\[ \def\N{\mathbb{N}} \def\R{\mathbb{R}} \def\Z{\mathbb{Z}} \def\E{\mathbb{E}} \def\label{\ensuremath{\;\;\;\;\;\;\;}} \notag \]
\[ \newcommand{\floor}[1]{\ensuremath{\lfloor #1 \rfloor}} \newcommand{\ceil}[1]{\ensuremath{\lceil #1 \rceil}} \newcommand{\expr}[1]{\ensuremath{\langle #1 \rangle}} \notag \]

★ See also the PDF version of this chapter (better formatting/references) ★

See any bugs/typos/confusing explanations? Open a GitHub issue. You can also comment below

Computation and Representation

  • Representating an object as a string (often of zeroes and ones).
  • Examples of representations for common objects such as numbers, vectors, lists, graphs.
  • Prefix-free representations.
  • Distinguish between specification and implementation, or equivalently between algorithms/programs and mathematical functions.

“The alphabet was a great invention, which enabled men 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 can be thought of as a process that maps an input to an output.

Figure 1: Our basic notion of computation is some process that maps an input to an output

When discussing computation, it is important 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. A priori this seems nontrivial, since computation today is applied to a huge 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.

Figure 2: We represent numbers, texts, images, networks and many other objects using strings of zeroes and ones. Writing the zeroes and ones themselves in green font over a black background is optional.

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 a deep insight with significant implications. Many animals can convey a particular fear or desire, but what’s 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’ve seen a revolution in what we are able to represent and convey in digital form. We can capture experiences with almost perfect fidelity, and disseminate it essentially instantaneously to an unlimited audience. What’s more, 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 this 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).There is nothing “holy” about using zero and one as the basic symbols, and we can (indeed sometimes people do) use any other finite set of two or more symbols as the fundamental “alphabet”. We use zero and one in this course mainly because it simplifies notation.

In later lectures, we will often fall back on taking this representation for granted, and hence write something like “program \(P\) takes \(x\) as input” when \(x\) might be a number, a vector, a graph, or any other objects, when we really mean that \(P\) takes as input the representation of \(x\) as a binary string. However, in this chapter, let us dwell a little bit on how such representations can be devised.

Examples of binary representations

In many instances, choosing the “right” string representation for a piece of data is highly nontrivial, 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, let us start by describing some simple representations for various natural objects.

Representing natural numbers

Perhaps the simplest object we want to represent is a natural number. That is, a member \(x\) of the set \(\N = \{0,1,2,3,\ldots \}\). We can represent a number \(x\in\N\) as a string using the binary basis. Specifically, every natural number \(x\) can be written in a unique way as \(x = x_02^0 + x_12^1 + \cdots + x_{n-1}2^{n-1}\) (or \(\sum_{i=0}^{n-1} x_i 2^i\) for short) where \(x_0,\ldots,x_{n-1}\) are zero/one and \(n\) is the smallest number such that \(2^n > x\) (and hence \(x_{n-1}=1\) for every nonzero \(x\)). We can then represent \(x\) as the string \((x_0,x_1,\ldots,x_{n-1})\).We can represent the number zero either as some string that contains only zeroes, or as the empty string. The choice will not make any difference for us. For example, the number 35 is represented as the string \((1,1,0,0,0,1)\).Typically when people write down the binary representation, they would print the string \(x\) in reverse order, with the least significant digit as the rightmost one. Representing the number \(x\) as \((x_{n-1},x_{n-2},\ldots,x_0)\) will of course work just as well. We chose the particular representation above for the sake of simplicity, so the the \(i\)-th bit corresponds to \(2^i\), but such low level choices will not make a difference in this course. A related, but not identical, distinction is the Big Endian vs Little Endian representation for integers in computing architecture.

We can think of a representation as consisting of encoding and decoding functions. In the case of the binary representation for integers, the encoding function \(E:\N \rightarrow \{0,1\}^*\) maps a natural number to the string representing it, and the decoding function \(D:\{0,1\}^* \rightarrow \N\) maps a string into the number it represents (i.e., \(D(x_0,\ldots,x_{n-1})= 2^0x_0 + 2^1x_1 +\ldots + 2^{n-1}x_{n-1}\) for every \(x_0,\ldots,x_{n-1} \in \{0,1\}\)). In the Python programming language, we can compute these encoding and decoding functions as follows:

from math import floor, log def int2bits(n): return [ floor(n / 2**i ) % 2 for i in range(floor(log(n,2))+1)] print(int2bits(236)) # [0, 0, 1, 1, 0, 1, 1, 1] print(int2bits(19)) # [1, 1, 0, 0, 1] def bits2int(L): return sum([2**i * L[i] for i in range(len(L))]) print(bits2int([0, 0, 1, 1, 0, 1, 1, 1])) # 236

In this book, we will often illustrate our points by using programming languages to present certain computations. Our examples will be fairly short, and our point will always be to emphasize that certain computations can be done concretely, rather than focus on a particular language feature. We often use Python, but that choice is rather arbitrary. Indeed, one of the messages of this course is that all programming language are in some sense equivalent to one another, and hence we could have just as well used JavaScript, C, COBOL, Visual Basic or even BrainF*ck.

This is not a programming course, and it is absolutely fine if you are not familiar with Python and don’t follow the fine points of code examples such as the above. Still you might find it instructive to try to parse them, with the help of websites such as Google or Stackoverflow. In particular, the function int2bits above uses the fact that the binary representation of a number \(n\) is the list \((\floor{\tfrac{n}{2^i}} \mod 2)_{i=0,\ldots,\floor{\log_2 n}}\), which in Python-speak is written as [ floor(n / 2**i ) % 2 for i in range(floor(log(n,2))+1)].

Well defined representations. For a representation to be well defined, we need every natural number to be represented by some string, where two distinct numbers must have distinct representations. This corresponds to requiring the encoding function to be one-to-one, and the decoding function to be onto.

If you don’t remember the definitions of one-to-one, onto, total and partial functions, now would be an excellent time to review them. Make sure you understand why the function \(E\) described above is one-to-one, and the function \(D\) is onto.

It is natural for us to think of \(236\) as a the “actual” number, and of \(00110111\) as “merely” its representation. However, for most Europeans in the middle ages CCXXXVI would be the “actual” number and \(236\) (if they have heard about it) would be the weird Hindu-Arabic positional representation.While the Babylonians already invented a positional system much earlier, the decimal positional system we use today was invented by Indian mathematicians around the third century. It was taken up by Arab mathematicians in the 8th century. It was mainly introduced to Europe in the 1202 book “Liber Abaci” by Leonardo of Pisa, also known as Fibonacci, but did not displace Roman numerals in common usage until the 15th century. When our AI robot overlords materialize, they will probably think of \(00110111\) 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 over the generations. 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). Thus 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 \(236\)”. Wittgenstein argued that mathematical objects don’t exist at all, and the only thing that exists are the actual splotches on paper that make up \(236\), \(00110111\) or CCXXXVI and mathematics are just about formal manipulation of symbols that don’t have any inherent meaning. You can also think of the “actual” number as (somewhat recursively) “that thing which is common to \(236\), \(00110111\) and CCXXXVI and all other past and future representations meant to capture the same object”. (Some mathematicians would say that the actual number can be thought of as an equivalence class of these representations.)

