Write brief notes to alert the database administrators to the difficulties that they may encounter. [8 marks] SECTION C 9 Semantics of Programming Languages (a) The integer expressions E of a programming language are given by E ::= n | X | E | E + E where n ranges over integer constants and X ranges over identifiers. Formalise and prove, using your second formula, that on the n th cycle the value of r is the sum of the values of a on all the cycles up to the n th. You may assume that values are natural numbers and ignore the possibility of overflow. [8 marks] 11 Advanced Graphics Give a parametric definition of a torus centred at the origin and aligned with the coordinate axes. [4 marks] During the development phase, one of the first decisions the developer must make is whether the application is to run over TCP or over UDP. Recall that TCP is connection oriented and provides a reliable bytestream channel through which data flows between two end systems. UDP is connectionless and sends independent packets of data from one end system to the other, without any guarantees about delivery. Recall also that when a client or server program implements a protocol defined by an RFC, it should use the well-known port number associated with the protocol; conversely, when developing a proprietary application, the developer must be careful to avoid using such well-known port numbers. (Port numbers were briefly discussed in Section 2.1. They are covered in more detail in Chapter 3.) We introduce UDP and TCP socket programming by way of a simple UDP application and a simple TCP application. We present the simple UDP and TCP applications in Python. We could have written the code in Java, C, or C++, but we chose Python mostly because Python clearly exposes the key socket concepts. With Python there are fewer lines of code, and each line can be explained to the novice programmer without difficulty. But there's no need to be frightened if you are not familiar with Python. Explain the principle of structural induction for proving that some property (E) holds for all integer expressions E. [5 marks] (b) Taking states to be finite partial functions mapping identifiers to integer constants, define a relation E, s n giving the result n (if any) of evaluating integer expression E in state s. [7 marks] (c) List the terminal images and non-terminal images, and count the creation rules both in the first syntax and in the sentence structure in your response to part (b). Demonstrate the beginning image in the two sentence structures. [2 marks]