Computation and Representation

“The alphabet (sic) was a great invention, which enabled men (sic) to store and to learn with little effort what others had learned the hard way – that is, to learn from books rather than from direct, possibly painful, contact with the real world.”, B.F. Skinner

“The name of the song is called `HADDOCK’S EYES.’” [said the Knight]

“Oh, that’s the name of the song, is it?” Alice said, trying to feel interested.

“No, you don’t understand,” the Knight said, looking a little vexed. “That’s what the name is CALLED. The name really is `THE AGED AGED MAN.’”

“Then I ought to have said `That’s what the SONG is called’?” Alice corrected herself.

“No, you oughtn’t: that’s quite another thing! The SONG is called `WAYS AND MEANS’: but that’s only what it’s CALLED, you know!”

“Well, what IS the song, then?” said Alice, who was by this time completely bewildered.

“I was coming to that,” the Knight said. “The song really IS `A-SITTING ON A GATE’: and the tune’s my own invention.”

Lewis Carroll, Through the Looking-Glass

To a first approximation, computation is a process that maps an input to an output.

When discussing computation, it is essential to separate the question of what is the task we need to perform (i.e., the specification) from the question of how we achieve this task (i.e., the implementation). For example, as we’ve seen, there is more than one way to achieve the computational task of computing the product of two integers.

In this chapter we focus on the what part, namely defining computational tasks. For starters, we need to define the inputs and outputs. Capturing all the potential inputs and outputs that we might ever want to compute seems challenging, since computation today is applied to a wide variety of objects. We do not compute merely on numbers, but also on texts, images, videos, connection graphs of social networks, MRI scans, gene data, and even other programs. We will represent all these objects as strings of zeroes and ones, that is objects such as \(0011101\) or \(1011\) or any other finite list of \(1\)’s and \(0\)’s. (This choice is for convenience: there is nothing “holy” about zeroes and ones, and we could have used any other finite collection of symbols.)

Today, we are so used to the notion of digital representation that we are not surprised by the existence of such an encoding. But it is actually a deep insight with significant implications. Many animals can convey a particular fear or desire, but what is unique about humans is language: we use a finite collection of basic symbols to describe a potentially unlimited range of experiences. Language allows transmission of information over both time and space and enables societies that span a great many people and accumulate a body of shared knowledge over time.

Over the last several decades, we have seen a revolution in what we can represent and convey in digital form. We can capture experiences with almost perfect fidelity, and disseminate it essentially instantaneously to an unlimited audience. Moreover, once information is in digital form, we can compute over it, and gain insights from data that were not accessible in prior times. At the heart of this revolution is the simple but profound observation that we can represent an unbounded variety of objects using a finite set of symbols (and in fact using only the two symbols 0 and 1).

In later chapters, we will typically take such representations for granted, and hence use expressions such as “program \(P\) takes \(x\) as input” when \(x\) might be a number, a vector, a graph, or any other object, when we really mean that \(P\) takes as input the representation of \(x\) as a binary string. However, in this chapter we will dwell a bit more on how we can construct such representations.

Defining representations

Every time we store numbers, images, sounds, databases, or other objects on a computer, what we actually store in the computer’s memory is the representation of these objects. Moreover, the idea of representation is not restricted to digital computers. When we write down text or make a drawing we are representing ideas or experiences as sequences of symbols (which might as well be strings of zeroes and ones). Even our brain does not store the actual sensory inputs we experience, but rather only a representation of them.

To use objects such as numbers, images, graphs, or others as inputs for computation, we need to define precisely how to represent these objects as binary strings. A representation scheme is a way to map an object \(x\) to a binary string \(E(x) \in \{0,1\}^*\). For example, a representation scheme for natural numbers is a function \(E:\N \rightarrow \{0,1\}^*\). Of course, we cannot merely represent all numbers as the string “\(0011\)” (for example). A minimal requirement is that if two numbers \(x\) and \(x'\) are different then they would be represented by different strings. Another way to say this is that we require the encoding function \(E\) to be one to one.

Representing natural numbers

We now show how we can represent natural numbers as binary strings. Over the years people have represented numbers in a variety of ways, including Roman numerals, tally marks, our own Hindu-Arabic decimal system, and many others. We can use any one of those as well as many others to represent a number as a string (see Figure 2.3). However, for the sake of concreteness, we use the binary basis as our default representation of natural numbers as strings. For example, we represent the number six as the string \(110\) since \(1\cdot 2^{2} + 1 \cdot 2^1 + 0 \cdot 2^0 = 6\), and similarly we represent the number thirty-five as the string \(y = 100011\) which satisfies \(\sum_{i=0}^5 y_i \cdot 2^{|y|-i-1} = 35\). Some more examples are given in the table below.