In this course you are free to choose your own philosophy of mathematics, as long as you maintain the distinction between the mathematical objects themselves to the particular choice of representing them, whether as splotches of ink, pixels on a screen, zeroes and one, or in another format.

Representing (potentially negative) integers

Now that we can represent natural numbers, 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. So, the string \((\sigma,x_0,\ldots,x_{n-1}) \in \{0,1\}^{n+1}\) will represent the number \[ (-1)^\sigma \left[ x_0 2^0 + \cdots x_{n-1} 2^{n} \right] \]

The decoding function of a representation should always be onto, since every object must be represented by some string. However, it does not always have to be one to one. For example, in this particular representation the two strings \(1\) and \(0\) both represent the number zero (since they can be thought of as representing \(-0\) and \(+0\) respectively, can you see why?). We can also allow a partial decoding function for representations. For example, in the representation above there is no number that is represented by the empty string. But this is still a fine representation, since the decoding partial function is onto and the encoding function is the one-to-one total function \(E:\Z \rightarrow \{0,1\}^*\) which maps an integer of the form \(a\times k\), where \(a\in \{\pm 1 \}\) and \(k\in \N\) to the bit \((-1)^a\) concatenated with the binary representation of \(k\). That is, every integer can be represented as a string, and every two distinct integers have distinct representations.

Given a string \(x\in \{0,1\}^*\), how do we know if it’s “supposed” to represent a (nonnegative) natural number or a (potentially negative) integer? For that matter, even if we know \(x\) is “supposed” to be an integer, how do we know what representation scheme it uses? The short answer is that we don’t necessarily know this information, unless it is supplied from the context.In programming language, the compiler or interpreter determines the representation of the sequence of bits corresponding to a variable based on the variable’s type. We can treat the same string \(x\) as representing a natural number, an integer, a piece of text, an image, or a green gremlin. Whenever we say a sentence such as “let \(n\) be the number represented by the string \(x\)”, we will assume that we are fixing some canonical representation scheme such as the ones above. The choice of the particular representation scheme will almost never matter, except that we want to make sure to stick with the same one for consistency.

Representing rational numbers

We can represent a rational number of the form \(a/b\) by representing the two numbers \(a\) and \(b\) (again, this is not a unique representation but this is fine). However, simply concatenating the representations of \(a\) and \(b\) will not work.Recall that the concatenation of two strings \(x\) and \(y\) is the string of length \(|x|+|y|\) obtained by writing \(y\) after \(x\). For example, recall that we represent \(4\) as \((0,0,1)\) and \(35\) as \((1,1,0,0,0,1)\), but the concatenation \((0,0,1,1,1,0,0,0,1)\) of these strings is also the concatenation of the representation \((0,0,1,1)\) of \(12\) and the representation \((1,0,0,0,1)\) of \(17\). Hence, if we used such simple concatenation then we would not be able to tell if the string \((0,0,1,1,1,0,0,0,1)\) is supposed to represent \(4/35\) or \(12/17\).The above assumes we use the simple binary representation of natural numbers as strings. If we want to handle negative numbers then we should add the sign bit as well, though it would not make any qualitative difference to this discussion.

The way to tackle this is to find a general representation for pairs of numbers. If we were using a pen and paper, we would simply use a separator such as the symbol \(\|\) to represent, for example, the pair consisting of the numbers represented by \((0,1)\) and \((1,1,0,0,0,1)\) as the length-\(9\) string \(s\) “\(01\|110001\)”. This is just like people add spaces and punctuation to separate words in English. By adding a little redundancy, we can do just that in the digital domain. The idea is that we will map the three element set \(\Sigma = \{0,1,\|\}\) to the four element set \(\{0,1\}^2\) via the one-to-one map that takes \(0\) to \(00\), \(1\) to \(11\) and \(\|\) to \(01\).

Consider the rational number \(r=19/236\). In our convention, we represent \(19\) as the string \(11001\) and \(236\) as the string \(00110111\), and so we could rerpresent \(r\) as the pair of strings \((11001,00110111)\). We can then represent this pair as the length \(14\) string \(11001\|00110111\) over the alphabet \(\{0,1,\|\}\). Now, applying the map \(0 \mapsto 00\), \(1\mapsto 11\), \(\| \mapsto 01\), we can represent the latter string as the length \(28\) string \(s=1111000011010000111100111111\) over the alphabet \(\{0,1\}\). So we represent the rational number \(r=19/36\) be the binary string \(s=1111000011010000111100111111\).

More generally, we obtained a representation of the non-negative rational numbers as binary strings by composing the following representations:

  1. Representing a non-negative rational number as a pair of natural numbers.
  2. Representing a natural number by a string via the binary representation. (We can use the representation of integers to handle rational numbers that can be negative. )
  3. Combining 1 and 2 to obtain 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 \(\{0,1\}\).

More generally, the above encoding yields a one-to-one map \(E\) from strings over the alphabet \(\Sigma\) to binary strings, such that for every \(s\in \Sigma^*\), \(|E(s)|=2|s|\). Using this, we get a one-to-one map \(E': (\{0,1\}^*)\times (\{0,1\}^*) \rightarrow \{0,1\}^*\) mapping pairs of binary strings into a single binary string. Given every pair \((a,b)\) of binary strings, we will first map it in a one-to-one way to a string \(s \in \Sigma^*\) using \(\|\) as a separator, and then map \(s\) to a single (longer) binary string using the encoding \(E\). The same idea can be used to represent triples, quadruples, and generally all tuples of strings as a single string (can you see why?).

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 so up to a small error 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 \(22/7\) with an error of about \(10^{-3}\) and if we wanted smaller error (e.g., about \(10^{-4}\)) then we can use \(311/99\) and so on and so forth.

