples of four units and a set of twenty other units,”universal“, while such correspondence is, probably, the most fundamental general feature of life on Earth”, Misha Gromov, 2013

A program is simply a sequence of symbols, each of which can be encoded as a string of \(0\)’s and \(1\)’s using (for example) the ASCII standard. Therefore we can represent every NAND-CIRC program (and hence also every Boolean circuit) as a binary string. This statement seems obvious but it is actually quite profound. It means that we can treat circuits or NAND-CIRC programs both as instructions to carrying computation and also as data that could potentially be used as inputs to other computations.

This correspondence between code and data is one of the most fundamental aspects of computing. It underlies the notion of general purpose computers, that are not pre-wired to compute only one task, and also forms the basis of our hope for obtaining general artificial intelligence. This concept finds immense use in all areas of computing, from scripting languages to machine learning, but it is fair to say that we haven’t yet fully mastered it. Many security exploits involve cases such as “buffer overflows” when attackers manage to inject code where the system expected only “passive” data (see Figure 5.1). The relation between code and data reaches beyond the realm of electronic computers. For example, DNA can be thought of as both a program and data (in the words of Schrödinger, who wrote before the discovery of DNA’s structure a book that inspired Watson and Crick, DNA is both “architect’s plan and builder’s craft”).

Representing programs as strings

We can represent programs or circuits as strings in a myriad of ways. For example, since Boolean circuits are labeled directed acyclic graphs, we can use the adjacency matrix or adjacency list representations for them. However, since the code of a program is ultimately just a sequence of letters and symbols, arguably the conceptually simplest representation of a program is as such a sequence. For example, the following NAND-CIRC program \(P\)

temp_0 = NAND(X[0],X[1])
temp_1 = NAND(X[0],temp_0)
temp_2 = NAND(X[1],temp_0)
Y[0] = NAND(temp_1,temp_2)

is simply a string of 107 symbols which include lower and upper case letters, digits, the underscore character _ and equality sign =, punctuation marks such as “(”,“)”,“,”, spaces, and “new line” markers (often denoted as “\ n” or “↵”). Each such symbol can be encoded as a string of \(7\) bits using the ASCII encoding, and hence the program \(P\) can be encoded as a string of length \(7 \cdot 107 = 749\) bits.

Nothing in the above discussion was specific to the program \(P\), and hence we can use the same reasoning to prove that every NAND-CIRC program can be represented as a string in \(\{0,1\}^*\). In fact, we can do a bit better. Since the names of the working variables of a NAND-CIRC program do not affect its functionality, we can always transform a program to have the form of \(P'\) where all variables apart from the inputs and outputs have the form temp_0, temp_1, temp_2, etc.. Moreover, if the program has \(s\) lines, then we will never need to use an index larger than \(3s\) (since each line involves at most three variables), and similarly the indices of the input and output variables will all be at most \(3s\). Since a number between \(0\) and \(3s\) can be expressed using at most \(\lceil \log_{10}(3s+1) \rceil = O(\log s)\) digits, each line in the program (which has the form foo = NAND(bar,blah)), can be represented using \(O(1) + O(\log s) = O(\log s)\) symbols, each of which can be represented by \(7\) bits. Hence an \(s\) line program can be represented as a string of \(O(s \log s)\) bits, resulting in the following theorem:

Counting programs, and lower bounds on the size of NAND-CIRC programs

One consequence of the representation of programs as strings is that the number of programs of certain length is bounded by the number of strings that represent them. This has consequences for the sets \(\ensuremath{\mathit{SIZE}}_{n,m}(s)\) that we defined in Section 4.6.

Theorem 5.2 has an important corollary. The number of functions that can be computed using small circuits/programs is much smaller than the total number of functions, and hence there exist functions that require very large (in fact exponentially large) circuits to compute. To see why this is the case, note that a function mapping \(\{0,1\}^2\) to \(\{0,1\}\) can be identified with the list of its four values on the inputs \(00,01,10,11\). A function mapping \(\{0,1\}^3\) to \(\{0,1\}\) can be identified with the list of its eight values on the inputs \(000,001,010,011,100,101,110,111\). More generally, every function \(F:\{0,1\}^n \rightarrow \{0,1\}\) can be identified with the list of its \(2^n\) values on the inputs \(\{0,1\}^n\). Hence the number of functions mapping \(\{0,1\}^n\) to \(\{0,1\}\) is equal to the number of possible \(2^n\) length lists of values which is exactly \(2^{2^n}\). Note that this is double exponential in \(n\), and hence even for small values of \(n\) (e.g., \(n=10\)) the number of functions from \(\{0,1\}^n\) to \(\{0,1\}\) is truly astronomical.² As mentioned, this yields the following corollary:

We have seen before that every function mapping \(\{0,1\}^n\) to \(\{0,1\}\) can be computed by an \(O(2^n /n)\) line program. Theorem 5.3 shows that this is tight in the sense that some functions do require such an astronomical number of lines to compute.