If \(n\) is even, then the least significant digit of \(n\)’s binary representation is \(0\), while if \(n\) is odd then this digit equals \(1\). Just like the number \(\floor{n/10}\) corresponds to “chopping off” the least significant decimal digit (e.g., \(\floor{457/10}=\floor{45.7}=45\)), the number \(\floor{n/2}\) corresponds to the “chopping off” the least significant binary digit. Hence the binary representation can be formally defined as the following function \(NtS:\N \rightarrow \{0,1\}^*\) (\(NtS\) stands for “natural numbers to strings”):

Throughout most of this book, the particular choices of representation of numbers as binary strings would not matter much: we just need to know that such a representation exists. In fact, for many of our purposes we can even use the simpler representation of mapping a natural number \(n\) to the length-\(n\) all-zero string \(0^n\).

def NtS(n):# natural numbers to strings
    if n > 1:
        return NtS(n // 2) + str(n % 2)
    else:
        return str(n % 2)

print(NtS(236))
# 11101100

print(NtS(19))
# 10011

def StN(x):# String to number
    k = len(x)-1
    return sum(int(x[i])*(2**(k-i)) for i in range(k+1))

print(StN(NtS(236)))
# 236

Meaning of representations (discussion)

It is natural for us to think of \(236\) as the “actual” number, and of \(11101100\) as “merely” its representation. However, for most Europeans in the middle ages CCXXXVI would be the “actual” number and \(236\) (if they have even heard about it) would be the weird Hindu-Arabic positional representation.¹ When our AI robot overlords materialize, they will probably think of \(11101100\) as the “actual” number and of \(236\) as “merely” a representation that they need to use when they give commands to humans.

So what is the “actual” number? This is a question that philosophers of mathematics have pondered throughout history. Plato argued that mathematical objects exist in some ideal sphere of existence (that to a certain extent is more “real” than the world we perceive via our senses, as this latter world is merely the shadow of this ideal sphere). In Plato’s vision, the symbols \(236\) are merely notation for some ideal object, that, in homage to the late musician, we can refer to as “the number commonly represented by \(236\)”.

The Austrian philosopher Ludwig Wittgenstein, on the other hand, argued that mathematical objects do not exist at all, and the only things that exist are the actual marks on paper that make up \(236\), \(11101100\) or CCXXXVI. In Wittgenstein’s view, mathematics is merely about formal manipulation of symbols that do not have any inherent meaning. You can think of the “actual” number as (somewhat recursively) “that thing which is common to \(236\), \(11101100\) and CCXXXVI and all other past and future representations that are meant to capture the same object”.

While reading this book, you are free to choose your own philosophy of mathematics, as long as you maintain the distinction between the mathematical objects themselves and the various particular choices of representing them, whether as splotches of ink, pixels on a screen, zeroes and ones, or any other form.

Representations beyond natural numbers

We have seen that natural numbers can be represented as binary strings. We now show that the same is true for other types of objects, including (potentially negative) integers, rational numbers, vectors, lists, graphs and many others. In many instances, choosing the “right” string representation for a piece of data is highly non-trivial, and finding the “best” one (e.g., most compact, best fidelity, most efficiently manipulable, robust to errors, most informative features, etc.) is the object of intense research. But for now, we focus on presenting some simple representations for various objects that we would like to use as inputs and outputs for computation.

Representing (potentially negative) integers

Since we can represent natural numbers as strings, we can represent the full set of integers (i.e., members of the set \(\Z=\{ \ldots, -3 , -2 , -1 , 0 , +1, +2, +3,\ldots \}\) ) by adding one more bit that represents the sign. To represent a (potentially negative) number \(m\), we prepend to the representation of the natural number \(|m|\) a bit \(\sigma\) that equals \(0\) if \(m \geq 0\) and equals \(1\) if \(m<0\). Formally, we define the function \(ZtS:\Z \rightarrow \{0,1\}^*\) as follows

While the encoding function of a representation needs to be one to one, it does not have to be onto. For example, in the representation above there is no number that is represented by the empty string but it is still a fine representation, since every integer is represented uniquely by some string.

Two’s complement representation (optional)

Section 2.2.1’s approach of representing an integer using a specific “sign bit” is known as the Signed Magnitude Representation and was used in some early computers. However, the two’s complement representation is much more common in practice. The two’s complement representation of an integer \(k\) in the set \(\{ -2^n , -2^n+1, \ldots, 2^n-1 \}\) is the string \(ZtS_n(k)\) of length \(n+1\) defined as follows:

Another way to say this is that we represent a potentially negative number \(k \in \{ -2^n,\ldots, 2^n-1 \}\) as the non-negative number \(k \mod 2^{n+1}\) (see also Figure 2.4). This means that if two (potentially negative) numbers \(k\) and \(k'\) are not too large (i.e., $ k + k’ { -2^n,, 2^n-1 }$), then we can compute the representation of \(k+k'\) by adding modulo \(2^{n+1}\) the representations of \(k\) and \(k'\) as if they were non-negative integers. This property of the two’s complement representation is its main attraction since, depending on their architectures, microprocessors can often perform arithmetic operations modulo \(2^w\) very efficiently (for certain values of \(w\) such as \(32\) and \(64\)). Many systems leave it to the programmer to check that values are not too large and will carry out this modular arithmetic regardless of the size of the numbers involved. For this reason, in some systems adding two large positive numbers can result in a negative number (e.g., adding \(2^n-100\) and \(2^n-200\) might result in \(-300\) since \((2^{n+1}-300) \mod 2^{n+1} = -300\), see also Figure 2.4).