This is a fine representation though a more common choice to represent real numbers is the floating point representation, where we represent \(x\) by the pair \((a,b)\) of (positive or negative) integers of some prescribed sizes (determined by the desired accuracy) such that \(a \times 2^{b}\) is closest to \(x\).You can think of this as related to scientific notation. In scientific notation we represent a number \(y\) as \(a \times 10^b\) for integers \(a,b\). Sometimes we write this as \(y=a \mathtt{E} b\). For example, in many programming languages 1.21E2 is the same as 121.0. In scientific notation, to represent \(\pi\) up to accuracy \(10^{-3}\) we will simply use \(3141 \times 10^{-3}\) and to represent it up to accuracy \(10^{-4}\) we will use \(31415 \times 10^{-4}\). The reader might be (rightly) worried about this issue of approximation. 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. This representation is called “floating point” because we can think of the number \(a\) as specifying a sequence of binary digits, and \(b\) 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. A floating point error has been implicated in the explosion of the Ariane 5 rocket, a bug that cost more than 370 million dollars, and the failure of a U.S. Patriot missile to intercept an Iraqi Scud missile, costing 28 lives. Floating point is often problematic in financial applications as well.

Can we represent reals exactly?

Given the issues with floating point representation, we could ask whether we could represent real numbers exactly as strings. Unfortunately, the following theorem says this cannot be done

There is no one-to-one function \(RtS:\R \rightarrow \{0,1\}^*\).\(RtS\) stands for “reals to strings”.

cantorthm was proven by Georg Cantor in 1874.Cantor used the set \(\N\) rather than \(\{0,1\}^*\), but one can show that these two result are equivalent using the one-to-one maps between those two sets, see naturalsstringsmapex. Saying that there is no one-to-one map from \(\R\) to \(\N\) is equivalent to saying that there is no onto map \(NtR:\N \rightarrow \R\) or, in other words, that there is no way to “count” all the real numbers as \(NtR(0),NtR(1),NtR(2),\ldots\). For this reason cantorthm is known as the uncountability of the reals. The 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 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 we will see later in this course, Cantor’s ideas also play a huge role in the theory of computation.

Now that we discussed the theorem’s importance, let us see the proof. cantorthm follows from the following two results:

Let \(\{0,1\}^\infty\) be the set \(\{ f \;|\; f:\N \rightarrow \{0,1\} \}\) of functions from \(\N\) to \(\{0,1\}\).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 \((f(0),f(1),f(2),\ldots )\). Then there is no one-to-one map \(FtS:\{0,1\}^\infty \rightarrow \{0,1\}^*\).\(FtS\) stands for “functions to strings”.

There does exist a one-to-one map \(FtR:\{0,1\}^\infty \rightarrow \R\).\(FtR\) stands for “functions to reals.”

sequencestostrings and sequencestoreals together imply cantorthm. To see why, suppose, for the sake of contradiction, that there did exist a one-to-one function \(RtS:\R \rightarrow \{0,1\}^*\). By sequencestoreals, 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 onetoonecompex), the function \(FtS:\{0,1\}^\infty \rightarrow \{0,1\}^*\) defined as \(FtS(f)=RtS(FtR(f))\) will be one to one, contradicting sequencestostrings. See proofofcantorfig for a graphical illustration of this argument.

Figure 3: We prove cantorthm by combining sequencestostrings and sequencestoreals. sequencestoreals, which uses standard calculus tools, shows the existence of a 1 to 1 map \(FtR\) from the set \(\{0,1\}^\infty\) to the real numbers. So, if a hypothetical 1 to 1 map \(RtS:\R \rightarrow \{0,1\}^*\) existed, then we could compose them to get a 1 to 1 map \(FtS:\{0,1\}^\infty \rightarrow \{0,1\}^*\). Yet this contradicts sequencestostrings- the heart of the proof- which rules out the existence of such a map.

Now all that is left is to prove these two lemmas. We start by proving sequencestostrings which is really the heart of cantorthm.

Let us assume, for the sake of contradiction, that there exists a one-to-one function \(FtS:\{0,1\}^\infty \rightarrow \{0,1\}^*\). Then, there is an onto function \(StF:\{0,1\}^* \rightarrow \{0,1\}^\infty\) (e.g., see onetooneimpliesonto). We will derive a contradiction by coming up with some function \(f^* : \N \rightarrow \{0,1\}\) such that \(f^* \neq StF(x)\) for every \(x\in \{0,1\}^*\).

The argument for this is short but subtle. We need to construct some function \(f^*:\N \rightarrow \{0,1\}\) such that for every \(x\in \{0,1\}^*\), if we let \(g=StF(x)\) then \(g \neq f^*\). Since two functions are identical if and only if they agree on every input, to do this we need to show that there is some \(n\in \N\) such that \(f^*(n) \neq g(n)\). (All these quantifiers can be confusing, so let’s again recap where we are and where we want to get to. We assumed by contradiction there is a one-to-one \(FtS\) and hence an onto \(StF\). To get our desired contradiction we need to show the existence of a single \(f^*\) such that for every \(x\in \{0,1\}^*\) there exists \(n\in \N\) on which \(f^*\) and \(g=StF(x)\) disagree.)

The idea is to construct \(f^*\) iteratively: for every \(x\in \{0,1\}^*\) we will “ruin” \(f^*\) in one input \(n(x)\in \N\) to ensure that \(f^*(n(x)) \neq g(n(x))\) where \(g=StF(x)\). If we are successful then this would ensure that \(f^* \neq StF(x)\) for every \(x\). Specifically, for every \(x\in \{0,1\}^*\), let \(n(x) \in N\) be the number \(x_0 + 2x_1 + 4x_2 + \cdots +2^{k-1}x_{k-1} + 2^{k}\) where \(k=|x|\). That is, \(n(x) = 2^k + \sum_{i=0}^{k-1}2^i x_i\). If \(x\neq x'\) then \(n(x) \neq n(x')\) (we leave verifying this as an exercise to you, the reader).

Now for every \(x\in \{0,1\}^*\), we define \[ f^*(n(x)) = 1 - g(n(x)) \label{eqcantordiagreals} \] where \(g=StF(x)\). For every \(n\) that is not of the form \(n=n(x)\) for some \(x\), we set \(f^*(n)=0\). \eqref{eqcantordiagreals} is well defined since the map \(x \mapsto n(x)\) is one-to-one and hence we will not try to give \(f^*(n)\) two different values.

Now by \eqref{eqcantordiagreals}, for every \(x\in \{0,1\}^*\), if \(g=StF(x)\) and \(n=n(x)\) then \(f^*(n) = 1 - g(n) \neq g(n)\). Hence \(StF(x) \neq f^*\) for every \(x\in \{0,1\}^*\), contradicting the assumption that \(StF\) is onto. This proof is known as the “diagonal” argument, as the construction of \(f^*\) can be thought of as going over the diagonal elements of a table that in the \(n\)-th row and \(m\)-column contains \(StF(x)(m)\) where \(x\) is the string such that \(n(x)=n\), see diagrealsfig.