In fact, as we explore in the exercises, this is the case for most functions. Hence functions that can be computed in a small number of lines (such as addition, multiplication, finding short paths in graphs, or even the \(\ensuremath{\mathit{EVAL}}\) function) are the exception, rather than the rule.

Size hierarchy theorem (optional)

By Theorem 4.15 the class \(\ensuremath{\mathit{SIZE}}_{n}(10 \cdot 2^n /n)\) contains all functions from \(\{0,1\}^n\) to \(\{0,1\}\), while by Theorem 5.3, there is some function \(f:\{0,1\}^n \rightarrow \{0,1\}\) that is not contained in \(\ensuremath{\mathit{SIZE}}_{n}(0.1 \cdot 2^n / n)\). In other words, for every sufficiently large \(n\),

The tuples representation

ASCII is a fine presentation of programs, but for some applications it is useful to have a more concrete representation of NAND-CIRC programs. In this section we describe a particular choice, that will be convenient for us later on. A NAND-CIRC program is simply a sequence of lines of the form

blah = NAND(baz,boo)

There is of course nothing special about the particular names we use for variables. Although they would be harder to read, we could write all our programs using only working variables such as temp_0, temp_1 etc. Therefore, our representation for NAND-CIRC programs ignores the actual names of the variables, and just associate a number with each variable. We encode a line of the program as a triple of numbers. If the line has the form foo = NAND(bar,blah) then we encode it with the triple \((i,j,k)\) where \(i\) is the number corresponding to the variable foo and \(j\) and \(k\) are the numbers corresponding to bar and blah respectively.

More concretely, we will associate every variable with a number in the set \([t]= \{0,1,\ldots,t-1\}\). The first \(n\) numbers \(\{0,\ldots,n-1\}\) correspond to the input variables, the last \(m\) numbers \(\{t-m,\ldots,t-1\}\) correspond to the output variables, and the intermediate numbers \(\{ n,\ldots, t-m-1\}\) correspond to the remaining “workspace” variables. Formally, we define our representation as follows:

The list of tuples representation is our default choice for representing NAND-CIRC programs. Since “list of tuples representation” is a bit of a mouthful, we will often call it simply “the representation” for a program \(P\). Sometimes, when the number \(n\) of inputs and number \(m\) of outputs are known from the context, we will simply represent a program as the list \(L\) instead of the triple \((n,m,L)\).

u = NAND(X[0],X[1])
v = NAND(X[0],u)
w = NAND(X[1],u)
Y[0] = NAND(v,w)

Transforming a NAND-CIRC program from its representation as code to the representation as a list of tuples is a fairly straightforward programming exercise, and in particular can be done in a few lines of Python.⁴ The list-of-tuples representation loses information such as the particular names we used for the variables, but this is OK since these names do not make a difference to the functionality of the program.

From tuples to strings

If \(P\) is a program of size \(s\), then the number \(t\) of variables is at most \(3s\) (as every line touches at most three variables). Hence we can encode every variable index in \([t]\) as a string of length \(\ell = \ceil{\log (3s)}\), by adding leading zeroes as needed. Since this is a fixed-length encoding, it is prefix free, and so we can encode the list \(L\) of \(s\) triples (corresponding to the encoding of the \(s\) lines of the program) as simply the string of length \(3\ell s\) obtained by concatenating all of these encodings.

We define \(S(s)\) to be the length of the string representing the list \(L\) corresponding to a size \(s\) program. By the above we see that

We can represent \(P=(n,m,L)\) as a string by prepending a prefix free representation of \(n\) and \(m\) to the list \(L\). Since \(n,m \leq 3s\) (a program must touch at least once all its input and output variables), those prefix free representations can be encoded using strings of length \(O(\log s)\). In particular, every program \(P\) of at most \(s\) lines can be represented by a string of length \(O(s\log s)\). Similarly, every circuit \(C\) of at most \(s\) gates can be represented by a string of length \(O(s \log s)\) (for example by translating \(C\) to the equivalent program \(P\)).

A NAND-CIRC interpreter in NAND-CIRC

Since we can represent programs as strings, we can also think of a program as an input to a function. In particular, for every natural number \(s,n,m>0\) we define the function \(\ensuremath{\mathit{EVAL}}_{s,n,m}:\{0,1\}^{S(s)+n} \rightarrow \{0,1\}^m\) as follows:

That is, \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) takes as input the concatenation of two strings: a string \(p\in \{0,1\}^{|S(s)|}\) and a string \(x\in \{0,1\}^n\). If \(p\) is a string that represents a list of triples \(L\) such that \((n,m,L)\) is a list-of-tuples representation of a size-\(s\) NAND-CIRC program \(P\), then \(\ensuremath{\mathit{EVAL}}_{s,n,m}(px)\) is equal to the evaluation \(P(x)\) of the program \(P\) on the input \(x\). Otherwise, \(\ensuremath{\mathit{EVAL}}_{s,n,m}(px)\) equals \(0^m\) (this case is not very important: you can simply think of \(0^m\) as some “junk value” that indicates an error).