Rational numbers and representing pairs of strings

We can represent a rational number of the form \(a/b\) by representing the two numbers \(a\) and \(b\). However, merely concatenating the representations of \(a\) and \(b\) will not work. For example, the binary representation of \(4\) is \(100\) and the binary representation of \(43\) is \(101011\), but the concatenation \(100101011\) of these strings is also the concatenation of the representation \(10010\) of \(18\) and the representation \(1011\) of \(11\). Hence, if we used such simple concatenation then we would not be able to tell if the string \(100101011\) is supposed to represent \(4/43\) or \(18/11\).

We tackle this by giving a general representation for pairs of strings. If we were using a pen and paper, we would just use a separator symbol such as \(\|\) to represent, for example, the pair consisting of the numbers represented by \(10\) and \(110001\) as the length-\(9\) string “\(10\|110001\)”. In other words, there is a one to one map \(F\) from pairs of strings \(x,y \in \{0,1\}^*\) into a single string \(z\) over the alphabet \(\Sigma = \{0,1,\| \}\) (in other words, \(z\in \Sigma^*\)). Using such separators is similar to the way we use spaces and punctuation to separate words in English. By adding a little redundancy, we achieve the same effect in the digital domain. We can map the three-element set \(\Sigma\) to the three-element set \(\{00,11,01 \} \subset \{0,1\}^2\) in a one-to-one fashion, and hence encode a length \(n\) string \(z\in \Sigma^*\) as a length \(2n\) string \(w\in \{0,1\}^*\).

Our final representation for rational numbers is obtained by composing the following steps:

1. Representing a (potentially negative) rational number as a pair of integers \(a,b\) such that \(r=a/b\).

2. Representing an integer by a string via the binary representation.

3. Combining 1 and 2 to obtain a representation of a rational number as a pair of strings.

4. Representing a pair of strings over \(\{0,1\}\) as a single string over \(\Sigma = \{0,1,\|\}\).

5. Representing a string over \(\Sigma\) as a longer string over \(\{0,1\}\).

The same idea can be used to represent triples of strings, quadruples, and so on as a string. Indeed, this is one instance of a very general principle that we use time and again in both the theory and practice of computer science (for example, in Object Oriented programming):

Repeating the same idea, once we can represent objects of type \(T\), we can also represent lists of lists of such objects, and even lists of lists of lists and so on and so forth. We will come back to this point when we discuss prefix free encoding in Section 2.5.2.

Representing real numbers

The set of real numbers \(\R\) contains all numbers including positive, negative, and fractional, as well as irrational numbers such as \(\pi\) or \(e\). Every real number can be approximated by a rational number, and thus we can represent every real number \(x\) by a rational number \(a/b\) that is very close to \(x\). For example, we can represent \(\pi\) by \(22/7\) within an error of about \(10^{-3}\). If we want a smaller error (e.g., about \(10^{-4}\)) then we can use \(311/99\), and so on and so forth.

The above representation of real numbers via rational numbers that approximate them is a fine choice for a representation scheme. However, typically in computing applications, it is more common to use the floating-point representation scheme (see Figure 2.5) to represent real numbers. In the floating-point representation scheme we represent \(x\in \R\) by the pair \((b,e)\) of (positive or negative) integers of some prescribed sizes (determined by the desired accuracy) such that \(b \times 2^{e}\) is closest to \(x\). Floating-point representation is the base-two version of scientific notation, where one represents a number \(y\in R\) as its approximation of the form \(b \times 10^e\) for \(b,e\). It is called “floating-point” because we can think of the number \(b\) as specifying a sequence of binary digits, and \(e\) as describing the location of the “binary point” within this sequence. The use of floating representation is the reason why in many programming systems, printing the expression 0.1+0.2 will result in 0.30000000000000004 and not 0.3, see here, here and here for more.

The reader might be (rightly) worried about the fact that the floating-point representation (or the rational number one) can only approximately represent real numbers. In many (though not all) computational applications, one can make the accuracy tight enough so that this does not affect the final result, though sometimes we do need to be careful. Indeed, floating-point bugs can sometimes be no joking matter. For example, floating-point rounding errors have been implicated in the failure of a U.S. Patriot missile to intercept an Iraqi Scud missile, costing 28 lives, as well as a 100 million pound error in computing payouts to British pensioners.

Cantor’s Theorem, countable sets, and string representations of the real numbers

“For any collection of fruits, we can make more fruit salads than there are fruits. If not, we could label each salad with a different fruit, and consider the salad of all fruits not in their salad. The label of this salad is in it if and only if it is not.”, Martha Storey.