Figure 4: We construct a function \(f^*\) such that \(f^* \neq StF(x)\) for every \(x\in \{0,1\}^*\) by ensuring that \(f^*(n(x)) \neq StF(x)(n(x))\) for every \(x\in \{0,1\}^*\). We can think of this as building a table where the columns correspond to numbers \(m\in \N\) and the rows correspond to \(x\in \{0,1\}^*\) (sorted according to \(n(x)\)). If the entry in the \(x\)-th row and the \(m\)-th column corresponds to \(g(m))\) where \(g=StF(x)\) then \(f^*\) is obtained by going over the “diagonal” elements in this table (the entries corresponding to the \(x\)-th row and \(n(x)\)-th column) and enduring that \(f^*(x)(n(x)) \neq StF(x)(n(x))\).

sequencestostrings doesn’t really have much to do with the natural numbers or the strings. An examination of the proof shows that it really shows that for every set \(S\), there is no one-to-one map \(F:\{0,1\}^S \rightarrow S\) where \(\{0,1\}^S\) denotes the set \(\{ f \;|\; f:S \rightarrow \{0,1\} \}\) of all Boolean functions with domain \(S\). Since we can identify a subset \(V \subseteq S\) with its characteristic function \(f=1_V\) (i.e., \(1_V(x)=1\) iff \(x\in V\)), we can think of \(\{0,1\}^S\) also as the set of all subsets of \(S\). This subset is sometimes called the power set of \(S\). The proof of sequencestostrings can be generalized to show that there is no one-to-one map between a set and its power set. In particular, it means that the set \(\{0,1\}^\R\) is “even bigger” than \(\R\). Cantor used these ideas to construct an infinite hierarchy of shades of infinity. The number of such shades turn out to be much larger than \(|\N|\) or even \(|\R|\). He denoted the cardinality of \(\N\) by \(\aleph_0\), where \(\aleph\) is the first letter in the Hebrew alphabet, and called the the next largest infinite number by \(\aleph_1\). Cantor also made the continuum hypothesis that \(|\R|=\aleph_1\). We will come back to the very interesting story of this hypothesis later on in this course. This lecture of Aaronson mentions some of these issues (see also this Berkeley CS 70 lecture).

To complete the proof of cantorthm, we need to show sequencestoreals. This requires some calculus background, but is otherwise straightforward. The idea is that we can construct a one-to-one map from \(\{0,1\}^\infty\) to the real numbers by mapping the function \(f:\N \rightarrow \{0,1\}\) to the number that has the infinite decimal expansion \(f(0).f(1)f(2)f(3)f(4)f(5)\ldots\) (i.e., the number between \(0\) and \(2\) that is \(\sum_{i=0}^\infty f(i)10^{-i}\)). We will now do this more formally. If you have not had much experience with limits of real series before, then the formal proof might be a little hard to follow. This part is not the core of Cantor’s argument, nor are such limits very crucial to this course, so feel free to also just take sequencestoreals on faith and skip the formal proof.

For every \(f\in \{0,1\}^\infty\) and \(n\in \N\), we define \(S(f)_n = \sum_{i=0}^n f(i)10^{-i}\). It is a known result (that we won’t repeat here) that for every \(f:\N \rightarrow \{0,1\}\), the sequence \(( S(f)_n )_{n=0}^\infty\) has a limit. That is, for every \(f\) there exists some value \(x(f)\) (often denoted as \(\sum_{i=0}^\infty f(i)10^{-i}\)) such that for every \(\epsilon>0\), if \(n\) is sufficiently large then \(|S_f(n)-x(f)| < \epsilon\). We define \(FtR(f)\) to be this value \(x(f)\). In other words, we define \[ FtR(f) = \sum_{i=0}^\infty f(i)10^{-i} \] which will be a number between \(0\) and \(2\).

To show that \(FtR\) is one to one, we need to show that \(FtR(f) \neq FtR(g)\) for every distinct \(f,g:\N \rightarrow \{0,1\}\). Let \(f \neq g\) be such functions, and let \(k\) be the smallest number for which \(f(k) \neq g(k)\). We will show that \(|FtR(f)-FtR(g)| > 0.5\cdot 10^{-k}\). This will complete the proof since in particular it implies \(FtR(f) \neq FtR(g)\).

Assume without loss of generality that \(f(k)=0\) and \(g(k)=1\) (otherwise switch the roles of \(f\) and \(g\)). Define \(S = \sum_{i=0}^{k-1} 10^{-i}f(i) = \sum_{i=0}^{k-1} 10^{-i}f(i)\) (the equality holds since \(f\) and \(g\) agree up to \(k\)). Now, since \(g(k)=1\), \[FtR(g) = \sum_{i=0}^\infty g(i)10^{-i} \geq \sum_{i=0}^k g(i)10^{-i} = S + 10^{-k} \;.\] On the other hand, since \(f(k)=0\) and \(f(k+1+j) \leq 1\) for every \(j\geq 0\), \[FtR(f) = \sum_{i=0}^\infty f(i)10^{-i} = S + \sum_{i=k+1}^\infty f(i) 10^{-i} \leq S + 10^{-(k-1)}\sum_{j=0}^\infty 10^{-j} \;.\] Now \(\sum_{j=0}^\infty 10^{-j}\) is simply the number \(1.11111\ldots = 11/9\), and hence we get that \(FtR(f) \leq S + 11/9 \cdot 10^{-k-1}\) while \(FtR(g) \geq S + 10^{-k}\) which means the difference between them is larger than \(0.5 \cdot 10^{-k}\).

Beyond numbers

We can of course represent objects other than numbers as binary strings. Let us give a general definition for representation:

Let \(\mathcal{O}\) be some set. A representation scheme for \(\mathcal{O}\) consists of a pair \((E,D)\) where \(E:\mathcal{O} \rightarrow \{0,1\}^*\) is a total one-to-one function, \(D:\{0,1\}^* \rightarrow_p \mathcal{O}\) is a (possibly partial) function, and such that \(D\) and \(E\) satisfy that \(D(E(o))=o\) for every \(o\in \mathcal{O}\). \(E\) is known as the encoding function and \(D\) is known as the decoding function.

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:

Suppose that \(E: \mathcal{O} \rightarrow \{0,1\}^*\) is one-to-one. Then there exists a function \(D:\{0,1\}^* \rightarrow \mathcal{O}\) such that \(D(E(o))=o\) for every \(o\in \mathcal{O}\).

Let \(o_0\) be some arbitrary element of \(O\). For every \(x \in \{0,1\}^*\), there exists either zero or a single \(o\in \mathcal{O}\) such that \(E(o)=x\) (otherwise \(E\) would not be one-to-one). We will define \(D(x)\) to equal \(o_0\) in the first case and this single object \(o\) in the second case. By definition \(D(E(o))=o\) for every \(o\in \mathcal{O}\).

