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Closed-world Assumption
The closed-world assumption (CWA), in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be true, is false. The same name also refers to a logical formalization of this assumption by Raymond Reiter. The opposite of the closed-world assumption is the open-world assumption (OWA), stating that lack of knowledge does not imply falsity. Decisions on CWA vs. OWA determine the understanding of the actual semantics of a conceptual expression with the same notations of concepts. A successful formalization of natural language semantics usually cannot avoid an explicit revelation of whether the implicit logical backgrounds are based on CWA or OWA. Negation as failure is related to the closed-world assumption, as it amounts to believing false every predicate that cannot be proved to be true. Example In the context of knowledge management, the closed-wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumscription (logic)
Circumscription is a non-monotonic logic created by John McCarthy to formalize the common sense assumption that things are as expected unless otherwise specified. Circumscription was later used by McCarthy in an attempt to solve the frame problem. To implement circumscription in its initial formulation, McCarthy augmented first-order logic to allow the minimization of the extension of some predicates, where the extension of a predicate is the set of tuples of values the predicate is true on. This minimization is similar to the closed-world assumption that what is not known to be true is false. The original problem considered by McCarthy was that of missionaries and cannibals: there are three missionaries and three cannibals on one bank of a river; they have to cross the river using a boat that can only take two, with the additional constraint that cannibals must never outnumber the missionaries on either bank (as otherwise the missionaries would be killed and, presumably, eate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique Name Assumption
The unique name assumption is a simplifying assumption made in some ontology languages and description logics. In logics with the unique name assumption, different names always refer to different entities in the world. It was included in Ray Reiter's discussion of the closed-world assumption often tacitly included in Database Management Systems (e.g. SQL) in his 1984 article "Towards a logical reconstruction of relational database theory" (in M. L. Brodie, J. Mylopoulos, J. W. Schmidt (editors), Data Modelling in Artificial Intelligence, Database and Programming Languages, Springer, 1984, pages 191–233). The standard ontology language OWL does not make this assumption, but provides explicit constructs to express whether two names denote the same or distinct entities. * owl: sameAs is the OWL property that asserts that two given names or identifiers (e.g., URIs) refer to the same individual or entity. * owl:differentFrom is the OWL property that asserts that two given names or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Model Semantics
The concept of a stable model, or answer set, is used to define a declarative semantics for logic programs with negation as failure. This is one of several standard approaches to the meaning of negation in logic programming, along with program completion and the well-founded semantics. The stable model semantics is the basis of answer set programming. Motivation Research on the declarative semantics of negation in logic programming was motivated by the fact that the behavior of SLDNF resolution — the generalization of SLD resolution used by Prolog in the presence of negation in the bodies of rules — does not fully match the truth tables familiar from classical propositional logic. Consider, for instance, the program :p\ :r \leftarrow p,\ q :s \leftarrow p,\ \operatorname q. Given this program, the query will succeed, because the program includes as a fact; the query will fail, because it does not occur in the head of any of the rules. The query will fail a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Default Logic
Default logic is a non-monotonic logic proposed by Raymond Reiter to formalize reasoning with default assumptions. Default logic can express facts like “by default, something is true”; by contrast, standard logic can only express that something is true or that something is false. This is a problem because reasoning often involves facts that are true in the majority of cases but not always. A classical example is: “birds typically fly”. This rule can be expressed in standard logic either by “all birds fly”, which is inconsistent with the fact that penguins do not fly, or by “all birds that are not penguins and not ostriches and ... fly”, which requires all exceptions to the rule to be specified. Default logic aims at formalizing inference rules like this one without explicitly mentioning all their exceptions. Syntax of default logic A default theory is a pair \langle W, D \rangle. is a set of logical formulas, called ''the background theory'', that formalize the fact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-monotonic Logic
A non-monotonic logic is a formal logic whose conclusion relation is not monotonic. In other words, non-monotonic logics are devised to capture and represent defeasible inferences (cf. defeasible reasoning), i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence. Most studied formal logics have a monotonic entailment relation, meaning that adding a formula to a theory never produces a pruning of its set of conclusions. Intuitively, monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known. A monotonic logic cannot handle various reasoning tasks such as reasoning by default (conclusions may be derived only because of lack of evidence of the contrary), abductive reasoning (conclusions are only deduced as most likely explanations), some important approaches to reasoning about knowledge (the ignorance of a conclusion must be retracted when the conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial-closed World Assumption