Given the issues with floating-point approximations for real numbers, a natural question is whether it is possible to represent real numbers exactly as strings. Unfortunately, the following theorem shows that this cannot be done:

Countable sets. We say that a set \(S\) is countable if there is an onto map \(C:\N \rightarrow S\), or in other words, we can write \(S\) as the sequence \(C(0),C(1),C(2),\ldots\). Since the binary representation yields an onto map from \(\{0,1\}^*\) to \(\N\), and the composition of two onto maps is onto, a set \(S\) is countable iff there is an onto map from \(\{0,1\}^*\) to \(S\). Using the basic properties of functions (see Section 1.4.3), a set is countable if and only if there is a one-to-one function from \(S\) to \(\{0,1\}^*\). Hence, we can rephrase Theorem 2.5 as follows:

Theorem 2.6 was proven by Georg Cantor in 1874. This result (and the theory around it) was quite shocking to mathematicians at the time. By showing that there is no one-to-one map from \(\R\) to \(\{0,1\}^*\) (or \(\N\)), Cantor showed that these two infinite sets have “different forms of infinity” and that the set of real numbers \(\R\) is in some sense “bigger” than the infinite set \(\{0,1\}^*\). The notion that there are “shades of infinity” was deeply disturbing to mathematicians and philosophers at the time. The philosopher Ludwig Wittgenstein (whom we mentioned before) called Cantor’s results “utter nonsense” and “laughable.” Others thought they were even worse than that. Leopold Kronecker called Cantor a “corrupter of youth,” while Henri Poincaré said that Cantor’s ideas “should be banished from mathematics once and for all.” The tide eventually turned, and these days Cantor’s work is universally accepted as the cornerstone of set theory and the foundations of mathematics. As David Hilbert said in 1925, “No one shall expel us from the paradise which Cantor has created for us.” As we will see later in this book, Cantor’s ideas also play a huge role in the theory of computation.

Now that we have discussed Theorem 2.5’s importance, let us see the proof. It is achieved in two steps:

1. Define some infinite set \(\mathcal{X}\) for which it is easier for us to prove that \(\mathcal{X}\) is not countable (namely, it’s easier for us to prove that there is no one-to-one function from \(\mathcal{X}\) to \(\{0,1\}^*\)).

2. Prove that there is a one-to-one function \(G\) mapping \(\mathcal{X}\) to \(\mathbb{R}\).

We can use a proof by contradiction to show that these two facts together imply Theorem 2.5. Specifically, if we assume (towards the sake of contradiction) that there exists some one-to-one \(F\) mapping \(\mathbb{R}\) to \(\{0,1\}^*\), then the function \(x \mapsto F(G(x))\) obtained by composing \(F\) with the function \(G\) from Step 2 above would be a one-to-one function from \(\mathcal{X}\) to \(\{0,1\}^*\), which contradicts what we proved in Step 1!

To turn this idea into a full proof of Theorem 2.5 we need to:

• Define the set \(\mathcal{X}\).

• Prove that there is no one-to-one function from \(\mathcal{X}\) to \(\{0,1\}^*\)

• Prove that there is a one-to-one function from \(\mathcal{X}\) to \(\R\).

We now proceed to do precisely that. That is, we will define the set \(\{0,1\}^\infty\), which will play the role of \(\mathcal{X}\), and then state and prove two lemmas that show that this set satisfies our two desired properties.

That is, \(\{0,1\}^\infty\) is a set of functions, and a function \(f\) is in \(\{0,1\}^\infty\) iff its domain is \(\N\) and its codomain is \(\{0,1\}\). We can also think of \(\{0,1\}^\infty\) as the set of all infinite sequences of bits, since a function \(f:\N \rightarrow \{0,1\}\) can be identified with the sequence \((f(0),f(1),f(2),\ldots )\). The following two lemmas show that \(\{0,1\}^\infty\) can play the role of \(\mathcal{X}\) to establish Theorem 2.5.

As we’ve seen above, Lemma 2.8 and Lemma 2.9 together imply Theorem 2.5. To repeat the argument more formally, suppose, for the sake of contradiction, that there did exist a one-to-one function \(RtS:\R \rightarrow \{0,1\}^*\). By Lemma 2.9, there exists a one-to-one function \(FtR:\{0,1\}^\infty \rightarrow \R\). Thus, under this assumption, since the composition of two one-to-one functions is one-to-one (see Exercise 2.12), the function \(FtS:\{0,1\}^\infty \rightarrow \{0,1\}^*\) defined as \(FtS(f)=RtS(FtR(f))\) will be one to one, contradicting Lemma 2.8. See Figure 2.7 for a graphical illustration of this argument.

Now all that is left is to prove these two lemmas. We start by proving Lemma 2.8 which is really the heart of Theorem 2.5.

Warm-up: “Baby Cantor”. The proof of Lemma 2.8 is rather subtle. One way to get intuition for it is to consider the following finite statement “there is no onto function \(f:\{0,\ldots,99\} \rightarrow \{0,1\}^{100}\)”. Of course we know it’s true since the set \(\{0,1\}^{100}\) is bigger than the set \([100]\), but let’s see a direct proof. For every \(f:\{0,\ldots,99\} \rightarrow \{0,1\}^{100}\), we can define the string \(\overline{d} \in \{0,1\}^{100}\) as follows: \(\overline{d} = (1-f(0)_0, 1-f(1)_1 , \ldots, 1-f(99)_{99})\). If \(f\) was onto, then there would exist some \(n\in [100]\) such that \(f(n) =\overline{d}\), but we claim that no such \(n\) exists. Indeed, if there was such \(n\), then the \(n\)-th coordinate of \(\overline{d}\) would equal \(f(n)_n\) but by definition this coordinate equals \(1-f(n)_n\). See also a “proof by code” of this statement.

