Review of Basic Computer Science Background Computer Science

Review of Basic Computer Science Background

Computer Science You’ll Need • Some models of computation – Finite state machines – Recursive functions / Turing machines – Logic circuit models – Cellular automata • Computability theory – Universal vs. non-universal models • Computational complexity theory – Polynomial vs. exponential problem classes • Computer architecture/engineering concepts – functional units, CPUs, instruction sets, etc. . .

Models of Computation • Non-universal models: – Finite state machines (w. read-only inputs) • Related to regular expression languages – Pushdown automata (FSM w. a stack) • Relation to context-free grammar languages • Universal models: – Recursive function theory (Church) – “File clerk model” / Turing machines (Turing) • FSM with unbounded-size read-write external memory – Logic circuits (Shannon) – RAM machines (von Neumann) – Cellular automata (von Neumann)

Church-Turing thesis • All the “universal” models were proved early on to be able to simulate each other – Straightforward to write “emulator” programs • “Church-Turing thesis”: – Anything that can be computed physically (in any way whatsoever) can be computed by machines in these models • Hence the term “universal” – This is in the form of a physical postulate. • Note that universality & the Church-Turing thesis say nothing about efficiency.

Computational Complexity Theory • Characterizes the “complexity” (defined in terms of resource usage) of computations – This “complexity” can be viewed as essentially the reciprocal of computational efficiency • efficiency would be amount of computation completed per unit resources consumed • Traditional focus in complexity theory only on either of two resources, in isolation: – “Time” (most popular) • Number of primitive ops. or parallel steps performed – “Space” • Max. number of bits of storage needed at any time

Cost-Efficiency • Resources (time, space, hardware, energy, etc. ) have some associated cost for their expenditure – Resource usage should be minimized • A 100% efficient process is one that wastes no resources. Want to maximize efficiency. – Let $min be the minimum cost required to carry out a process, $ the actual cost. – So %$ = $min/$, or cost-efficiency, is the fraction of cost that is well spent. • Note that %$ is inversely proportional to cost $ (or “complexity”), even if $min is unknown.

Complexity Classes • In complexity theory, it is useful to classify computational problems or tasks on a given machine in terms of the asymptotic order of growth of the complexity. – Allows ignoring the effects of minor implementation details • Further, similar orders of growth are often grouped together, e. g. : – P - The class of problems solvable in polynomial (O(nk) for some k) time • This class was thought to be independent of the machine model used, until the discovery of quantum computing!

Order of Growth Notation