Take-away points. The fine details of \(\ensuremath{\mathit{EVAL}}_{s,n,m}\)’s definition are not very crucial. Rather, what you need to remember about \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) is that:

• \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) is a finite function taking a string of fixed length as input and outputting a string of fixed length as output.

• \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) is a single function, such that computing \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) allows to evaluate arbitrary NAND-CIRC programs of a certain length on arbitrary inputs of the appropriate length.

• \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) is a function, not a program (recall the discussion in Section 3.7.2). That is, \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) is a specification of what output is associated with what input. The existence of a program that computes \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) (i.e., an implementation for \(\ensuremath{\mathit{EVAL}}_{s,n,m}\)) is a separate fact, which needs to be established (and which we will do in Theorem 5.9, with a more efficient program shown in Theorem 5.10).

One of the first examples of self circularity we will see in this book is the following theorem, which we can think of as showing a “NAND-CIRC interpreter in NAND-CIRC”:

That is, the NAND-CIRC program \(U_{s,n,m}\) takes the description of any other NAND-CIRC program \(P\) (of the right length and inputs/outputs) and any input \(x\), and computes the result of evaluating the program \(P\) on the input \(x\). Given the equivalence between NAND-CIRC programs and Boolean circuits, we can also think of \(U_{s,n,m}\) as a circuit that takes as input the description of other circuits and their inputs, and returns their evaluation, see Figure 5.6. We call this NAND-CIRC program \(U_{s,n,m}\) that computes \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) a bounded universal program (or a universal circuit, see Figure 5.6). “Universal” stands for the fact that this is a single program that can evaluate arbitrary code, where “bounded” stands for the fact that \(U_{s,n,m}\) only evaluates programs of bounded size. Of course this limitation is inherent for the NAND-CIRC programming language, since a program of \(s\) lines (or, equivalently, a circuit of \(s\) gates) can take at most \(2s\) inputs. Later, in Chapter 7, we will introduce the concept of loops (and the model of Turing machines), that allow to escape this limitation.

Efficient universal programs

Theorem 5.9 establishes the existence of a NAND-CIRC program for computing \(\ensuremath{\mathit{EVAL}}_{s,n,m}\), but it provides no explicit bound on the size of this program. Theorem 4.12, which we used to prove Theorem 5.9, guarantees the existence of a NAND-CIRC program whose size can be as large as exponential in the length of its input. This would mean that even for moderately small values of \(s,n,m\) (for example \(n=100,s=300,m=1\)), computing \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) might require a NAND program with more lines than there are atoms in the observable universe! Fortunately, we can do much better than that. In fact, for every \(s,n,m\) there exists a NAND-CIRC program for computing \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) with size that is polynomial in its input length. This is shown in the following theorem.

Unlike Theorem 5.9, Theorem 5.10 is not a trivial corollary of the fact that every finite function can be computed by some circuit. Proving Theorem 5.10 requires us to present a concrete NAND-CIRC program for computing the function \(\ensuremath{\mathit{EVAL}}_{s,n,m}\). We will do so in several stages.

1. First, we will describe the algorithm to evaluate \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) in “pseudo code”.

2. Then, we will show how we can write a program to compute \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) in Python. We will not use much about Python, and a reader that has familiarity with programming in any language should be able to follow along.

3. Finally, we will show how we can transform this Python program into a NAND-CIRC program.

This approach yields much more than just proving Theorem 5.10: we will see that it is in fact always possible to transform (loop free) code in high level languages such as Python to NAND-CIRC programs (and hence to Boolean circuits as well).

A NAND-CIRC interpeter in “pseudocode”

To prove Theorem 5.10 it suffices to give a NAND-CIRC program of \(O(s^2 \log s)\) lines that can evaluate NAND-CIRC programs of \(s\) lines. Let us start by thinking how we would evaluate such programs if we weren’t restricted to only performing NAND operations. That is, let us describe informally an algorithm that on input \(n,m,s\), a list of triples \(L\), and a string \(x\in \{0,1\}^n\), evaluates the program represented by \((n,m,L)\) on the string \(x\).

We will now describe such an algorithm. We assume that we have access to a bit array data structure that can store for every \(i\in [t]\) a bit \(T_i \in \{0,1\}\). Specifically, if Table is a variable holding this data structure, then we assume we can perform the operations:

• GET(Table,i) which retrieves the bit corresponding to i in Table. The value of i is assumed to be an integer in \([t]\).