To complete the proof of Theorem 2.5, we need to show Lemma 2.9. This requires some calculus background but is otherwise straightforward. If you have not had much experience with limits of a real series before, then the formal proof below might be a little hard to follow. This part is not the core of Cantor’s argument, nor are such limits important to the remainder of this book, so you can feel free to take Lemma 2.9 on faith and skip the proof.

Corollary: Boolean functions are uncountable

Cantor’s Theorem yields the following corollary that we will use several times in this book: the set of all Boolean functions (mapping \(\{0,1\}^*\) to \(\{0,1\}\)) is not countable:

Equivalent conditions for countability

The results above establish many equivalent ways to phrase the fact that a set is countable. Specifically, the following statements are all equivalent:

1. The set \(S\) is countable

2. There exists an onto map from \(\N\) to \(S\)

3. There exists an onto map from \(\{0,1\}^*\) to \(S\).

4. There exists a one-to-one map from \(S\) to \(\N\)

5. There exists a one-to-one map from \(S\) to \(\{0,1\}^*\).

6. There exists an onto map from some countable set \(T\) to \(S\).

7. There exists a one-to-one map from \(S\) to some countable set \(T\).

Representing objects beyond numbers

Numbers are of course by no means the only objects that we can represent as binary strings. A representation scheme for representing objects from some set \(\mathcal{O}\) consists of an encoding function that maps an object in \(\mathcal{O}\) to a string, and a decoding function that decodes a string back to an object in \(\mathcal{O}\). Formally, we make the following definition:

Note that the condition \(D(E(o))=o\) for every \(o\in\mathcal{O}\) implies that \(D\) is onto (can you see why?). It turns out that to construct a representation scheme we only need to find an encoding function. That is, every one-to-one encoding function has a corresponding decoding function, as shown in the following lemma:

Finite representations

If \(\mathcal{O}\) is finite, then we can represent every object in \(\mathcal{O}\) as a string of length at most some number \(n\). What is the value of \(n\)? Let us denote by \(\{0,1\}^{\leq n}\) the set \(\{ x\in \{0,1\}^* : |x| \leq n \}\) of strings of length at most \(n\). The size of \(\{0,1\}^{\leq n}\) is equal to

To obtain a representation of objects in \(\mathcal{O}\) as strings of length at most \(n\) we need to come up with a one-to-one function from \(\mathcal{O}\) to \(\{0,1\}^{\leq n}\). We can do so, if and only if \(|\mathcal{O}| \leq 2^{n+1}-1\) as is implied by the following lemma:

Prefix-free encoding

When showing a representation scheme for rational numbers, we used the “hack” of encoding the alphabet \(\{ 0,1, \|\}\) to represent tuples of strings as a single string. This is a special case of the general paradigm of prefix-free encoding. The idea is the following: if our representation has the property that no string \(x\) representing an object \(o\) is a prefix (i.e., an initial substring) of a string \(y\) representing a different object \(o'\), then we can represent a list of objects by merely concatenating the representations of all the list members. For example, because in English every sentence ends with a punctuation mark such as a period, exclamation, or question mark, no sentence can be a prefix of another and so we can represent a list of sentences by merely concatenating the sentences one after the other. (English has some complications such as periods used for abbreviations (e.g., “e.g.”) or sentence quotes containing punctuation, but the high level point of a prefix-free representation for sentences still holds.)

It turns out that we can transform every representation to a prefix-free form. This justifies Big Idea 1, and allows us to transform a representation scheme for objects of a type \(T\) to a representation scheme of lists of objects of the type \(T\). By repeating the same technique, we can also represent lists of lists of objects of type \(T\), and so on and so forth. But first let us formally define prefix-freeness:

Recall that for every set \(\mathcal{O}\), the set \(\mathcal{O}^*\) consists of all finite length tuples (i.e., lists) of elements in \(\mathcal{O}\). The following theorem shows that if \(E\) is a prefix-free encoding of \(\mathcal{O}\) then by concatenating encodings we can obtain a valid (i.e., one-to-one) representation of \(\mathcal{O}^*\):

Making representations prefix-free

Some natural representations are prefix-free. For example, every fixed output length representation (i.e., one-to-one function \(E:\mathcal{O} \rightarrow \{0,1\}^n\)) is automatically prefix-free, since a string \(x\) can only be a prefix of an equal-length \(x'\) if \(x\) and \(x'\) are identical. Moreover, the approach we used for representing rational numbers can be used to show the following:

The proof of Lemma 2.20 is not the only or even the best way to transform an arbitrary representation into prefix-free form. Exercise 2.10 asks you to construct a more efficient prefix-free transformation satisfying \(|\overline{E}(o)| \leq |E(o)| + O(\log |E(o)|)\).

“Proof by Python” (optional)

The proofs of Theorem 2.18 and Lemma 2.20 are constructive in the sense that they give us:

• A way to transform the encoding and decoding functions of any representation of an object \(O\) to encoding and decoding functions that are prefix-free, and

• A way to extend prefix-free encoding and decoding of single objects to encoding and decoding of lists of objects by concatenation.

