The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. Shaw, as well as algorithmic methods, such as the resolution principle developed by John Alan Robinson.It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science.
Well-defined problems allow for more initial planning than ill-defined problems.
Solving problems sometimes involves dealing with pragmatics, the way that context contributes to meaning, and semantics, the interpretation of the problem.
Researchers' underlying assumption was that simple tasks such as the Tower of Hanoi correspond to the main properties of "real world" problems and thus the characteristic cognitive processes within participants' attempts to solve simple problems are the same for "real world" problems too; simple problems were used for reasons of convenience and with the expectation that thought generalizations to more complex problems would become possible.
Perhaps the best-known and most impressive example of this line of research is the work by Allen Newell and Herbert A. In computer science and in the part of artificial intelligence that deals with algorithms ("algorithmics"), problem solving includes techniques of algorithms, heuristics and root cause analysis.
There are two different types of problems, ill-defined and well-defined: different approaches are used for each.
Well-defined problems have specific goals and clear expected solutions, while ill-defined problems do not.Problem solving consists of using generic or ad hoc methods in an orderly manner to find solutions to problems.Some of the problem-solving techniques developed and used in philosophy, artificial intelligence, computer science, engineering, mathematics, or medicine are related to mental problem-solving techniques studied in psychology.The process starts with problem finding and problem shaping, where the problem is discovered and simplified.The next step is to generate possible solutions and evaluate them.The ability to understand what the goal of the problem is, and what rules could be applied, represents the key to solving the problem.Sometimes the problem requires abstract thinking or coming up with a creative solution.In these disciplines, problem solving is part of a larger process that encompasses problem determination, de-duplication, analysis, diagnosis, repair, and other steps.Other problem solving tools are linear and nonlinear programming, queuing systems, and simulation.Much of computer science involves designing completely automatic systems that will later solve some specific problem -- systems to accept input data and, in a reasonable amount of time, calculate the correct response or a correct-enough approximation.In addition, people in computer science spend a surprisingly large amount of human time finding and fixing problems in their programs -- debugging.