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The halting problem is a decision problem about properties of computer programs on a fixed Turing-complete model of computation, i.e., all programs that can be written in some given programming language that is general enough to be equivalent to a Turing machine. The problem is to determine, given a program and an input to the program, whether ...
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
The problem of determining if a given set of Wang tiles can tile the plane. The problem of determining the Kolmogorov complexity of a string. Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers.
A decision problem whose input consists of strings or more complex values is formalized as the set of numbers that, via a specific Gödel numbering, correspond to inputs that satisfy the decision problem's criteria. A decision problem A is called decidable or effectively solvable if the formalized set of A is a recursive set.
The problem to determine all positive integers such that the concatenation of and in base uses at most distinct characters for and fixed [citation needed] and many other problems in the coding theory are also the unsolved problems in mathematics.
For example, if s=2, then 𝜁(s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly 𝜋²/6. When s is a complex number—one that looks like a+b𝑖, using ...
The algorithm for deciding this is conceptually simple: it constructs (the description of) a new program t taking an argument n, which (1) first executes program a on input i (both a and i being hard-coded into the definition of t), and (2) then returns the square of n. If a(i) runs forever, then t never gets to step (2), regardless of n.
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0 , the line is the graph of the function of x that has been defined in the preceding section.