On the Work of Madhu Sudan Avi Wigderson Madhu Sudan is the recipient of the 2002 Nevan- “eyeglasses” were key in this study of the no- linna Prize. Sudan has made fundamental contri- tions proof (again) and error correction. butions to two major areas of research, the con- • Theoretical computer science is an extremely nections between them, and their applications. interactive and collaborative community. The first area is coding theory. Established by Sudan’s work was not done in a vacuum, and Shannon and Hamming over fifty years ago, it is much of the background to it, conceptual and technical, was developed by other people. The the mathematical study of the possibility of, and space I have does not allow me to give proper the limits on, reliable communication over noisy credit to all these people. A much better job media. The second area is probabilistically check- has been done by Sudan himself; his homepage able proofs (PCPs). By contrast, it is only ten years (http://theory.lcs.mit.edu/~madhu/) con- old. It studies the minimal resources required for tains several surveys of these areas which give probabilistic verification of standard mathemati- proper historical accounts and references. In cal proofs. particular see [13] for a survey on PCPs and [15] My plan is to briefly introduce these areas, their for a survey on the work on error correction. motivation, and foundational questions and then to explain Sudan’s main contributions to each. Be- Probabilistic Checking of Proofs fore we get to the specific works of Madhu Sudan, One informal variant of the celebrated P versus let us start with a couple of comments that will set NP question asks, Can mathematicians, past and up the context of his work. future, be replaced by an efficient computer pro- gram? We first define these notions and then ex- • Madhu Sudan works in computational com- plexity theory. This research discipline at- plain the PCP theorem and its impact on this foun- dational question. tempts to rigorously define and study efficient versions of objects and notions arising in com- Efficient Computation Throughout, by an efficient algorithm (or program, putational settings. This focus on efficiency is machine, or procedure) we mean an algorithm which of course natural when studying computation runs at most some fixed polynomial time1 in the itself, but it also has proved extremely fruit- length of its input. The input is always a finite ful in studying other fundamental notions such as proof, randomness, knowledge, and more. 1Time refers to the number of elementary steps taken by Here I will try to explain how the efficiency the algorithm. The choice of “polynomial” to represent ef- ficiency is both small enough to often imply practicality Avi Wigderson is professor of mathematics at the Institute and large enough to make the definition independent of for Advanced Study, Princeton, and The Hebrew Univer- particular aspects of the model, e.g., the choice of allowed sity, Jerusalem. His email address is [email protected]. “elementary operations”. JANUARY 2003 NOTICES OF THE AMS 45 string of symbols from a fixed, finite alphabet. must be short, and the verification procedure must Note that an algorithm is an object of fixed size be efficient. It is important to note that all standard which is supposed to solve a problem on all inputs (logical) proof systems used in mathematics con- of all (finite) lengths. A problem is efficiently com- form to the second restriction: since only “local in- putable if it can be solved by an efficient algo- ferences” are made from “easily recognizable” ax- rithm. ioms, verification is always efficient in the total length of statement and proof. The first restriction, Definition 1. The class P is the class of all problems on the length of the proof, is natural, since we solvable by efficient algorithms. want the verification to be efficient in terms of the For example, the problems Integer Multiplication, length of the statement. Determinant, Linear Programming, Univariate Poly- An excellent, albeit informal, example is the lan- nomial Factorization, and (recently established) guage MATH of all mathematical statements, whose Testing Primality are in P. proof verification is defined by the well-known ef- Let us restrict attention (for a while) to algo- ficient (anonymous) algorithm REFEREE.2 As hu- rithms whose output is always “accept” or “reject”. mans we are simply not interested in theorems Such an algorithm A solves a decision problem. The whose proofs take, say, longer than our lifetime (or set L of inputs which are accepted by A is called the three-month deadline given by EDITOR) to read the language recognized by A. Statements of the and verify. form “x ∈ L” are correctly classified as “true” or But is this notion of efficient verification—read- “false” by the efficient