• Table = UPDATE(Table,i,b) which updates Table so the bit corresponding to i is now set to b. The value of i is assumed to be an integer in \([t]\) and b is a bit in \(\{0,1\}\).

Algorithm 5.11 evaluates the program given to it as input one line at a time, updating the Vartable table to contain the value of each variable. At the end of the execution it outputs the variables at positions \(t-m,t-m+1,\ldots,t-1\) which correspond to the input variables.

A NAND interpreter in Python

To make things more concrete, let us see how we implement Algorithm 5.11 in the Python programming language. (There is nothing special about Python. We could have easily presented a corresponding function in JavaScript, C, OCaml, or any other programming language.) We will construct a function NANDEVAL that on input \(n,m,L,x\) will output the result of evaluating the program represented by \((n,m,L)\) on \(x\). To keep things simple, we will not worry about the case that \(L\) does not represent a valid program of \(n\) inputs and \(m\) outputs. The code is presented in Figure 5.7.

def NANDEVAL(n,m,L,X):
    # Evaluate a NAND-CIRC program from list of tuple representation.
    s = len(L) # num of lines
    t = max(max(a,b,c) for (a,b,c) in L)+1 # max index in L + 1
    Vartable = [0] * t # initialize array

    # helper functions
    def GET(V,i): return V[i]
    def UPDATE(V,i,b):
        V[i]=b
        return V

    # load input values to Vartable:
    for i in range(n):
        Vartable = UPDATE(Vartable,i,X[i])

    # Run the program
    for (i,j,k) in L:
        a = GET(Vartable,j)
        b = GET(Vartable,k)
        c = NAND(a,b)
        Vartable = UPDATE(Vartable,i,c)

    # Return outputs Vartable[t-m], Vartable[t-m+1],....,Vartable[t-1]
    return [GET(Vartable,t-m+j) for j in range(m)]

# Test on XOR (2 inputs, 1 output)
L = ((2, 0, 1), (3, 0, 2), (4, 1, 2), (5, 3, 4))
print(NANDEVAL(2,1,L,(0,1))) # XOR(0,1)
# [1]
print(NANDEVAL(2,1,L,(1,1))) # XOR(1,1)
# [0]

Accessing an element of the array Vartable at a given index takes a constant number of basic operations. Hence (since \(n,m \leq s\) and \(t \leq 3s\)), the program above will use \(O(s)\) basic operations.⁵

Constructing the NAND-CIRC interpreter in NAND-CIRC

We now turn to describing the proof of Theorem 5.10. To prove the theorem it is not enough to give a Python program. Rather, we need to show how we compute the function \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) using a NAND-CIRC program. In other words, our job is to transform, for every \(s,n,m\), the Python code of Section 5.4.3 to a NAND-CIRC program \(U_{s,n,m}\) that computes the function \(\ensuremath{\mathit{EVAL}}_{s,n,m}\).

Our construction will follow very closely the Python implementation of EVAL above. We will use variables Vartable[\(0\)],\(\ldots\),Vartable[\(2^\ell-1\)], where \(\ell = \ceil{\log 3s}\) to store our variables. However, NAND doesn’t have integer-valued variables, so we cannot write code such as Vartable[i] for some variable i. However, we can implement the function GET(Vartable,i) that outputs the i-th bit of the array Vartable. Indeed, this is nothing but the function \(\ensuremath{\mathit{LOOKUP}}_\ell\) that we have seen in Theorem 4.10!

We saw that we can compute \(\ensuremath{\mathit{LOOKUP}}_\ell\) in time \(O(2^\ell) = O(s)\) for our choice of \(\ell\).

For every \(\ell\), let \(\ensuremath{\mathit{UPDATE}}_\ell:\{0,1\}^{2^\ell + \ell +1} \rightarrow \{0,1\}^{2^\ell}\) correspond to the UPDATE function for arrays of length \(2^\ell\). That is, on input \(V\in \{0,1\}^{2^\ell}\) , \(i\in \{0,1\}^\ell\), \(b\in \{0,1\}\), \(\ensuremath{\mathit{UPDATE}}_\ell(V,i,b)\) is equal to \(V' \in \{0,1\}^{2^\ell}\) such that

1. For every \(j\in [2^\ell]\), there is an \(O(\ell)\) line NAND-CIRC program to compute the function \(\ensuremath{\mathit{EQUALS}}_j: \{0,1\}^\ell \rightarrow \{0,1\}\) that on input \(i\) outputs \(1\) if and only if \(i\) is equal to (the binary representation of) \(j\). (We leave verifying this as Exercise 5.2 and Exercise 5.3.)

2. We have seen that we can compute the function \(\ensuremath{\mathit{IF}}:\{0,1\}^3 \rightarrow \{0,1\}\) such that \(\ensuremath{\mathit{IF}}(a,b,c)\) equals \(b\) if \(a=1\) and \(c\) if \(a=0\).

Together, this means that we can compute UPDATE (using some “syntactic sugar” for bounded length loops) as follows:

def UPDATE_ell(V,i,b):
    # Get V[0]...V[2^ell-1], i in {0,1}^ell, b in {0,1}
    # Return NewV[0],...,NewV[2^ell-1]
    # updated array with NewV[i]=b and all
    # else same as V
    for j in range(2**ell): # j = 0,1,2,....,2^ell -1
        a = EQUALS_j(i)
        NewV[j] = IF(a,b,V[j])
    return NewV

Since the loop over j in UPDATE is run \(2^\ell\) times, and computing EQUALS_j takes \(O(\ell)\) lines, the total number of lines to compute UPDATE is \(O(2^\ell \cdot \ell) = O(s \log s)\). Once we can compute GET and UPDATE, the rest of the implementation amounts to “book keeping” that needs to be done carefully, but is not too insightful, and hence we omit the full details. Since we run GET and UPDATE \(s\) times, the total number of lines for computing \(\ensuremath{\mathit{EVAL}}_{s,n,m}\) is \(O(s^2) + O(s^2 \log s) = O(s^2 \log s)\). This completes (up to the omitted details) the proof of Theorem 5.10.

A Python interpreter in NAND-CIRC (discussion)

To prove Theorem 5.10 we essentially translated every line of the Python program for EVAL into an equivalent NAND-CIRC snippet. However, none of our reasoning was specific to the particular function \(\ensuremath{\mathit{EVAL}}\). It is possible to translate every Python program into an equivalent NAND-CIRC program of comparable efficiency. (More concretely, if the Python program takes \(T(n)\) operations on inputs of length at most \(n\) then there exists NAND-CIRC program of \(O(T(n) \log T(n))\) lines that agrees with the Python program on inputs of length \(n\).) Actually doing so requires taking care of many details and is beyond the scope of this book, but let me try to convince you why you should believe it is possible in principle.

For starters, one can use CPython (the reference implementation for Python), to evaluate every Python program using a C program. We can combine this with a C compiler to transform a Python program to various flavors of “machine language”. So, to transform a Python program into an equivalent NAND-CIRC program, it is enough to show how to transform a machine language program into an equivalent NAND-CIRC program. One minimalistic (and hence convenient) family of machine languages is known as the ARM architecture which powers many mobile devices including essentially all Android devices.⁶ There are even simpler machine languages, such as the LEG architecture for which a backend for the LLVM compiler was implemented (and hence can be the target of compiling any of the large and growing list of languages that this compiler supports). Other examples include the TinyRAM architecture (motivated by interactive proof systems that we will discuss in Chapter 22) and the teaching-oriented Ridiculously Simple Computer architecture. Going one by one over the instruction sets of such computers and translating them to NAND snippets is no fun, but it is a feasible thing to do. In fact, ultimately this is very similar to the transformation that takes place in converting our high level code to actual silicon gates that are not so different from the operations of a NAND-CIRC program. Indeed, tools such as MyHDL that transform “Python to Silicon” can be used to convert a Python program to a NAND-CIRC program.

The NAND-CIRC programming language is just a teaching tool, and by no means do I suggest that writing NAND-CIRC programs, or compilers to NAND-CIRC, is a practical, useful, or enjoyable activity. What I do want is to make sure you understand why it can be done, and to have the confidence that if your life (or at least your grade) depended on it, then you would be able to do this. Understanding how programs in high level languages such as Python are eventually transformed into concrete low-level representation such as NAND is fundamental to computer science.

The astute reader might notice that the above paragraphs only outlined why it should be possible to find for every particular Python-computable function \(f\), a particular comparably efficient NAND-CIRC program \(P\) that computes \(f\). But this still seems to fall short of our goal of writing a “Python interpreter in NAND” which would mean that for every parameter \(n\), we come up with a single NAND-CIRC program \(\ensuremath{\mathit{UNIV}}_s\) such that given a description of a Python program \(P\), a particular input \(x\), and a bound \(T\) on the number of operations (where the lengths of \(P\) and \(x\) and the value of \(T\) are all at most \(s\)) returns the result of executing \(P\) on \(x\) for at most \(T\) steps. After all, the transformation above takes every Python program into a different NAND-CIRC program, and so does not yield “one NAND-CIRC program to rule them all” that can evaluate every Python program up to some given complexity. However, we can in fact obtain one NAND-CIRC program to evaluate arbitrary Python programs. The reason is that there exists a Python interpreter in Python: a Python program \(U\) that takes a bit string, interprets it as Python code, and then runs that code. Hence, we only need to show a NAND-CIRC program \(U^*\) that computes the same function as the particular Python program \(U\), and this will give us a way to evaluate all Python programs.