Specifically, we could transform any pair of Python functions encode and decode to functions pfencode and pfdecode that correspond to a prefix-free encoding and decoding. Similarly, given pfencode and pfdecode for single objects, we can extend them to encoding of lists. Let us show how this works for the case of the NtS and StN functions we defined above.

We start with the “Python proof” of Lemma 2.20: a way to transform an arbitrary representation into one that is prefix free. The function prefixfree below takes as input a pair of encoding and decoding functions, and returns a triple of functions containing prefix-free encoding and decoding functions, as well as a function that checks whether a string is a valid encoding of an object.

# takes functions encode and decode mapping
# objects to lists of bits and vice versa,
# and returns functions pfencode and pfdecode that
# maps objects to lists of bits and vice versa
# in a prefix-free way.
# Also returns a function pfvalid that says
# whether a list is a valid encoding
def prefixfree(encode, decode):
    def pfencode(o):
        L = encode(o)
        return [L[i//2] for i in range(2*len(L))]+[0,1]
    def pfdecode(L):
        return decode([L[j] for j in range(0,len(L)-2,2)])
    def pfvalid(L):
        return (len(L) % 2 == 0 ) and all(L[2*i]==L[2*i+1] for i in range((len(L)-2)//2)) and L[-2:]==[0,1]

    return pfencode, pfdecode, pfvalid

pfNtS, pfStN , pfvalidN = prefixfree(NtS,StN)

NtS(234)
# 11101010
pfNtS(234)
# 111111001100110001
pfStN(pfNtS(234))
# 234
pfvalidM(pfNtS(234))
# true

We now show a “Python proof” of Theorem 2.18. Namely, we show a function represlists that takes as input a prefix-free representation scheme (implemented via encoding, decoding, and validity testing functions) and outputs a representation scheme for lists of such objects. If we want to make this representation prefix-free then we could fit it into the function prefixfree above.

def represlists(pfencode,pfdecode,pfvalid):
    """
    Takes functions pfencode, pfdecode and pfvalid,
    and returns functions encodelists, decodelists
    that can encode and decode lists of the objects
    respectively.
    """

    def encodelist(L):
        """Gets list of objects, encodes it as list of bits"""
        return "".join([pfencode(obj) for obj in L])

    def decodelist(S):
        """Gets lists of bits, returns lists of objects"""
        i=0; j=1 ; res = []
        while j<=len(S):
            if pfvalid(S[i:j]):
                res += [pfdecode(S[i:j])]
                i=j
            j+= 1
        return res

    return encodelist,decodelist


LtS , StL = represlists(pfNtS,pfStN,pfvalidN)

LtS([234,12,5])
# 111111001100110001111100000111001101
StL(LtS([234,12,5]))
# [234, 12, 5]

Representing letters and text

We can represent a letter or symbol by a string, and then if this representation is prefix-free, we can represent a sequence of symbols by merely concatenating the representation of each symbol. One such representation is the ASCII that represents \(128\) letters and symbols as strings of \(7\) bits. Since the ASCII representation is fixed-length, it is automatically prefix-free (can you see why?). Unicode is the representation of (at the time of this writing) about 128,000 symbols as numbers (known as code points) between \(0\) and \(1,114,111\). There are several types of prefix-free representations of the code points, a popular one being UTF-8 that encodes every codepoint into a string of length between \(8\) and \(32\).

int      2    : 00000010 00000000 00000000 00000000
int      4    : 00000100 00000000 00000000 00000000
int      513  : 00000001 00000010 00000000 00000000
long     513  : 00000001 00000010 00000000 00000000 00000000 00000000 00000000 00000000
int      -1   : 11111111 11111111 11111111 11111111
int      -2   : 11111110 11111111 11111111 11111111
string   Hello: 01001000 01100101 01101100 01101100 01101111 00000000
string   abcd : 01100001 01100010 01100011 01100100 00000000
float    33.0 : 00000000 00000000 00000100 01000010
float    66.0 : 00000000 00000000 10000100 01000010
float    132.0: 00000000 00000000 00000100 01000011
double   132.0: 00000000 00000000 00000000 00000000 00000000 10000000 01100000 01000000

Representing vectors, matrices, images

Once we can represent numbers and lists of numbers, then we can also represent vectors (which are just lists of numbers). Similarly, we can represent lists of lists, and thus, in particular, can represent matrices. To represent an image, we can represent the color at each pixel by a list of three numbers corresponding to the intensity of Red, Green and Blue. (We can restrict to three primary colors since most humans only have three types of cones in their retinas; we would have needed 16 primary colors to represent colors visible to the Mantis Shrimp.) Thus an image of \(n\) pixels would be represented by a list of \(n\) such length-three lists. A video can be represented as a list of images. Of course these representations are rather wasteful and much more compact representations are typically used for images and videos, though this will not be our concern in this book.