algorithm A, deterministi- ing through the statement and proof, and check- cally (and without any “outside help”). ing that every new lemma indeed follows from pre- Efficient Verification vious ones (and known results)—the best we can In contrast, allowing an efficient algorithm to use hope for? Certainly as referees we would love some “outside help” (a guess or an alleged proof) natu- shortcuts as long as they do not change our notion rally defines a proof system. We say that a language of mathematical truth too much. Are there such L is efficiently verifiable if there is an efficient al- shortcuts? gorithm V (for “Verifier”) and a fixed polynomial Efficient Probabilistic Verification p for which the following completeness and sound- A major paradigm in computational complexity is ness conditions hold: allowing algorithms to flip coins. We postulate ac- • For every x ∈ L there exists a string π of length cess to a supply of independent unbiased random |π|≤p(|x|) such that V accepts the joint input variables which the probabilistic (or randomized) (x, π). algorithm can use in its computation on a given • For every x ∈ L, for every string π of length input. We comment that the very rich theories |π|≤p(|x|), V rejects the joint input (x, π). (which we have no room to discuss) of pseudo- Naturally, we can view all strings x in L as the- randomness and of weak random sources attempt orems of the proof system V. Those strings π to bridge the gap between this postulate and “real- which cause V to accept x are legitimate proofs of life” generation of random bits in computer pro- the theorem x ∈ L in this system. grams. The notion of efficiency remains the same: prob- Definition 2. The class NP is the class of all lan- abilistic algorithms can make only a polynomial guages that are efficiently verifiable. number of steps in the input length. However, the It is clear that P⊆NP. Are they equal? This is output becomes a random variable. We demand that the “P versus NP” question [5], one of the most the probability of error, on every input, never ex- important open scientific problems today. Not only ceed a given small bound . (Note that can be mathematicians but scientists and engineers as taken to be, e.g., 1/3, since repeating the algorithm well daily attempt to perform tasks (create theo- with fresh independent randomness and taking ries and designs) whose success can hopefully be majority vote of the answers can decrease the error efficiently verified. Reflect on the practical and exponentially in the number of repetitions.) philosophical impact of a positive answer to the Returning to proof systems, we now allow the question: if P = NP, then much of their (creative!) verifier V to be a probabilistic algorithm. As above, work can be performed efficiently by one com- we allow it to err (namely, accept false “proofs”) puter program. with extremely small probability. The gain would Many important computational problems, like be extreme efficiency: the verifier will access only the Travelling Salesman, Integer Programming, Map a constant number of symbols in the alleged proof. Coloring, Systems of Quadratic Equations, and In- Naturally, the positions of the viewed symbols can teger Factorization are (when properly coded as lan- guages) in NP. 2This system can, of course, be formalized. However, it is We stress two aspects of efficient verification. better to have the social process of mathematics in mind The purported “proof” π for the statement “x ∈ L” before we plunge into the notions of the next subsection. 46 NOTICES OF THE AMS VOLUME 50, NUMBER 1 be randomly chosen. What kind of theorems can The conversion above is efficient and deter- be proved in the resulting proof system? First, let ministic. So in principle an efficient program can us formalize it. be written to convert standard mathematical proofs We say that a language L has a probabilistically to robust ones which can be refereed in a jiffy. checkable proof if there is an efficient probabilis- The PCP theorem challenges the classical belief tic algorithm V, a fixed polynomial p, and a fixed that proofs have to be read and verified fully for constant c for which the following completeness one to be confident of the validity of the theorem. and probabilistic soundness conditions hold. Of course one does not expect the PCP theorem to • For every x ∈ L there exists a string π , of dramatically alter the process of writing and veri- length |π|≤p(|x|), such that V accepts the fying proofs (any more than one would expect au- joint input (x, π) with probability 1. tomated verifiers of proof systems to replace the • For every x ∈ L, for every string π of length REFEREE for journal papers). In this sense the PCP |π|≤p(|x|), V rejects the joint input (x, π) theorem is just a statement of philosophical im- with probability ≥ 1/2.