Note that, while in general we allowed the decoding function to be partial. This proof shows that we can always obtain a total decoding function if we need to. This observation can sometimes be useful.

Finite representations

If \(\mathcal{O}\) is finite, then we can represent every object in \(o\) as a string of length at most some number \(n\). What is the value of \(n\)? Let us denote the set \(\{ x\in \{0,1\}^* : |x| \leq n \}\) of strings of length at most \(n\) by \(\{0,1\}^{\leq n}\). To obtain a representation of objects in \(\mathcal{O}\) as strings in \(\{0,1\}^{\leq n}\) we need to come up with a one-to-one function from the former set to the latter. We can do so, if and only if \(|\mathcal{O}| \leq 2^{n+1}-1\) as is implied by the following lemma:

For every two finite sets \(S,T\), there exists a one-to-one \(E:S \rightarrow T\) if and only if \(|S| \leq |T|\).

Let \(k=|S|\) and \(m=|T|\) and so write the elements of \(S\) and \(T\) as \(S = \{ s_0 , s_1, \ldots, s_{k-1} \}\) and \(T= \{ t_0 , t_1, \ldots, t_{m-1} \}\). We need to show that there is a one-to-one function \(E: S \rightarrow T\) iff \(k \leq m\). For the “if” direction, if \(k \leq m\) we can simply define \(E(s_i)=t_i\) for every \(i\in [k]\). Clearly for \(i \neq j\), \(t_i = E(s_i) \neq E(s_j) = t_j\), and hence this function is one-to-one. In the other direction, suppose that \(k>m\) and \(E: S \rightarrow T\) is some function. Then \(E\) cannot be one-to-one. Indeed, for \(i=0,1,\ldots,m-1\) let us “mark” the element \(t_j=E(s_i)\) in \(T\). If \(t_j\) was marked before, then we have found two objects in \(S\) mapping to the same element \(t_j\). Otherwise, since \(T\) has \(m\) elements, when we get to \(i=m-1\) we mark all the objects in \(T\). Hence, in this case \(E(s_m)\) must map to an element that was already marked before.This direction is sometimes known as the “Pigeon Hole Principle”: the principle that if you have a pigeon coop with \(m\) holes, and \(k>m\) pigeons, then there must be two pigeons in the same hole.

Now the size of \(\{0,1\}^n\) is \(2^n\), and the size of \(\{0,1\}^{\leq n}\) is only slightly bigger: \(2^0 + 2^1 + \ldots + 2^n = 2^{n+1}-1\) by the formula for a geometric series.

Prefix-free encoding

In our discussion of the representation of rational numbers, we used the “hack” of encoding the alphabet \(\{ 0,1, \|\}\) to represent tuples of strings as a single string. This turns out to be a special case of the general paradigm of prefix-free encoding. An encoding function \(E:\mathcal{O} \rightarrow \{0,1\}^*\) is prefix-free if there are no two objects \(o \neq o'\) such that the representation \(E(o)\) is a prefix of the representation \(E(o')\). The definition of prefix is as you would expect: a length \(n\) string \(x\) is a prefix of a length \(n' \geq n\) string \(x'\) if \(x_i=x'_i\) for every \(1 \leq i \leq n\). Given a representation scheme for \(\mathcal{O}\) with a prefix-free encoding map, we can use simple concatenation to encode tuples of objects in \(\mathcal{O}\):

Suppose that \((E,D)\) is a representation scheme for \(\mathcal{O}\) and \(E\) is prefix free. Then there exists a representation scheme \((E',D')\) for \(\mathcal{O}^*\) such that for every \((o_0,\ldots,o_{k-1}) \in \mathcal{O}^*\), \(E'(o_0,\ldots,o_{k-1}) = E(o_0)E(o_1)\cdots E(o_{k-1})\).

prefixfreethm is one of those statements that are a little hard to parse, but in fact are fairly straightforward to prove once you understand what they mean. Thus I highly recommend that you pause here, make sure you understand statement of the theorem, and try to prove it yourself before proceeding further.

Figure 5: If we have a prefix-free representation of each object then we can concatenate the representations of \(k\) objects to obtain a representation for the tuple \((o_1,\ldots,o_k)\).

The idea behind the proof is simple. Suppose that for example we want to decode a triple \((o_0,o_1,o_2)\) from its representation \(x= E'(o_0,o_1,o_2)=E(o_0)E(o_1)E(o_2)\). We will do so by first finding the first prefix \(x_0\) of \(x\) such is a representation of some object. Then we will decode this object, remove \(x_0\) from \(x\) to obtain a new string \(x'\), and continue onwards to find the first prefix \(x_1\) of \(x'\) and so on and so forth (see prefix-free-tuples-ex). The prefix-freeness property of \(E\) will ensure that \(x_0\) will in fact be \(E(o_0)\), \(x_1\) will be \(E(o_1)\) etc.

We now show the formal proof. By decodelem, to prove the theorem it suffices to show that \(E'\) is one-to-one. Suppose, towards the sake of contradiction that there exist two distinct tuples \((o_0,\ldots,o_{k-1})\) and \((o'_0,\ldots,o'_{k'-1})\) such that

\[ E'(o_0,\ldots,o_{k-1})=E'(o'_0,\ldots,o'_{k'-1}) \;. \label{prefixfreeassump} \]

We denote \(x_i = E(o_i)\) and \(x'_i = E(o'_i)\). By our assumption and the definition of \(E'\), \(x_0x_1 \cdots x_{k-1} = x'_0 x'_1 \cdots x'_{k'-1}\). Without loss of generality we can assume \(k' \leq k\). Let \(i\) be the larget number such that \(x_j = x'_j\) for all \(j<i\). (If \(x'_0 \neq x_0\) then \(i=0\); if \(x_j = x'_j\) for all \(j<k\) then we let \(i=k\); note that the fact that the concatenation of \(x_0,\ldots,x_{k-1}\) is equal to the concatenation of \(x'_0,\ldots,x'_{k'-1}\) does not mean the individual components have to agree.)

Since \(x_j=x'_j\) for all \(j<i\), the strings \(x_{i}\cdots x_{k-1}\) and \(x'_i \cdots x'_{k'-1}\) are identical, and we denote this string by \(s\). If \(i<k\) then both \(x_i\) and \(x'_i\) are prefixes of \(s\) which means that one of them is a prefix of the other, since by the choice of \(i\), \(x_i \neq x'_i\) we get that both of them are valid representation of distinct objects which contradicts prefix-freeness. If \(i=k\) then the string \(s\) must be empty, but this would mean that \(i=k'\) as well, which means that \(x_i=x'_i\) for all \(i\), which means that the original tuples of objects must have been the same.