In a formal system of logic used for knowledge representation, the open-world assumption is the assumption that the truth value of a statement may be true irrespective of whether or not it is ''known'' to be true. It is the opposite of the closed-world assumption, which holds that any statement that is true is also known to be true. Origin An open-world assumption was first developed by Ancient Greek philosophers as a means to explain varying degrees of validity amongst mathematical and philosophical concepts proposed at the time of inception. Logical implication The open-world assumption (OWA) codifies the informal notion that in general no single agent or observer has complete knowledge, and therefore cannot make the closed-world assumption. The OWA limits the kinds of inference and deductions an agent can make to those that follow from statements that are known to the agent to be true. In contrast, the closed world assumption allows an agent to infer from the lack of knowled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open-world Assumption
In a formal system of logic used for knowledge representation, the open-world assumption is the assumption that the truth value of a statement may be true irrespective of whether or not it is ''known'' to be true. It is the opposite of the closed-world assumption, which holds that any statement that is true is also known to be true. Origin An open-world assumption was first developed by Ancient Greek philosophers as a means to explain varying degrees of validity amongst mathematical and philosophical concepts proposed at the time of inception. Logical implication The open-world assumption (OWA) codifies the informal notion that in general no single agent or observer has complete knowledge, and therefore cannot make the closed-world assumption. The OWA limits the kinds of inference and deductions an agent can make to those that follow from statements that are known to the agent to be true. In contrast, the closed world assumption allows an agent to infer from the lack of knowledge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial-closed World Assumption
In a formal system of logic used for knowledge representation, the open-world assumption is the assumption that the truth value of a statement may be true irrespective of whether or not it is ''known'' to be true. It is the opposite of the closed-world assumption, which holds that any statement that is true is also known to be true. Origin An open-world assumption was first developed by Ancient Greek philosophers as a means to explain varying degrees of validity amongst mathematical and philosophical concepts proposed at the time of inception. Logical implication The open-world assumption (OWA) codifies the informal notion that in general no single agent or observer has complete knowledge, and therefore cannot make the closed-world assumption. The OWA limits the kinds of inference and deductions an agent can make to those that follow from statements that are known to the agent to be true. In contrast, the closed world assumption allows an agent to infer from the lack of knowled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oracle Machine
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to solve certain problems in a single operation. The problem can be of any complexity class. Even undecidable problems, such as the halting problem, can be used. Oracles An oracle machine can be conceived as a Turing machine connected to an oracle. The oracle, in this context, is an entity capable of solving some problem, which for example may be a decision problem or a function problem. The problem does not have to be computable; the oracle is not assumed to be a Turing machine or computer program. The oracle is simply a "black box" that is able to produce a solution for any instance of a given computational problem: * A decision problem is represented as a set ''A'' of natural numbers (or strings). An instance of the problem is an arbitrary natural number (or strin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CoNP
In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP precisely if only ''no''-instances have a polynomial-length "certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate. That is, co-NP is the set of decision problems where there exists a polynomial ''p(n)'' and a polynomial-time bounded Turing machine ''M'' such that for every instance ''x'', ''x'' is a ''no''-instance if and only if: for some possible certificate ''c'' of length bounded by ''p(n)'', the Turing machine ''M'' accepts the pair (''x'', ''c''). Complementary Problems While an NP problem asks whether a given instance is a ''yes''-instance, its ''complement'' asks whether an instance is a ''no''-instance, which means the complement is in co-NP. Any ''yes''-instance for the original NP p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P (complexity)
In computational complexity theory, P, also known as PTIME or DTIME(''n''O(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or " tractable". This is inexact: in practice, some problems not known to be in P have practical solutions, and some that are in P do not, but this is a useful rule of thumb. Definition A language ''L'' is in P if and only if there exists a deterministic Turing machine ''M'', such that * ''M'' runs for polynomial time on all inputs * For all ''x'' in ''L'', ''M'' outputs 1 * For all ''x'' not in ''L'', ''M'' outputs 0 P can also be viewed as a uniform family of boolean circuits. A language ''L'' is in P if and only if there exists a polynomial-time uniform family of boolean circuits \, such that * For all n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |