From Wikipedia, the free encyclopedia
A undecidable problem is a decision problem that is not decidable.
A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as: the set of inputs for which the problem returns yes.
These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as Gödel numbers, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is phrased in terms of subsets of the natural numbers.
Formally, a decision problem is a subset of the natural numbers. The corresponding informal problem is that of deciding whether a given number is in the set.
A decision problem A is called decidable or effectively solvable if A is a recursive set. A problem is called partially decidable, semidecidable, solvable, or provable if A is a recursively enumerable set. Partially decidable problems and any other problems that are not decidable are called undecidable.
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[edit] The undecidable problem in Computability Theory
In computability theory, the halting problem is a decision problem which can be stated as follows:
- Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input.
Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. We say that the halting problem is undecidable over Turing machines.
[edit] Relationship with Gödel’s incompleteness theorem
The concepts raised by Gödel’s incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that a complete, consistent and sound axiomatization of all statements about natural numbers is unachievable. The “sound” part is the weakening: it means that we require the axiomatic system in question to prove only true statements about natural numbers (it’s very important to observe that the statement of the standard form of Gödel’s First Incompleteness Theorem is completely unconcerned with the question of truth, but only concerns the issue of whether it can be proven).
The weaker form of the theorem can be proved from the undecidability of the halting problem as follows. Assume that we have a consistent and complete axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that enumerates all these statements. This means that there is an algorithm N(n) that, given a natural number n, computes a true first-order logic statement about natural numbers such that, for all the true statements, there is at least one n such that N(n) yields that statement. Now suppose we want to decide if the algorithm with representation a halts on input i. We know that this statement can be expressed with a first-order logic statement, say H(a, i). Since the axiomatization is complete it follows that either there is an n such that N(n) = H(a, i) or there is an n’ such that N(n’) = ¬ H(a, i). So if we iterate over all n until we either find H(a, i) or its negation, we will always halt. This means that this gives us an algorithm to decide the halting problem. Since we know that there cannot be such an algorithm, it follows that the assumption that there is a consistent and complete axiomatization of all true first-order logic statements about natural numbers must be false.
[edit] List of undecidable problems
In computability theory, an undecidable problem is a problem whose language is not a recursive set. More informally, such problems cannot be solved in general by computers; see decidability. This is a list of undecidable problems. Note that there are uncountably many undecidable problems, so this list is necessarily incomplete. Though undecidable languages are not recursive languages, they may be a subset of Turing recognizable languages.
- The halting problem (determining whether a specified machine halts or runs forever).
- The busy beaver problem (determining the length of the longest halting computation among machines of a specified size).
- Rice’s theorem states that for all non-trivial properties of partial functions, it is undecidable whether a machine computes a partial function with that property.
[edit] Other problems
[edit] Examples of undecidable statements
There are two distinct senses of the word “undecidable” in contemporary use. The first of these is the sense used in relation to Gödel’s theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set. The connection between these two is that if a decision problem is undecidable (in the recursion theoretical sense) then there is no consistent, effective formal system which proves for every question A in the problem either “the answer to A is yes” or “the answer to A is no”.
Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the “neither provable nor refutable” sense. The usage of “independent” is also ambiguous, however. Some use it to mean just “not provable”, leaving open whether an independent statement might be refuted.
Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called “absolutely undecidable” statements, whose truth value can never be known or is ill-specified, is a controversial point among various philosophical schools.
One of the first problems suspected to be undecidable, in the second sense of the term, was the word problem for groups, first posed by Max Dehn in 1911, which asks if there is a finitely presented group for which no algorithm exists to determine whether two words are equivalent. This was shown to be the case in 1952.
The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proven from ZFC.
In 1970, Soviet mathematician Yuri Matiyasevich showed that Hilbert’s Tenth Problem, posed in 1900 as a challenge to the next century of mathematicians, cannot be solved. Hilbert’s challenge sought an algorithm which finds all solutions of a Diophantine equation. A Diophantine equation is a more general case of Fermat’s Last Theorem; we seek the rational roots of a polynomial in any number of variables with integer coefficients. Since we have only one equation but n variables, infinite solutions exist (and are easy to find) in the complex plane; the problem becomes difficult (impossible) by constraining solutions to rational values only. Matiyasevich showed this problem to be unsolvable by mapping a Diophantine equation to a recursively enumerable set and invoking Gödel’s Incompleteness Theorem.[1]
In 1936, Alan Turing proved that the halting problem—the question of whether or not a Turing machine halts on a given program—is undecidable, in the second sense of the term. This result was later generalized to Rice’s theorem.
In 1973, the Whitehead problem in group theory was shown to be undecidable, in the first sense of the term, in standard set theory.
In 1977, Paris and Harrington proved that the Paris-Harrington principle, a version of the Ramsey theorem, is undecidable in the axiomatization of arithmetic given by the Peano axioms but can be proven to be true in the larger system of second-order arithmetic.
