Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although

this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR-85-11713, CCR-86-05353, and CCR-88-14977, and by DARPA contract N00039-84-C-0165.

We investigate the computational complexity of an optically inspired model of computation. The model is called the continuous space machine and operates in discrete timesteps over a number of two-dimensional complex-valued images of constant size and arbitrary spatial resolution. We define a number of optically inspired complexity measures and data representations for the model. We show the growth of each complexity measure under each of the model's operations. We characterise the power of an important discrete restriction of the model. Parallel time on this variant of the model is shown to correspond, within a polynomial, to sequential space on Turing machines, thus verifying the parallel computation thesis. We also give a characterisation of the class NC. As a result the model has computational power equivalent to that of many well-known parallel models. These characterisations give a method to translate parallel algorithms to optical algorithms and facilitate the application of the complexity theory toolbox to optical computers. Finally we show that another variation on the model is very powerful

We present lower bounds on the computational power of an optical model of computation called the C2-CSM.

An addition circuit by Ladner and Fisher is analysed. It has size bounded by (8+6 \Delta 2 \Gammak )n and depth bounded by 2dlog 2 ne+2k+2, where 0 k dlog 2 ne and n is the bit length of the input numbers. Moreover the implementation of an instance of this adder capable of adding two numbers of 8-bit length was timed. The average computing time of its logical design and its layout is 26:8 ns and 94:2 ns, respectively. The layout of this circuit was done on an FPGA by Atmel. Contents 1 Introduction 2 2 Complexity of addition circuits 2 2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Bounds on the complexity of the addition circuit . . . . . . . . . . . 3 3 Design and analysis of the adder by Ladner and Fisher 4 3.1 Design of the adder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Parallel prefix computation . . . . . . . . . . . . . . . . . . . . . . . 6 3.3 Complexity of Ladner's and Fisher's adder . . . . . . . . . . . . . ....