Abstract
This chapter provides the basic definition of what we mean by “computable” and what we mean by “problem.” A short historical perspective is given. The term “effective procedure” is introduced that describes a program executable in a finite number of simple steps in a mechanical way. Sets and structures on sets, i.e. relations and functions , together with their basic operations, are defined and some basic reasoning principles reviewed.
What does computable mean? What problems do we consider?
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Notes
- 1.
The Association for Computing Machinery (ACM) , “the world’s largest educational and scientific computing society, delivers resources that advance computing as a science and a profession” [1]. It was founded in 1947 and has its headquarters in New York.
- 2.
And every reader may have their own problems, i.e. their own idea of what a “problem” is.
- 3.
The name “algorithm” dates back to the name of the ninth century Persian mathematician Al-Khwarizmi.
- 4.
Euclid (ca. 300 BC) was a Greek mathematician often called the “Father of Geometry” who also worked in number theory. He is the inventor of the common divisor algorithm.
- 5.
Gottfried Wilhelm von Leibniz (July 1, 1646–November 14, 1716) was a German mathematician and philosopher, credited with the independent invention of differential and integral calculus. He also invented calculating machines.
- 6.
Charles Babbage, (December 26, 1791–October 18, 1871) was an English mathematician, philosopher, inventor and mechanical engineer (and a Fellow of the Royal Society). He is also known for originating the concept of a programmable calculating machine. The London Science Museum has constructed two of his machines where they are on display.
- 7.
David Hilbert (January 23, 1862–February 14, 1943) was a world-renowned German mathematician.
- 8.
Kurt Friedrich Gödel (April 28, 1906–January 14, 1978) was an Austrian (and American) logician, mathematician, and philosopher and is considered one of the most significant logicians in history. He proved many important results, relevant here is the Incompleteness Theorem.
- 9.
Alonzo Church (June 14, 1903–August 11, 1995) was an important American mathematician and logician.
- 10.
Church’s first version that the computable functions are those definable by \(\lambda \)-terms [3] was initially rejected.
- 11.
This type may be a set again.
- 12.
The Cartesian product is named in honour of Reneé Descartes (31 March 1596–11 February 1650), a French mathematician and philosopher, who spent most of his life in the Netherlands, and is famous for his saying “I think therefore I am” as well as for the development of (Cartesian) analytical geometry. He was invited to the court of Queen Christina of Sweden in 1649. “In Sweden—where, Descartes said, in winter men’s thoughts freeze like the water—the 22-year-old Christina perversely made the 53-year-old Descartes rise before 5:00 am to give her philosophy lessons, even though she knew of his habit of lying in bed until 11 o’clock in the morning” [9]. Consequently, Descartes caught pneumonia and died.
- 13.
The symbol itself is called a “perp”.
- 14.
At first glance, it is not obvious whether \(S\times T\times U\) means \(S\times (T\times U)\) or \((S\times T)\times U\). We assume cartesian products to be associative, identifying these two definitions, thus dropping the extra parentheses and writing (s, t, u) for triples, and similarly for n-tuples where \(n>3\).
- 15.
Which may possibly be the same type.
- 16.
Mathematical objects.
- 17.
Boolean values are named after George Boole (2 November 1815–8 December 1864), an English mathematician and logician famous for his work on differential equations and algebraic logic. He is most famous for what is called Boolean algebra. Throughout this book, we will use the term “boolean” to indicate a truth value for which the corresponding algebra operations are available.
- 18.
This question is easily confused with the one famously asked in Douglas Adam’s masterpiece: “The Hitchhiker’s Guide to the Galaxy” [2] which actually is called: “Ultimate Question of Life, The Universe, and Everything”. The computer in question, Deep Thought, after a considerable 7.5 million years answered famously: “42”. Alas, nobody understood the question. So Deep Thought suggested to build an even more powerful super-computer to produce the question to the answer. This computer was later revealed to be planet “Earth” which was unfortunately destroyed 5 min before completion of the calculations.
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Reus, B. (2016). Problems and Effective Procedures. In: Limits of Computation. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-27889-6_2
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