Busy beaverIn theoretical computer science, the busy beaver game aims at finding a terminating program of a given size that produces the most output possible. Since an endlessly looping program producing infinite output is easily conceived, such programs are excluded from the game. More precisely, the busy beaver game consists of designing a halting Turing machine with alphabet {0,1} which writes the most 1s on the tape, using only a given set of states.
Gödel's incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.
Diophantine setIn mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(, ) = 0) where P(, ) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns. A Diophantine set is a subset S of , the set of all j-tuples of natural numbers, so that for some Diophantine equation P(, ) = 0, That is, a parameter value is in the Diophantine set S if and only if the associated Diophantine equation is satisfiable under that parameter value.
Rice's theoremIn computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is non-trivial if it is neither true for every partial computable function, nor false for every partial computable function.
Theory of computationIn theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., approximate solutions versus precise ones). The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".
EntscheidungsproblemIn mathematics and computer science, the Entscheidungsproblem; ɛntˈʃaɪ̯dʊŋspʁoˌbleːm) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "yes" or "no" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
Kőnig's lemmaKőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic, especially in computability theory. This theorem also has important roles in constructive mathematics and proof theory. Let be a connected, locally finite, infinite graph.
Universal Turing machineIn computer science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Common sense might say that a universal machine is impossible, but Turing proves that it is possible. He suggested that we may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q 1: q 2 . ....
Arithmetical hierarchyIn mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy was invented independently by Kleene (1943) and Mostowski (1946). The arithmetical hierarchy is important in computability theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.
Alan TuringAlan Mathison Turing (ˈtjʊərɪŋ; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. He is widely considered to be the father of theoretical computer science and artificial intelligence.