Kruskal’s tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal’s tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.
Goodstein’s theorem is a statement about the Ramsey theory of the natural numbers that Kirby and Paris showed is undecidable in Peano arithmetic.
Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin’s theorem states that for any theory that can represent enough arithmetic, there is an upper bound c such that no specific number can be proven in that theory to have Kolmogorov complexity greater than c. While Gödel’s theorem is related to the liar paradox, Chaitin’s result is related to Berry’s paradox.
Douglas Hofstadter gives a notable alternative proof of incompleteness, inspired by Gödel, in his book Gödel, Escher, Bach.
From Wikipedia, the free encyclopedia
[edit] From Logic to Ontology: The limit of “The Semantic Web”
The limits of the semantic web are not set by the use of machines themselves and biological systems could be used to reach this goal, but as the logic that is being used to construct it does not contemplate the concept of time, since it is purely formal logic and metonymic lacks the metaphor, and that is what Gödel’s theorems remark, the final tautology of each construction or metonymic language (mathematical), which leads to inconsistencies. The construction of the Semantic Web is a undecidible problem.
This consistent logic is completely opposite to the logic that makes inconsistent use of time, inherent of human unconscious, but the use of time is built on the lack, not on positive things, it is based on denials and absences, and that is impossible to reflect on a machine because of the perceived lack of the required self-awareness is acquired with the absence.
The problem is we are trying to build an intelligent system to replace our way of thinking, at least in the information search, but the special nature of human mind is the use of time which lets human beings reach a conclusion, therefore does not exist in the human mind the halting problem or stop of calculation.
So all efforts faced toward semantic web are doomed to failure a priori if the aim is to extend our human way of thinking into machines, they lack the metaphorical speech, because only a mathematical construction, which will always be tautological and metonymic, and lacks the use of the time that is what leads to the conclusion or “stop”.
As a demonstration of that, if you suppose it is possible to construct the semantic web, as a language with capabilities similar to human language, which has the use of time, should we face it as a theorem, we can prove it to be false with a counter example, and it is given in the particular case of the Turing machine and “the halting problem”.
One solution for the problem to build the semantic web could be the use of Non-formal or Inconsistency Logic.
From Wikipedia, the free encyclopedia
Completeness, incompleteness, consistency, inconsistency, decidable and undecidable are concepts of meta logic which can be attributed to certain features of the formal logical systems, more precisely axiomatic systems. These are concepts that are attributed to Kurt Gödel from their theorems from the beginning of the previous century. They emerge in a very particular context of mathematics as opposed to the ideal of David Hilbert who believed that everything in that area could be proof.
Kurt Gödel was born on April 28, 1906 in Brünn, Moravia. It became part of the Vienna Circle, and from that moment they begin to develop their most important theories on the completeness of the formal systems from two publications: his doctoral thesis written in 1929, and the theorem (formally on propositions undecidable in the Principia Mathematica and related systems) published in 1931.
In 1931, Gödel published About propositions …, article that called into question the agenda D Hilbert, because not only showed that the system Russel and Whitehead had cracks, but the entire system would be axiomatic. An axiomatic system consists of a set of formulas set forth or allowed without demonstration-axioms-from which all others are derived assertions theory called theorems. The set of axioms over the definition of phrasing or formula System (definition preceding statement of the axioms) and the set of rules for obtaining theorems from the axioms (transformation rules) are the basis of the primitive system.
K. Gödel proved that it is impossible to establish consistency internal logic of a broad class of deductive systems, unless it is taken early so complex reasoning that its internal consistency remains as subject to the doubt as to the systems themselves, putting at stake the impossibility proofing certain propositions. Consistency, inconsistency, completeness and incompleteness.What is a system, which means that it is consistently inconsistent, complete or incomplete, which is a proposition, etc.?
A system is a set of axioms and rules of inference, a claim that a proposition can be true or false. When a system is complete? Once inside it can be determined by the value of truth or falsity of any proposition The completeness assures us that there is no truth in our system that we will not be able to find But we can only be sure of being able to reach the whole truth if our system is complete.Change is incomplete when it contains proposals on which we are unable to determine their truth or falsity. Moreover, a system is consistent when no contradictions of any kind nor does it have any paradox, and is inconsistent when we run into contradictions and paradoxes. A system is consistent if it is clean of paradoxes and contradictions and complete if any proposition can be proved or disproved sign him. Gödel believed that if it is consistent is incomplete and if it is completely inconsistent.
In that sense, consistency means that it is not possible to deduce from the same set of axioms, two theorems which are contradictory. When it comes to contradiction semantics, the system is inconsistent.
The principle of inconsistency then assumed that the truth-value of a system can not be determined from a set of axioms, but only from a foreign axiom. That is a system that is inconsistent when it can not get rid of its internal contradictions semantic.