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:

Let \(E:\mathcal{O} \rightarrow \{0,1\}^*\) be a one-to-one function. Then there is a one-to-one prefix-free encoding \(\overline{E}\) such that \(|\overline{E}(o)| \leq 2|E(o)|+2\) for every \(o\in \mathcal{O}\).

For the sake of completeness, we will include the proof below, but it is a good idea for you to pause here and try to prove it yourself, using the same technique we used for representing rational numbers.

Define the function \(PF:\{0,1\}^* \rightarrow \{0,1\}^*\) as follows \(PF(x)=x_0 x_0 x_1 x_1 \ldots x_{n-1}x_{n-1}01\) for every \(x\in \{0,1\}^*\). If \(E:\mathcal{O} \rightarrow \{0,1\}^*\) is the (potentially not prefix-free) representation for \(\mathcal{O}\), then we transform it into a prefix-free representation \(\overline{E}:\mathcal{O} \rightarrow \{0,1\}^*\) by defining \(\overline{E}(o)=PF(E(o))\).

To prove the lemma we need to show that (1) \(\overline{E}\) is one-to-one and (2) \(\overline{E}\) is prefix-free. In fact (2) implies (1), since if \(\overline{E}(o)\) is never a prefix of \(\overline{E}(o')\) for every \(o \neq o'\) then in particular \(\overline{E}\) is one-to-one. Now suppose, toward a contradiction, that there are \(o \neq o'\) in \(\mathcal{O}\) such that \(\overline{E}(o)\) is a prefix of \(\overline{E}(o')\). (That is, if \(y=\overline{E}(o)\) and \(y'=\overline{E}(o')\), then \(y_j = y'_j\) for every \(j<|y|\).)

Define \(x = E(o)\) and \(x'=E(o')\). Note that since \(E\) is one-to-one, \(x \neq x'\). (Recall that two strings \(x,x'\) are distinct if they either differ in length or have at least one distinct coordinate.) Under our assumption, \(|PF(x)| \leq |PF(x')|\), and since by construction \(|PF(x)|=2|x|+2\), it follows that \(|x| \leq |x'|\). If \(|x|=|x'|\) then, since \(x \neq x'\), there must be a coordinate \(i\in \{0,\ldots, |x|-1\}\) such that \(x_i \neq x'_i\). But since \(PF(x)_{2i}=x_i\), we get that \(PF(x)_{2i} \neq PF(x')_{2i}\) and hence \(\overline{E}(o)=PF(x)\) is not a prefix of \(\overline{E}(o')=PF(x')\). Otherwise (if \(|x| \neq |x'|\)) then it must be that \(|x| < |x'|\), and hence if \(n=|x|\), then \(PF(x)_{2n}=0\) and \(PF(x)_{2n+1}=1\). But since \(n<|x'|\), \(PF(x')_{2n},PF(x')_{2n+1}\) is equal to either \(00\) or \(11\), and in any case we get that \(\overline{E}(o)=PF(x)\) is not a prefix of \(\overline{E}(o')=PF(x')\).

In fact, we can even obtain a more efficient transformation where \(|E'(o)| \leq |o| + O(\log |o|)\). We leave proving this as an exercise (see prefix-free-ex).

“Proof by Python” (optional)

The proofs of prefixfreethm and predixfreeransformation 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 a encoding and decoding functions that are prefix free;
  • 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 int2bits and bits2int functions we defined above.

# 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 L[-2:]==[0,1] return pfencode, pfdecode, pfvalid pfint2bits, pfbits2int , pfvalidint = prefixfree(int2bits,bits2int) print(int2bits(23)) # [1, 1, 1, 0, 1] print(pfint2bits(23)) # [1, 1, 1, 1, 1, 1, 0, 0, 0, 1] print(pfbits2int(pfint2bits(23))) # 23 print(pfvalidint(pfint2bits(23))) # true print(pfvalidint([1,1,1,1])) #false

Note that Python function prefixfree above takes two Python functions as input and outputs three Python functions as output.When it’s not too awkward, I use the term “Python function” or “subroutine” to distinguish between such snippets of python programs and mathematical functions. However, in comments in python source I use “functions” to denote python functions, just as I use “integers” to denote python int objects. You don’t have to know Python in this course, but you do need to get comfortable with the idea of functions as mathematical objects in their own right, that can be used as inputs and outputs of other functions.

# Takes functions pfencode, pfdecode and pfvalid, # and returns functions encodelists, decodelists # that can encode and decode # lists of the objects respectively def represlists(pfencode,pfdecode,pfvalid): def encodelist(L): """Gets list of objects, encodes it as list of bits""" return [bit for obj in L for bit in pfencode(obj)] 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 intlist2bits, bits2intlist = represlists(pfint2bits,pfbits2int,pfvalidint) print(intlist2bits([12,8])) # [0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1] print(bits2intlist(intlist2bits([12,8]))) # [12, 8]

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 simply concatenating the representation of each symbol. One such representation is the ASCII that represents \(128\) letters and symbols as strings of \(7\) bits. Since it is a fixed-length representation it is automatically prefix-free (can you see why?). Unicode is a representation of (at the time of this writing) about 128,000 symbols into 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 \(32\).

We can use programming languages to probe how our computing environment represents various values. This is easier to do in “unsafe” programming languages such as C that allow direct access to the memory.

Using a simple C program we have produced the following representations of various values. One can see that for integers, multiplying by 2 corresponds to a “left shift” inside each byte. In contrast, for floating point numbers, multiplying by two corresponds to adding one to the exponent part of the representation. A negative number is represented using the two’s complement approach. Strings are represented in a prefix-free form by ensuring that a zero byte is at their end.

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

If you are curious, the code for this program (which you can run here) is the following:

#include <stdio.h> #include <stdlib.h> #include <string.h> char *bytes(void *p,int n){ int i; int j; char *a = (char *) p; char *s = malloc(9*n+2); s[9*n] = '\n'; s[9*n+1] = 0; j = 0; for(i=0;i< n*8;i++){ s[j] = a[i/8] & (128 >> (i % 8)) ? '1' : '0'; if (i% 8 == 7) { s[++j] = ' '; } ++j; } return s; } void printint(int a) { printf("%-8s %-5d: %s", "int", a, bytes(&a,sizeof(int))); } void printlong(long a) { printf("%-8s %-5d: %s", "long", a, bytes(&a,sizeof(long))); } void printstring(char *s) { printf("%-8s %-5s: %s", "string", s, bytes(s,strlen(s)+1)); } void printfloat(float f) { printf("%-8s %-5.1f: %s", "float", f, bytes(&f,sizeof(float))); } void printdouble(double f) { printf("%-8s %-5.1f: %s", "double", f, bytes(&f,sizeof(double))); } int main(void) { printint(2); printint(4); printint(513); printlong(513); printint(-1); printint(-2); printstring("Hello"); printstring("abcd"); printfloat(33); printfloat(66); printfloat(132); printdouble(132); return 0; }

Representing vectors, matrices, images

Once we can represent numbers, and lists of numbers, then we can obviously 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 basic colors since (most) humans only have three types of cones in their retinas. We would have needed 16 basic 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 course.

