Algorithm
The word "algorithm" itself derives from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized "Algoritmi". A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert in 1928. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–37 and 1939.
Algorithmic program debugging
Algorithmic debugging was first developed by Ehud Shapiro during his PhD research at Yale University, as introduced in his PhD thesis, selected as a 1982 ACM Distinguished Dissertation. Shapiro implemented the method of algorithmic debugging in Prolog (a general purpose logic programming language) for the debugging of logic programs. In case of logic programs, the intended behavior of the program is a model (a set of simple true statements) and bugs are manifested as program incompleteness (inability to prove a true statement) or incorrectness (ability to prove a false statement). The algorithm would identify a false statement in the program and provide a counter-example to it or a missing true statement that it or its generalization should be added to the program. A method to handle non-termination was also developed. 
The research and development in the field of Algorithmic debugging has made major improvements over the original algorithms for debugging Prolog and other and extended the ideas to other language paradigms such as functional languages and object oriented languages.
Three decades since its introduction, algorithmic debugging is still an active field of computer science research and will probably remain so for decades as no panacea is in sight.