Representing graphs

A graph on \(n\) vertices can be represented as an \(n\times n\) adjacency matrix whose \((i,j)^{th}\) entry is equal to \(1\) if the edge \((i,j)\) is present and is equal to \(0\) otherwise. That is, we can represent an \(n\) vertex directed graph \(G=(V,E)\) as a string \(A\in \{0,1\}^{n^2}\) such that \(A_{i,j}=1\) iff the edge \(\overrightarrow{i\;j}\in E\). We can transform an undirected graph to a directed graph by replacing every edge \(\{i,j\}\) with both edges \(\overrightarrow{i\; j}\) and \(\overleftarrow{i\;j}\)

Another representation for graphs is the adjacency list representation. That is, we identify the vertex set \(V\) of a graph with the set \([n]\) where \(n=|V|\), and represent the graph \(G=(V,E)\) as a list of \(n\) lists, where the \(i\)-th list consists of the out-neighbors of vertex \(i\). The difference between these representations can be significant for some applications, though for us would typically be immaterial.

Representing lists and nested lists

If we have a way of representing objects from a set \(\mathcal{O}\) as binary strings, then we can represent lists of these objects by applying a prefix-free transformation. Moreover, we can use a trick similar to the above to handle nested lists. The idea is that if we have some representation \(E:\mathcal{O} \rightarrow \{0,1\}^*\), then we can represent nested lists of items from \(\mathcal{O}\) using strings over the five element alphabet \(\Sigma = \{\) 0,1,[ , ] , , \(\}\). For example, if \(o_1\) is represented by 0011, \(o_2\) is represented by 10011, and \(o_3\) is represented by 00111, then we can represent the nested list \((o_1,(o_2,o_3))\) as the string "[0011,[10011,00111]]" over the alphabet \(\Sigma\). By encoding every element of \(\Sigma\) itself as a three-bit string, we can transform any representation for objects \(\mathcal{O}\) into a representation that enables representing (potentially nested) lists of these objects.

Notation

We will typically identify an object with its representation as a string. For example, if \(F:\{0,1\}^* \rightarrow \{0,1\}^*\) is some function that maps strings to strings and \(n\) is an integer, we might make statements such as “\(F(n)+1\) is prime” to mean that if we represent \(n\) as a string \(x\), then the integer \(m\) represented by the string \(F(x)\) satisfies that \(m+1\) is prime. (You can see how this convention of identifying objects with their representation can save us a lot of cumbersome formalism.) Similarly, if \(x,y\) are some objects and \(F\) is a function that takes strings as inputs, then by \(F(x,y)\) we will mean the result of applying \(F\) to the representation of the ordered pair \((x,y)\). We use the same notation to invoke functions on \(k\)-tuples of objects for every \(k\).

This convention of identifying an object with its representation as a string is one that we humans follow all the time. For example, when people say a statement such as “\(17\) is a prime number”, what they really mean is that the integer whose decimal representation is the string “17”, is prime.

When we say

\(A\) is an algorithm that computes the multiplication function on natural numbers.

what we really mean is that

\(A\) is an algorithm that computes the function \(F:\{0,1\}^* \rightarrow \{0,1\}^*\) such that for every pair \(a,b \in \N\), if \(x\in \{0,1\}^*\) is a string representing the pair \((a,b)\) then \(F(x)\) will be a string representing their product \(a\cdot b\).

Defining computational tasks as mathematical functions

Abstractly, a computational process is some process that takes an input which is a string of bits and produces an output which is a string of bits. This transformation of input to output can be done using a modern computer, a person following instructions, the evolution of some natural system, or any other means.

In future chapters, we will turn to mathematically defining computational processes, but, as we discussed above, at the moment we focus on computational tasks. That is, we focus on the specification and not the implementation. Again, at an abstract level, a computational task can specify any relation that the output needs to have with the input. However, for most of this book, we will focus on the simplest and most common task of computing a function. Here are some examples:

• Given (a representation of) two integers \(x,y\), compute the product \(x\times y\). Using our representation above, this corresponds to computing a function from \(\{0,1\}^*\) to \(\{0,1\}^*\). We have seen that there is more than one way to solve this computational task, and in fact, we still do not know the best algorithm for this problem.

• Given (a representation of) an integer \(z>1\), compute its factorization; i.e., the list of primes \(p_1 \leq \cdots \leq p_k\) such that \(z = p_1\cdots p_k\). This again corresponds to computing a function from \(\{0,1\}^*\) to \(\{0,1\}^*\). The gaps in our knowledge of the complexity of this problem are even larger.

• Given (a representation of) a graph \(G\) and two vertices \(s\) and \(t\), compute the length of the shortest path in \(G\) between \(s\) and \(t\), or do the same for the longest path (with no repeated vertices) between \(s\) and \(t\). Both these tasks correspond to computing a function from \(\{0,1\}^*\) to \(\{0,1\}^*\), though it turns out that there is a vast difference in their computational difficulty.

• Given the code of a Python program, determine whether there is an input that would force it into an infinite loop. This task corresponds to computing a partial function from \(\{0,1\}^*\) to \(\{0,1\}\) since not every string corresponds to a syntactically valid Python program. We will see that we do understand the computational status of this problem, but the answer is quite surprising.