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 \(G=(V,E)\) a 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 important for some applications, though for us would typically be immaterial.

Figure 6: Representing the graph \(G=(\{0,1,2,3,4\},\{ (1,0),(4,0),(1,4),(4,1),(2,1),(3,2),(4,3) \})\) in the adjacency matrix and adjacency list representations.

Once again, we can also define these encoding and decoding functions in python:

from graphviz import Graph # get n by n matrix (as list of n lists) # return graph corresponding to it def matrix2graph(M): G = Graph(); n = len(M) for i in range(n): G.node(str(i)) # add vertex i for j in range(n): G.node(str(j)) if M[i][j]: G.edge(str(i),str(j)) # if M[i][j] is nonzero then add edge between i and j return G matrix2graph([[0,1,0],[0,0,1],[1,0,0]])


Representing lists

If we have a way of represent 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,[1011,00111]]" 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 allows to represent (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 \(x\) is an integer, we might make statements such as “\(F(x)+1\) is prime” to mean that if we represent \(x\) as a string \(\underline{x}\) and let \(\underline{y}=F(\underline{x})\), then the integer \(y\) represented by the string \(\underline{y}\) satisfies that \(y+1\) 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 order pair \((x,y)\). We will 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.

Defining computational tasks

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.

Figure 7: A computational process

In future chapters, we will turn to mathematically defining computational process, but, as we discussed above for now we want to focus on computational tasks; i.e., 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. But for most of this course, 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’ve seen that there is more than one way to solve this computational task, and in fact, we still don’t know the best algorithm for this problem.
  • Given (a representation of) an integer \(z\), 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 longer.
  • 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 huge 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 corresponds to computing a partial function from \(\{0,1\}^*\) to \(\{0,1\}\); though it is easy to make it into a total function by mapping every string into the trivial Python program that stops without doing anything. 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 correspond to computing some (partial) function from \(\{0,1\}^*\) to \(\{0,1\}\).

An important special case of computational tasks corresponds to computing Boolean functions, whose output is a single bit \(\{0,1\}\). Computing such functions corresponds to answering a YES/NO question, and hence this task is also known as a decision problem. Given any function \(F:\{0,1\}^* \rightarrow \{0,1\}\) and \(x\in \{0,1\}^*\), the task of computing \(F(x)\) corresponds to the task of deciding whether or not \(x\in L\) where \(L = \{ x : F(x)=1 \}\) is known as the language that corresponds to the function \(F\).The language terminology is due to historical connections between the theory of computation and formal linguistics as developed by Noam Chomsky. Hence many texts refer to such as computational task as deciding a language.

Figure 8: A subset \(L \subseteq \{0,1\}^*\) can be identified with the function \(F:\{0,1\}^* \rightarrow \{0,1\}\) such that \(F(x)=1\) if \(x\in L\) and \(F(x)=0\) if \(x\not\in L\). Functions with a single bit of output are called Boolean functions, while subsets of strings are called languages. The above shows that the two are essentially the same object, and we can identify the task of deciding membership in \(L\) (known as deciding a language in the literature) with the task of computing the function \(F\).

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 can’t 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’ll do in compchap.

Distinguish functions from programs

You should always watch out for potential confusions between specification and implementation or equivalently between mathematical functions and algorithms/programs. It does not help that programming languages (my favorite 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 \(MULT:\N\times \N \rightarrow \N\) 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 = int2bits(x) b = int2bits(y) res = [0]*(len(a)+len(b)) for i in range(len(a)): for j in range(len(b)): res[i+j] += a[i]*b[j] return bits2int(res) # use a bit of a cheat that bits2int can handle trailing zeroes # as well as lists with elements in 0,1,2 instead of 0,1 print(mult1(12,7)) # 84 print(mult2(12,7)) # 84

Both mult1 and mult2 produce the same output given the same pair of 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 functionornotfig).

Figure 9: A function is a mapping of inputs to outputs. A program is a set of instructions of how to obtain an output given an input. A program computes a function, but it is not the same as a function, popular programming language terminology notwithstanding.

Advanced note: beyond computing functions

Functions capture quite a lot of computational tasks, but one can consider more general settings as well. For starters, we can and will talk about partial functions, that are not defined on all inputs. When computing a partial function, we only need to worry about the inputs on which the function is defined. Another way to say it is that we can design an algorithm for a partial function \(F\) under the assumption that someone “promised” us that all inputs \(x\) would be such that \(F(x)\) is defined (as otherwise we don’t care about the result). Hence such tasks are also known as promise problems.

Another generalization is to consider relations that may have more than one possible admissible output. For example, consider the task of finding any solution for a given set of equations. A relation \(R\) maps a string \(x\in \{0,1\}^*\) into a set of strings \(R(x)\) (for example, \(x\) might describe a set of equations, in which case \(R(x)\) would correspond to the set of all solutions to \(x\)). We can also identify a relation \(R\) with the set of pairs of strings \((x,y)\) where \(y\in R(x)\). A computational process solves a relation if for every \(x\in \{0,1\}^*\), it outputs some string \(y\in R(x)\).

Later on in this course we will consider even more general tasks, including interactive tasks, such as finding good strategy in a game, tasks defined using probabilistic notions, and others. However, for much of this course we will focus on the task of computing a function, and often even a Boolean function, that has only a single bit of output. It turns out that a great deal of the theory of computation can be studied in the context of this task, and the insights learned are applicable in the more general settings.

  • We can represent essentially every object we want to compute on using binary strings.
  • A representation scheme for a set of objects \(\mathcal{O}\) is a one-to-one map from \(\mathcal{O}\) to \(\{0,1\}^*\).
  • A basic computational task is the task of computing a function \(F:\{0,1\}^* \rightarrow \{0,1\}^*\). This encompasses not just arithmetical computations such as multiplication, factoring, etc. but a great many other tasks arising in areas as diverse as scientific computing, artificial intelligence, image processing, data mining and many many more.
  • We will study the question of finding (or at least giving bounds on) what is the best algorithm for computing \(F\) for various interesting functions \(F\).

Exercises

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.

Which one of these objects can be represented by a binary string?