What we are seeing time and again is the notion of universality or self reference of computation, which is the sense that all reasonably rich models of computation are expressive enough that they can “simulate themselves”. The importance of this phenomenon to both the theory and practice of computing, as well as far beyond it, including the foundations of mathematics and basic questions in science, cannot be overstated.

The physical extended Church-Turing thesis (discussion)

We’ve seen that NAND gates (and other Boolean operations) can be implemented using very different systems in the physical world. What about the reverse direction? Can NAND-CIRC programs simulate any physical computer?

We can take a leap of faith and stipulate that Boolean circuits (or equivalently NAND-CIRC programs) do actually encapsulate every computation that we can think of. Such a statement (in the realm of infinite functions, which we’ll encounter in Chapter 7) is typically attributed to Alonzo Church and Alan Turing, and in that context is known as the Church-Turing Thesis. As we will discuss in future lectures, the Church-Turing Thesis is not a mathematical theorem or conjecture. Rather, like theories in physics, the Church-Turing Thesis is about mathematically modeling the real world. In the context of finite functions, we can make the following informal hypothesis or prediction:

“Physical Extended Church-Turing Thesis” (PECTT): If a function \(F:\{0,1\}^n \rightarrow \{0,1\}^m\) can be computed in the physical world using \(s\) amount of “physical resources” then it can be computed by a Boolean circuit program of roughly \(s\) gates.

A priori it might seem rather extreme to hypothesize that our meager model of NAND-CIRC programs or Boolean circuits captures all possible physical computation. But yet, in more than a century of computing technologies, no one has yet built any scalable computing device that challenges this hypothesis.

We now discuss the “fine print” of the PECTT in more detail, as well as the (so far unsuccessful) challenges that have been raised against it. There is no single universally-agreed-upon formalization of “roughly \(s\) physical resources”, but we can approximate this notion by considering the size of any physical computing device and the time it takes to compute the output, and ask that any such device can be simulated by a Boolean circuit with a number of gates that is a polynomial (with not too large exponent) in the size of the system and the time it takes it to operate.

In other words, we can phrase the PECTT as stipulating that any function that can be computed by a device that takes a certain volume \(V\) of space and requires \(t\) time to complete the computation, must be computable by a Boolean circuit with a number of gates \(p(V,t)\) that is polynomial in \(V\) and \(t\).

The exact form of the function \(p(V,t)\) is not universally agreed upon but it is generally accepted that if \(f:\{0,1\}^n \rightarrow \{0,1\}\) is an exponentially hard function, in the sense that it has no NAND-CIRC program of fewer than, say, \(2^{n/2}\) lines, then a demonstration of a physical device that can compute in the real world \(f\) for moderate input lengths (e.g., \(n=500\)) would be a violation of the PECTT.

Attempts at refuting the PECTT

One of the admirable traits of mankind is the refusal to accept limitations. In the best case this is manifested by people achieving longstanding “impossible” challenges such as heavier-than-air flight, putting a person on the moon, circumnavigating the globe, or even resolving Fermat’s Last Theorem. In the worst case it is manifested by people continually following the footsteps of previous failures to try to do proven-impossible tasks such as build a perpetual motion machine, trisect an angle with a compass and straightedge, or refute Bell’s inequality. The Physical Extended Church-Turing thesis (in its various forms) has attracted both types of people. Here are some physical devices that have been speculated to achieve computational tasks that cannot be done by not-too-large NAND-CIRC programs:

• Spaghetti sort: One of the first lower bounds that Computer Science students encounter is that sorting \(n\) numbers requires making \(\Omega(n \log n)\) comparisons. The “spaghetti sort” is a description of a proposed “mechanical computer” that would do this faster. The idea is that to sort \(n\) numbers \(x_1,\ldots,x_n\), we could cut \(n\) spaghetti noodles into lengths \(x_1,\ldots,x_n\), and then if we simply hold them together in our hand and bring them down to a flat surface, they will emerge in sorted order. There are a great many reasons why this is not truly a challenge to the PECTT hypothesis, and I will not ruin the reader’s fun in finding them out by her or himself.