• Given (a representation of) an image \(I\), decide if \(I\) is a photo of a cat or a dog. This corresponds to computing some (partial) function from \(\{0,1\}^*\) to \(\{0,1\}\).

For every particular function \(F\), there can be several possible algorithms to compute \(F\). We will be interested in questions such as:

• For a given function \(F\), can it be the case that there is no algorithm to compute \(F\)?

• If there is an algorithm, what is the best one? Could it be that \(F\) is “effectively uncomputable” in the sense that every algorithm for computing \(F\) requires a prohibitively large amount of resources?

• If we cannot answer this question, can we show equivalence between different functions \(F\) and \(F'\) in the sense that either they are both easy (i.e., have fast algorithms) or they are both hard?

• Can a function being hard to compute ever be a good thing? Can we use it for applications in areas such as cryptography?

In order to do that, we will need to mathematically define the notion of an algorithm, which is what we will do in Chapter 3.

Distinguish functions from programs!

You should always watch out for potential confusions between specifications and implementations or equivalently between mathematical functions and algorithms/programs. It does not help that programming languages (Python included) use the term “functions” to denote (parts of) programs. This confusion also stems from thousands of years of mathematical history, where people typically defined functions by means of a way to compute them.

For example, consider the multiplication function on natural numbers. This is the function \(\ensuremath{\mathit{MULT}}:\N\times \N \rightarrow \N\) that maps a pair \((x,y)\) of natural numbers to the number \(x \cdot y\). As we mentioned, it can be implemented in more than one way:

def mult1(x,y):
    res = 0
    while y>0:
        res += x
        y   -= 1
    return res

def mult2(x,y):
    a = str(x) # represent x as string in decimal notation
    b = str(y) # represent y as string in decimal notation
    res = 0
    for i in range(len(a)):
        for j in range(len(b)):
            res += int(a[len(a)-i])*int(b[len(b)-j])*(10**(i+j))
    return res

print(mult1(12,7))
# 84
print(mult2(12,7))
# 84

Both mult1 and mult2 produce the same output given the same pair of natural number inputs. (Though mult1 will take far longer to do so when the numbers become large.) Hence, even though these are two different programs, they compute the same mathematical function. This distinction between a program or algorithm \(A\), and the function \(F\) that \(A\) computes will be absolutely crucial for us in this course (see also Figure 2.13).

Distinguishing functions from programs (or other ways for computing, including circuits and machines) is a crucial theme for this course. For this reason, this is often a running theme in questions that I (and many other instructors) assign in homework and exams (hint, hint).

Exercises

Bibliographical notes

The study of representing data as strings, including issues such as compression and error corrections falls under the purview of information theory, as covered in the classic textbook of Cover and Thomas (Cover, Thomas, 2006) . Representations are also studied in the field of data structures design, as covered in texts such as (Cormen, Leiserson, Rivest, Stein, 2009) .

The question of whether to represent integers with the most significant digit first or last is known as Big Endian vs. Little Endian representation. This terminology comes from Cohen’s (Cohen, 1981) entertaining and informative paper about the conflict between adherents of both schools which he compared to the warring tribes in Jonathan Swift’s “Gulliver’s Travels”. The two’s complement representation of signed integers was suggested in von Neumann’s classic report (von Neumann, 1945) that detailed the design approaches for a stored-program computer, though similar representations have been used even earlier in abacus and other mechanical computation devices.

The idea that we should separate the definition or specification of a function from its implementation or computation might seem “obvious,” but it took quite a lot of time for mathematicians to arrive at this viewpoint. Historically, a function \(F\) was identified by rules or formulas showing how to derive the output from the input. As we discuss in greater depth in Chapter 9, in the 1800s this somewhat informal notion of a function started “breaking at the seams,” and eventually mathematicians arrived at the more rigorous definition of a function as an arbitrary assignment of input to outputs. While many functions may be described (or computed) by one or more formulas, today we do not consider that to be an essential property of functions, and also allow functions that do not correspond to any “nice” formula.

We have mentioned that all representations of the real numbers are inherently approximate. Thus an important endeavor is to understand what guarantees we can offer on the approximation quality of the output of an algorithm, as a function of the approximation quality of the inputs. This question is known as the question of determining the numerical stability of given equations. The Floating-Point Guide website contains an extensive description of the floating-point representation, as well the many ways in which it could subtly fail, see also the website 0.30000000000000004.com.

Dauben (Dauben, 1990) gives a biography of Cantor with emphasis on the development of his mathematical ideas. (Halmos, 1960) is a classic textbook on set theory, also including Cantor’s theorem. Cantor’s Theorem is also covered in many texts on discrete mathematics, including (Lehman, Leighton, Meyer, 2018) (Lewis, Zax, 2019) .

The adjacency matrix representation of graphs is not merely a convenient way to map a graph into a binary string, but it turns out that many natural notions and operations on matrices are useful for graphs as well. (For example, Google’s PageRank algorithm relies on this viewpoint.) The notes of Spielman’s course are an excellent source for this area, known as spectral graph theory. We will return to this view much later in this book when we talk about random walks.

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Compiled on 12/06/2023 00:06:46

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