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Lecture № 1 Theory of Algorithms Lecture № 1 Theory of Algorithms

Plan of lecture 1. The notion of algorithm 2. Elementary, primitive recursive and partial Plan of lecture 1. The notion of algorithm 2. Elementary, primitive recursive and partial recursive functions 3. Church Thesis 4. Algorithmic problems

1. The notion of algorithm The main properties of the algorithm: ü algorithm - 1. The notion of algorithm The main properties of the algorithm: ü algorithm - sequential process of building values, which occurs in discrete time so that at the initial time given rising finite set of variables, and each subsequent time the system variables is well defined by law (the program) from the system of values that were in the previous time (discrete of algorithm) ; ü Determinacy of algorithm - an organization of steps in which always clear how to make the transition from one stage to another;

1. The notion of algorithm ü Algorithm finite - for achievement the result in 1. The notion of algorithm ü Algorithm finite - for achievement the result in the application of the algorithm to solve a specific task performed a finite sequence of steps of the algorithm; ü Algorithm mass - the applicability of the algorithm to the class of problems, not only to the one problem.

Partial recursive functions, Post machines, Turing machines, Markov algorithms are the refinement of the Partial recursive functions, Post machines, Turing machines, Markov algorithms are the refinement of the intuitive notion of algorithm.

2. Elementary, primitive recursive and partial recursive functions 2. Elementary, primitive recursive and partial recursive functions

2. Elementary, primitive recursive and partial recursive functions 2. Elementary, primitive recursive and partial recursive functions

2. Elementary, primitive recursive and partial recursive functions 2. Elementary, primitive recursive and partial recursive functions

2. Elementary, primitive recursive and partial recursive functions 2. Elementary, primitive recursive and partial recursive functions

2. Elementary, primitive recursive and partial recursive functions 2. Elementary, primitive recursive and partial recursive functions

2. Elementary, primitive recursive and partial recursive functions 2. Elementary, primitive recursive and partial recursive functions

2. Elementary, primitive recursive and partial recursive functions 2. Elementary, primitive recursive and partial recursive functions

2. Elementary, primitive recursive and partial recursive functions 2. Elementary, primitive recursive and partial recursive functions

3. Church Thesis Any partial function is computed by the algorithm if and only 3. Church Thesis Any partial function is computed by the algorithm if and only if it is partial recursive.

4. Algorithmic problems Ø Identity problem of elementary algebra; Ø Identity problem for semigroups; 4. Algorithmic problems Ø Identity problem of elementary algebra; Ø Identity problem for semigroups; Ø Identity problem of words for semigroups.

Thank you for your attention!!! Thank you for your attention!!!

Homework v To work out the material of the lecture. Homework v To work out the material of the lecture.

References References