|
|
| Line 1: |
Line 1: |
| The '''autoepistemic logic''' is a [[logic|formal logic]] for the representation and reasoning of knowledge about knowledge. While [[propositional logic]] can only express facts, autoepistemic logic can express knowledge and lack of knowledge about facts.
| | Greetings! I am Marvella and I really feel comfy when people use the full name. His family lives in South Dakota but his wife wants them to move. To do aerobics is a thing that I'm totally addicted to. My working day job is a meter reader.<br><br>Feel free to surf to my web blog ... std testing at home, [http://nxnn.info/user/G8498 Read the Full Guide], |
| | |
| The [[stable model semantics]], which is used to give a semantics to [[logic programming]] with [[negation as failure]], can be seen as a simplified form of autoepistemic logic.
| |
| | |
| ==Syntax==
| |
| | |
| The [[syntax]] of autoepistemic logic extends that of propositional logic by a modal operator <math>\Box</math> indicating knowledge: if <math>F</math> is a formula, <math>\Box F</math> indicates that <math>F</math> is known. As a result, <math>\Box \neg F</math> indicates that <math>\neg F</math> is known and <math>\neg \Box F</math> indicates that <math>F</math> is not known.
| |
| | |
| This syntax is used for allowing reasoning based on knowledge of facts. For example, <math>\neg \Box F \rightarrow \neg F</math> means that <math>F</math> is assumed false if it is not known to be true. This is a form of [[negation as failure]].
| |
| | |
| ==Semantics==
| |
| | |
| The semantics of autoepistemic logic is based on the ''expansions'' of a theory, which have a role similar to models in [[propositional logic]]. While a propositional model specifies which axioms are true or false, an expansion specifies which formulae <math>\Box F</math> are true and which ones are false. In particular, the expansions of an autoepistemic formula <math>T</math> makes this distinction for every subformula <math>\Box F</math> contained in <math>T</math>. This distinction allows <math>T</math> to be treated as a [[propositional formula]], as all its subformulae containing <math>\Box</math> are either true or false. In particular, checking whether <math>T</math> [[logical consequence|entails]] <math>F</math> in this condition can be done using the rules of the propositional calculus. In order for an initial assumption to be an expansion, it must be that a subformula <math>F</math> is entailed if and only if <math>\Box F</math> has been initially assumed true.
| |
| | |
| For example, in the formula <math>T = \Box x \rightarrow x</math>, there is only a single “boxed subformula”, which is <math>\Box x</math>. Therefore, there are only two candidate expansions, assuming it true or false, respectively. The check for them being actual expansions is as follows.
| |
| | |
| <math>\Box x</math> is false : with this assumption, <math>T</math> becomes tautological, as <math>\Box x \rightarrow x</math> is equivalent to <math>\neg \Box x \vee x</math>, and <math>\neg \Box x</math> is assumed true; therefore, <math>x</math> is not entailed. This result confirms the assumption implicit in <math>\Box x</math> being false, that is, that <math>x</math> is not currently known. Therefore, the assumption that <math>\Box x</math> is false is an expansion. | |
| | |
| <math>\Box x</math> is true : together with this assumption, <math>T</math> entails <math>x</math>; therefore, the initial assumption that is implicit in <math>\Box x</math> being true, i.e., that <math>x</math> is known to be true, is satisfied. As a result, this is another expansion.
| |
| | |
| The formula <math>T</math> has therefore two expansions, one in which <math>x</math> is not known and one in which <math>x</math> is known. The second one has been regarded as unintuitive, as the initial assumption that <math>\Box x</math> is true is the only reason why <math>x</math> is true, which confirms the assumption. In other words, this is a self-supporting assumption. A logic allowing such a self-support of beliefs is called ''not strongly grounded'' to differentiate them from ''strongly grounded'' logics, in which self-support is not possible. Strongly grounded variants of autoepistemic logic exist.
| |
| | |
| ==Generalizations==
| |
| In [[uncertain inference]], the known/unknown duality of truth values is replaced by a degree of certainty of a fact or deduction; certainty may vary from 0 (completely uncertain/unknown) to 1 (certain/known). In [[probabilistic logic network]]s, truth values are also given a probabilistic interpretation (''i.e.'' truth values may be uncertain, and, even if almost certain, they may still be "probably" true (or false).)
| |
| | |
| ==See also==
| |
| | |
| * [[Non-monotonic logic]]
| |
| * [[Modal logic]]
| |
| | |
| ==References==
| |
| {{refbegin}}
| |
| *{{cite journal |first=G. |last=Gottlob |title=Translating default logic into standard autoepistemic logic |journal=Journal of the ACM |volume=42 |issue=4 |pages=711–740 |date=July 1995 |doi=10.1145/210332.210334 |url=http://dl.acm.org/citation.cfm?id=210332.210334}}
| |
| *{{cite book |first=T. |last=Janhunen |chapter=On the intertranslatability of autoepistemic, default and priority logics |editor1-first=Jürgen |editor1-last=Dix |editor2-first=Luís Fariñas |editor2-last=del Cerro |editor3-first=Ulrich |editor3-last=Furbach |title=Logics in Artificial Intelligence: European Workshop, JELIA '98, Dagstuhl, Germany, October 12–15, 1998 : Proceedings |publisher=Springer |year=1998 |isbn=3540495452 |pages=216–232 |series=Lecture notes in computer science: Lecture notes in artificial intelligence}}
| |
| *{{cite journal |first1=W. |last1=Marek |first2=M. |last2=Truszczyński |title=Autoepistemic logic |journal=Journal of the ACM |volume=38 |issue=3 |pages=588–618 |date=July 1991 |doi=10.1145/116825.116836 |url=http://dl.acm.org/citation.cfm?id=116825.116836}}
| |
| *{{cite journal |last=Moore |first=R.C. |title=Semantical considerations on nonmonotonic logic |journal=Artificial Intelligence |volume=25 |issue= |pages=75–94 |date=January 1985 |doi=10.1016/0004-3702(85)90042-6 |url=http://www.sciencedirect.com/science/article/pii/0004370285900426}}
| |
| *{{cite book |first=I. |last=Niemelä |chapter=Decision procedure for autoepistemic logic |editor1-first=Ewing |editor1-last=Lusk |editor2-first=Ross |editor2-last=Overbeek |title=9th International Conference on Automated Deduction: Argonne, Illinois, USA, May 23-26, 1988. Proceedings |chapterurl=http://books.google.com/books?id=xd57Kjl60q0C&pg=PA675 |year=1988 |publisher=Springer |isbn=978-3-540-19343-2 |pages=675–684 |volume=310 |series=Lecture Notes in Computer Science}}
| |
| {{refend}}
| |
| | |
| [[Category:Logic programming]]
| |
| [[Category:Modal logic]]
| |
| [[Category:Knowledge]]
| |
Greetings! I am Marvella and I really feel comfy when people use the full name. His family lives in South Dakota but his wife wants them to move. To do aerobics is a thing that I'm totally addicted to. My working day job is a meter reader.
Feel free to surf to my web blog ... std testing at home, Read the Full Guide,