• Soap bubbles: One function \(F:\{0,1\}^n \rightarrow \{0,1\}\) that is conjectured to require a large number of NAND lines to solve is the Euclidean Steiner Tree problem. This is the problem where one is given \(m\) points in the plane \((x_1,y_1),\ldots,(x_m,y_m)\) (say with integer coordinates ranging from \(1\) till \(m\), and hence the list can be represented as a string of \(n=O(m \log m)\) size) and some number \(K\). The goal is to figure out whether it is possible to connect all the points by line segments of total length at most \(K\). This function is conjectured to be hard because it is NP complete - a concept that we’ll encounter later in this course - and it is in fact reasonable to conjecture that as \(m\) grows, the number of NAND lines required to compute this function grows exponentially in \(m\), meaning that the PECTT would predict that if \(m\) is sufficiently large (such as few hundreds or so) then no physical device could compute \(F\). Yet, some people claimed that there is in fact a very simple physical device that could solve this problem, that can be constructed using some wooden pegs and soap. The idea is that if we take two glass plates, and put \(m\) wooden pegs between them in the locations \((x_1,y_1),\ldots,(x_m,y_m)\) then bubbles will form whose edges touch those pegs in a way that will minimize the total energy which turns out to be a function of the total length of the line segments. The problem with this device is that nature, just like people, often gets stuck in “local optima”. That is, the resulting configuration will not be one that achieves the absolute minimum of the total energy but rather one that can’t be improved with local changes. Aaronson has carried out actual experiments (see Figure 5.8), and saw that while this device often is successful for three or four pegs, it starts yielding suboptimal results once the number of pegs grows beyond that.

• DNA computing. People have suggested using the properties of DNA to do hard computational problems. The main advantage of DNA is the ability to potentially encode a lot of information in a relatively small physical space, as well as compute on this information in a highly parallel manner. At the time of this writing, it was demonstrated that one can use DNA to store about \(10^{16}\) bits of information in a region of radius about a millimeter, as opposed to about \(10^{10}\) bits with the best known hard disk technology. This does not posit a real challenge to the PECTT but does suggest that one should be conservative about the choice of constant and not assume that current hard disk + silicon technologies are the absolute best possible.⁷

• Continuous/real computers. The physical world is often described using continuous quantities such as time and space, and people have suggested that analog devices might have direct access to computing with real-valued quantities and would be inherently more powerful than discrete models such as NAND machines. Whether the “true” physical world is continuous or discrete is an open question. In fact, we do not even know how to precisely phrase this question, let alone answer it. Yet, regardless of the answer, it seems clear that the effort to measure a continuous quantity grows with the level of accuracy desired, and so there is no “free lunch” or way to bypass the PECTT using such machines (see also this paper). Related to that are proposals known as “hypercomputing” or “Zeno’s computers” which attempt to use the continuity of time by doing the first operation in one second, the second one in half a second, the third operation in a quarter second and so on.. These fail for a similar reason to the one guaranteeing that Achilles will eventually catch the tortoise despite the original Zeno’s paradox.

• Relativity computer and time travel. The formulation above assumed the notion of time, but under the theory of relativity time is in the eye of the observer. One approach to solve hard problems is to leave the computer to run for a lot of time from his perspective, but to ensure that this is actually a short while from our perspective. One approach to do so is for the user to start the computer and then go for a quick jog at close to the speed of light before checking on its status. Depending on how fast one goes, few seconds from the point of view of the user might correspond to centuries in computer time (it might even finish updating its Windows operating system!). Of course the catch here is that the energy required from the user is proportional to how close one needs to get to the speed of light. A more interesting proposal is to use time travel via closed timelike curves (CTCs). In this case we could run an arbitrarily long computation by doing some calculations, remembering the current state, and then travelling back in time to continue where we left off. Indeed, if CTCs exist then we’d probably have to revise the PECTT (though in this case I will simply travel back in time and edit these notes, so I can claim I never conjectured it in the first place…)

• Humans. Another computing system that has been proposed as a counterexample to the PECTT is a 3 pound computer of about 0.1m radius, namely the human brain. Humans can walk around, talk, feel, and do other things that are not commonly done by NAND-CIRC programs, but can they compute partial functions that NAND-CIRC programs cannot? There are certainly computational tasks that at the moment humans do better than computers (e.g., play some video games, at the moment), but based on our current understanding of the brain, humans (or other animals) have no inherent computational advantage over computers. The brain has about \(10^{11}\) neurons, each operating at a speed of about \(1000\) operations per seconds. Hence a rough first approximation is that a Boolean circuit of about \(10^{14}\) gates could simulate one second of a brain’s activity.⁸ Note that the fact that such a circuit (likely) exists does not mean it is easy to find it. After all, constructing this circuit took evolution billions of years. Much of the recent efforts in artificial intelligence research is focused on finding programs that replicate some of the brain’s capabilities and they take massive computational effort to discover, these programs often turn out to be much smaller than the pessimistic estimates above. For example, at the time of this writing, Google’s neural network for machine translation has about \(10^4\) nodes (and can be simulated by a NAND-CIRC program of comparable size). Philosophers, priests and many others have since time immemorial argued that there is something about humans that cannot be captured by mechanical devices such as computers; whether or not that is the case, the evidence is thin that humans can perform computational tasks that are inherently impossible to achieve by computers of similar complexity.⁹

• Quantum computation. The most compelling attack on the Physical Extended Church-Turing Thesis comes from the notion of quantum computing. The idea was initiated by the observation that systems with strong quantum effects are very hard to simulate on a computer. Turning this observation on its head, people have proposed using such systems to perform computations that we do not know how to do otherwise. At the time of this writing, scalable quantum computers have not yet been built, but it is a fascinating possibility, and one that does not seem to contradict any known law of nature. We will discuss quantum computing in much more detail in Chapter 23. Modeling quantum computation involves extending the model of Boolean circuits into Quantum circuits that have one more (very special) gate. However, the main takeaway is that while quantum computing does suggest we need to amend the PECTT, it does not require a complete revision of our worldview. Indeed, almost all of the content of this book remains the same regardless of whether the underlying computational model is Boolean circuits or quantum circuits.

