Theory of Computability Giorgi Japaridze Theory of Computability The class P We saw mapping reducibility. But does it capture our intuitive concept of reducibility in the most general way? It doesn’t. E.g., consider ATM and its complement. Intuitively, they are reducible to one another because a solution to either could be used to solve the other by simply reversing the answer. However, we know that the complement of ATM is not mapping reducible to ATM because the latter is Turing recognizable while the former is not. Now we’ll look at a very general form of reducibility, called Turing reducibility, which captures our intuitive concept of reducibility more closely. Section 7.2

The definition of P and the importance of P Giorgi Japaridze Theory of Computability Definition 7.12 P is the class of languages that are decidable in polynomial time on a deterministic (single-tape) TM. In other words, P = TIME(n1)  TIME(n2)  TIME(n3)  TIME(n4)  … Importance P is invariant for all models of computation that are polynomially equivalent to the deterministic single-tape TM. Here polynomial equivalence means equivalence with only a polynomial difference in running time. Thus, P is a mathematically robust class, not affected by the particulars of the model of computation that we are using. P roughly corresponds to the class of problems that are realistically solvable on a computer. Thus, P is relevant from a practical standpoint.

Analyzing polynomial-time algorithms 7.2.b Giorgi Japaridze Theory of Computability 1. When we only care about polynomiality, algorithms can be described at high level without reference to features of a particular implementation model. Doing so avoids tedious details of tapes and head motions. 2. We describe algorithms with numbered stages. The notion of a stage is Analogous to a step of a TM, though of course, implementing one stage will usually require many TM steps. Asymptotic analysis allows us to ignore this difference. 3. To show that an algorithm runs in polynomial time, it is sufficient to show that: a) There is a polynomial upper bound (usually in big-O notation) on the number of stages that the algorithm uses when it runs on an input of length n, and b) Each stage takes a polynomial number of (actual TM) steps on a reasonable deterministic model. 4. The underlying (and usually non-specified) encoding of objects should be reasonable, and polynomially equivalent to other reasonable encodings. E.g., encoding 12 as 111111111111 is unreasonable as it is exponentially bigger than the binary (or decimal) encoding. 5. Among the reasonable encodings for graphs are encodings of their adjacency matrices. Since the size of such a matrix only polynomially differs from the number of nodes, it is OK if we show the polynomiality of an algorithm in the number of its nodes rather than in the size of its adjacency matrix.

The PATH problem 7.2.c Giorgi Japaridze Theory of Computability PATH = {<G,s,t> | G is a directed graph that has a directed path from s to t} 2 3 1 4 5 8 s 7 6 t 9 12 10 11 14 15 13 16

A polynomial-time algorithm for the PATH problem 7.2.d Giorgi Japaridze Theory of Computability M = “On input <G,s,t> where G is a directed graph and s,t are nodes of G: 1. Mark s. 2. Repeat until no additional nodes are marked: 3. Scan all the edges of G. If an edge (a,b) is found going from a marked node a to an unmarked node b, mark node b. 4. If t is marked, accept. Otherwise reject.” 2 3 1 4 5 8 s 7 6 t 9 12 10 11 14 15 13 16

RELPRIME = {<x,y> | x and y are relatively prime} The RELPRIME problem 7.2.e Giorgi Japaridze Theory of Computability Two numbers are relatively prime iff 1 is the largest integer that evenly divides both of them. Are the following numbers relatively prime? 15 and 27 8 and 9 11 and 19 RELPRIME = {<x,y> | x and y are relatively prime} Is RELPRIME decidable? What is the time complexity of an ad hoc decision algorithm (TM)? But there is a smarter algorithm that runs in polynomial time. It is based on the Euclidean algorithm for finding the greatest common divisor.

A polynomial-time algorithm for the RELPRIME problem Giorgi Japaridze Theory of Computability The Euclidean algorithm: E = “On input <x,y>, where x and y are natural numbers, x>y: 1. Repeat until y=0. 2. Assign x  x mod y. 3. Exchange x and y. 4. Output x.” Testing on <33,15>: Output: 3 What is the time complexity of this algorithm? --- It can be shown to be polynomial because on every iteration of Step 1, the value of x is at most half of the previous value. Now, the following algorithm R solves RELPRIME in polynomial time: R = “On input <x,y>, where x and y are natural numbers: 1. Swap x and y if necessary so that x>y. 2. Run E on <x,y>. 3. If the result is 1, accept. Otherwise reject.

The time complexity of context-free languages Giorgi Japaridze Theory of Computability Theorem 7.16 Every context-free language is a member of P. Specifically, is of complexity O(n3). Proof omitted (and will not be asked).