  1. An integer \(x\)
  2. An undirected graph \(G\).
  3. A directed graph \(H\)
  4. All of the above.

Recall that the grade-school algorithm for multiplying two numbers requires \(O(n^2)\) operations. Suppose that instead of using decimal representation, we use one of the following representations \(R(x)\) to represent a number \(x\) between \(0\) and \(10^n-1\). For which one of these representations you can still multiply the numbers in \(O(n^2)\) operations?

  1. The standard binary representation: \(B(x)=(x_0,\ldots,x_{k})\) where \(x = \sum_{i=0}^{k} x_i 2^i\) and \(k\) is the largest number s.t. \(x \geq 2^k\).
  2. The reverse binary representation: \(B(x) = (x_{k},\ldots,x_0)\) where \(x_i\) is defined as above for \(i=0,\ldots,k-1\).
  3. Binary coded decimal representation: \(B(x)=(y_0,\ldots,y_{n-1})\) where \(y_i \in \{0,1\}^4\) represents the \(i^{th}\) decimal digit of \(x\) mapping \(0\) to \(0000\), \(1\) to \(0001\), \(2\) to \(0010\), etc. (i.e. \(9\) maps to \(1001\))
  4. All of the above.

Suppose that \(R:\N \rightarrow \{0,1\}^*\) corresponds to representing a number \(x\) as a string of \(x\) \(1\)’s, (e.g., \(R(4)=1111\), \(R(7)=1111111\), etc.). If \(x,y\) are numbers between \(0\) and \(10^n -1\), can we still multiply \(x\) and \(y\) using \(O(n^2)\) operations if we are given them in the representation \(R(\cdot)\)?

Recall that if \(F\) is a one-to-one and onto function mapping elements of a finite set \(U\) into a finite set \(V\) then the sizes of \(U\) and \(V\) are the same. Let \(B:\N\rightarrow\{0,1\}^*\) be the function such that for every \(x\in\N\), \(B(x)\) is the binary representation of \(x\).

  1. Prove that \(x < 2^k\) if and only if \(|B(x)| \leq k\).
  2. Use a. to compute the size of the set \(\{ y \in \{0,1\}^* : |y| \leq k \}\) where \(|y|\) denotes the length of the string \(y\).
  3. Use a. and b. to prove that \(2^k-1 = 1 + 2 + 4+ \cdots + 2^{k-1}\).

Suppose that \(F:\N\rightarrow\{0,1\}^*\) is a one-to-one function that is prefix-free in the sense that there is no \(a\neq b\) s.t. \(F(a)\) is a prefix of \(F(b)\).

  1. Prove that \(F_2:\N\times \N \rightarrow \{0,1\}^*\), defined as \(F_2(a,b) = F(a)F(b)\) (i.e., the concatenation of \(F(a)\) and \(F(b)\)) is a one-to-one function.
  2. Prove that \(F_*:\N^*\rightarrow\{0,1\}^*\) defined as \(F_*(a_1,\ldots,a_k) = F(a_1)\cdots F(a_k)\) is a one-to-one function, where \(\N^*\) denotes the set of all finite-length lists of natural numbers.

Suppose that \(F:O\rightarrow\{0,1\}^*\) is some (not necessarily prefix-free) representation of the objects in the set \(O\), and \(G:\N\rightarrow\{0,1\}^*\) is a prefix-free representation of the natural numbers. Define \(F'(o)=G(|F(o)|)F(o)\) (i.e., the concatenation of the representation of the length \(F(o)\) and \(F(o)\)).

  1. Prove that \(F'\) is a prefix-free representation of \(O\).
  2. Show that we can transform any representation to a prefix-free one by a modification that takes a \(k\) bit string into a string of length at most \(k+O(\log k)\).
  3. Show that we can transform any representation to a prefix-free one by a modification that takes a \(k\) bit string into a string of length at most \(k+ \log k + O(\log\log k)\).Hint: Think recursively how to represent the length of the string.

Suppose that \(S \subseteq \{0,1\}^n\) is some finite prefix-free set.

  1. For every \(k \leq n\) and length-\(k\) string \(x\in S\), let \(L(x) \subseteq \{0,1\}^n\) denote all the length-\(n\) strings whose first \(k\) bits are \(x_0,\ldots,x_{k-1}\). Prove that (1) \(|L(x)|=2^{n-|x|}\) and (2) If \(x \neq x'\) then \(L(x)\) is disjoint from \(L(x')\).
  2. Prove that \(\sum_{x\in S}2^{-|x|} \leq 1\).
  3. Prove that there is no prefix-free encoding of strings with less than logarithmic overhead. That is, prove that there is no function \(PF:\{0,1\}^* \rightarrow \{0,1\}^*\) s.t. \(|PF(x)| \leq |x|+0.9\log |x|\) for every \(x\in \{0,1\}^*\) and such that the set \(\{ PF(x) : x\in \{0,1\}^* \}\) is prefix-free. The factor \(0.9\) is arbitrary; all that matters is that it is less than \(1\).

Prove that for every two one-to-one functions \(F:S \rightarrow T\) and \(G:T \rightarrow U\), the function \(H:S \rightarrow U\) defined as \(H(x)=G(F(x))\) is one to one.

  1. We have shown that the natural numbers can be represented as strings. Prove that the other direction holds as well: that there is a one-to-one map \(StN:\{0,1\}^* \rightarrow \N\). (\(StN\) stands for “strings to numbers”.)
  2. Recall that Cantor proved that there is no one-to-one map \(RtN:\R \rightarrow \N\). Show that Cantor’s result implies cantorthm.

Recall that for every set \(S\), the set \(S^*\) is defined as the set of all finite sequences of members of \(S\) (i.e., \(S^* = \{ (x_0,\ldots,x_{n-1}) \;|\; n\in\mathbb{N} \;,\; \forall_{i\in [n]} x_i \in S \}\) ). Prove that there is a one-one-map from \(\mathbb{Z}^*\) to \(\mathbb{N}\) where \(\mathbb{Z}\) is the set of \(\{ \ldots, -3 , -2 , -1,0,+1,+2,+3,\ldots \}\) of all integers.

Bibliographical notes

The idea that we should separate the definition or specification of a function from its implementation or computation might seem “obvious”, but it took some 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 chapcomputable, in the 1800’s 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.

Gromov and Pomerantz’s quotes are lifted from Doron Zeilberger’s page.

Further explorations

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

  • Succinct data structures. These are representations that map objects from some set \(\mathcal{O}\) into strings of length not much larger than the minimum of \(\log_2 |\mathcal{O}|\) but still enable fast access to certain queries, see for example this paper.
  • We’ve 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 is known as the question of numerical stability.
  • The linear algebraic view of graphs: 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 this course are an excellent source for this area, known as spectral graph theory. We might discuss this view much later in this course when we talk about random walks.

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