Recap of Part I: Finite Computation

This chapter concludes the first part of this book that deals with finite computation (computing functions that map a fixed number of Boolean inputs to a fixed number of Boolean outputs). The main take-aways from Chapter 3, Chapter 4, and Chapter 5 are as follows (see also Figure 5.9):

• We can formally define the notion of a function \(f:\{0,1\}^n \rightarrow \{0,1\}^m\) being computable using \(s\) basic operations. Whether these operations are AND/OR/NOT, NAND, or some other universal basis does not make much difference. We can describe such a computation either using a circuit or using a straight-line program.

• We define \(\ensuremath{\mathit{SIZE}}_{n,m}(s)\) to be the set of functions that are computable by NAND circuits of at most \(s\) gates. This set is equal to the set of functions computable by a NAND-CIRC program of at most \(s\) lines and up to a constant factor in \(s\) (which we will not care about); this is also the same as the set of functions that are computable by a Boolean circuit of at most \(s\) AND/OR/NOT gates. The class \(\ensuremath{\mathit{SIZE}}_{n,m}(s)\) is a set of functions, not of programs/circuits.

• Every function \(f:\{0,1\}^n \rightarrow \{0,1\}^m\) can be computed using a circuit of at most \(O(m \cdot 2^n / n)\) gates. Some functions require at least \(\Omega(m \cdot 2^n /n)\) gates. We define \(\ensuremath{\mathit{SIZE}}_{n,m}(s)\) to be the set of functions from \(\{0,1\}^n\) to \(\{0,1\}^m\) that can be computed using at most \(s\) gates.

• We can describe a circuit/program \(P\) as a string. For every \(s\), there is a universal circuit/program \(U_s\) that can evaluate programs of length \(s\) given their description as strings. We can use this representation also to count the number of circuits of at most \(s\) gates and hence prove that some functions cannot be computed by circuits of smaller-than-exponential size.

• If there is a circuit of \(s\) gates that computes a function \(f\), then we can build a physical device to compute \(f\) using \(s\) basic components (such as transistors). The “Physical Extended Church-Turing Thesis” postulates that the reverse direction is true as well: if \(f\) is a function for which every circuit requires at least \(s\) gates then that every physical device to compute \(f\) will require about \(s\) “physical resources”. The main challenge to the PECTT is quantum computing, which we will discuss in Chapter 23.

Sneak preview: In the next part we will discuss how to model computational tasks on unbounded inputs, which are specified using functions \(F:\{0,1\}^* \rightarrow \{0,1\}^*\) (or \(F:\{0,1\}^* \rightarrow \{0,1\}\)) that can take an unbounded number of Boolean inputs.

Exercises

Bibliographical notes

The \(\ensuremath{\mathit{EVAL}}\) function is usually known as a universal circuit. The implementation we describe is not the most efficient known. Valiant (Valiant, 1976) first showed a universal circuit of size \(O(n \log n)\) where \(n\) is the size of the input. Universal circuits have seen in recent years new motivations due to their applications for cryptography, see (Lipmaa, Mohassel, Sadeghian, 2016) (Günther, Kiss, Schneider, 2017) .

While we’ve seen that “most” functions mapping \(n\) bits to one bit require circuits of exponential size \(\Omega(2^n/n)\), we actually do not know of any explicit function for which we can prove that it requires, say, at least \(n^{100}\) or even \(100n\) size. At the moment, the strongest such lower bound we know is that there are quite simple and explicit \(n\)-variable functions that require at least \((5-o(1))n\) lines to compute, see this paper of Iwama et al as well as this more recent work of Kulikov et al. Proving lower bounds for restricted models of circuits is an extremely interesting research area, for which Jukna’s book (Jukna, 2012) (see also Wegener (Wegener, 1987) ) provides a very good introduction and overview. I learned of the proof of the size hierarchy theorem (Theorem 5.5) from Sasha Golovnev.

Scott Aaronson’s blog post on how information is physical is a good discussion on issues related to the physical extended Church-Turing Physics. Aaronson’s survey on NP complete problems and physical reality (Aaronson, 2005) discusses these issues as well, though it might be easier to read after we reach Chapter 15 on \(\mathbf{NP}\) and \(\mathbf{NP}\)-completeness.

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