Nodira Khoussainova

Deterministic finite automata

In this lecture we introduce deterministic finite automata, one of the foundational concepts in computing sciences. Finite automata are the simplest mathematical model of computers. Informally, a finite automaton is a system that consists of states and transitions. Each state represents a finite amount of information gathered from the start of the system to the present moment. Transitions represent state changes described by the system rules. Practical applications of finite automata include digital circuits, language design and implementations, image processing, modeling and building reliable software, and theoretical computing.

Lectures on discrete mathematics for computer science

Definitions, Theorems, and Proofs Arithmetic Graphs Circuits Trees Basics of Sets Relations and Databases Induction Reachability Games on Graphs Functions and Transitions Propositional Logic Finite Automata Regular Expressions Counting Probability.

Equivalence relations and partial orders

Relations abound in mathematics and in regular life too. We could speak of relations on the set of people like “A is a brother of B” or “A is B’s aunt” or “A and B are neighbors”. In mathematics, we have relations on sets of numbers like “≤”, “>”, and “sum to a rational number”. Another familiar relation is that of “⊆” when dealing with sets. It is quite useful to abstract the concept of equality. Relations which behave like “equals” are called “equivalence relations” (which are defined below). Another important kind of relation abstracts the properties of ≤ and ⊆. We call such relations “partial orders”. Let us give names to some familiar properties. Let R be a relation on a set X.

Semantics of propositional logic

An interpretation (also truth-assignment, valuation) of a set of propositional formulas S is a function that assigns elements of {f , t} to the propositional variables in S. The function can be partial, but it must assign values to the propositional variables in S.