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Topic: Negation normal form

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DeMorgans Law

	Florida Tech, CSE3001: ML Project (Fall 2002)
		A literal is an atom or its negation.
		A \/ B A \/ B (negation normal form) A \/ B (conjunctive normal form, one conjunct)
		Since all conjuncts contain an atom and its negation, the proposition is a tautology.
	www.cs.fit.edu /~ryan/cse4250/proj-taut.html (710 words)

	Normal Forms and Skolem Functions (Site not responding. Last check: 2007-10-10)
		Normal forms are based on the expressing formulas in terms of negation, conjunction, disjunction, and the quantifiers, {¬, /\, \/, /\x, \/x}.
		The procedure to convert a formula to negation normal form is to recursively replace formulas appearing on the left with formulas appearing on the right.
		A formula is placed in prenix normal form by recursively moving quantifiers outward so that all quantifiers appear at the beginning of the formula.
	cs.wwc.edu /~aabyan/Logic/normal.html (671 words)

	Andrews. To Truth through Proof. (Site not responding. Last check: 2007-10-10)
		Absolute consistency (there is an unprovable wff) and consistency with respect to negation (there is no wff such that it and its negation are provable) are defined for general logical systems and a condition is given under which the two become equivalent (the provability of "A implies.
		Negation normal form is defined, and the two-dimensional representation is introduced (conjuncts=vertical paths and disjuncts=horizontal paths).
		The system F is modified to include existential quantifiers as primitive and called system E. A wff G in E is in prenex normal form iff no vacuous quantifier occurs and no quantifier occurs in the scope of a propositional connective.
	www.andrew.cmu.edu /user/cebrown/notes/tttp.html (6301 words)

	[No title]
		A rules is either a literal or of the form {\em H $\leftarrow$ Body} (written ``{\tt if(H,Body)}'') where $H$ is a literal and $Body$ is a conjunction and disjunction of literals.
		Negation normal form of a formula is an equivalent formula consisting of conjunctions and disjunctions of literals (either an atom or of the form $n(A)$ where $A$ is an atom).
		Usually we want to find the negation normal form of the negation of the formula, as this is the form suitable for use in the body of a rule.
	www.cs.uu.nl /docs/vakken/mbr/2002-2003/theorist.pl (5031 words)

	negation - W3C RIF-WG Wiki (Site not responding. Last check: 2007-10-10)
		In the ongoing discussion about monotonic vs. non-monotonic negation (better known as negation as failure), it is often overlooked that not only non-monotonicity is an issue but actually there are several definitions of both monotonic and non-monotonic forms of negation in the literature) with subtle but important differences.
		This overview is by no means to be understood to be exhaustive, but shall cover and clarify the most prominent forms of negation in the context of logics and rule languages, especially those mentioned in discussions within the RIF working group.
		Stable models are not necessarily unique, a set of normal rules may have several stable models (as in the case of the even loop), or it may have no stable model (as in the case of the odd loop).
	www.w3.org /2005/rules/wg/wiki/negation (1873 words)

	Normal Forms and Skolem Functions (Site not responding. Last check: 2007-10-10)
		Normal forms are based on the expressing formulas in terms of negation, conjunction, disjunction, and the quantifiers, {
		A formula in NNF is placed in the conjunctive normal form by recursively moving disjunctions inward and conjunctions outward using the following rewriting rules.
		A formula in NNF is placed in the disjunctive normal form by recursively moving conjunctions inward and disjunctions outward using the following rewriting rules.
	cs.wwc.edu /~aabyan/Logic/Book/book/node33.html (561 words)

	'DYNAMIC' INFERENCING WITH GENERALIZED RESOLUTION
		A conjunctive normal form (CNF) is a conjunction of the form A1 and A2 and...
		In its simplest form, binary resolution relies upon a single rule of inference - a generalization of the law of disjunctive syllogism (from formulas of the form A v B and -A we may conclude B), the general form of which is:
		Haken [3] proves that "...for infinitely many disjunctive normal form propositional calculus tautologies e, the length of the shortest resolution proof of e cannot be bounded by any polynomial of the length of e." Haken further explains that the tautologies which he used were introduced by Cook and Reckhow [1] and encode the pigeonhole principle.
	berghel.net /publications/atp2/atp2.php (4708 words)

	Deduction in Classical and Multiple-Valued Logic
		Although such normalization simplifies analysis and implementation, it can be computationally expensive: Normalizing a formula may cause a great increase in size.
		(These normal forms are often collectively referred to as clause form.) The fundamental ideas behind such frameworks had their genesis in my 1982 Artificial Intelligence Journal paper ``Completely Non-Clausal Theorem Proving.'' It was one of the first formal treatments of mechanized theorem proving without requiring CNF or DNF.
		Formulas in DNNF are linkless, in negation normal form (NNF), and have the property that atoms are not shared across conjunctions.
	www.cs.albany.edu /~nvm/research_lay-abbrev.html (1674 words)

	Research Interests and Selected Publications (Site not responding. Last check: 2007-10-10)
		The development of inference techniques for negation normal form (NNF) formulas and related tableau-based techniques is central.
		These techniques are relevant not only for automated theorem provers, but for other systems; examples are deductive databases, systems based on logic programming, and other AI systems with an inferencing component such as deduction-based program synthesis and non-monotonic reasoning systems.
		These issues are closely related to recent developments in ``Decomposable Negation Normal Form (DNNF)'' (Darwiche, J.ACM (48,4), July 2001), and to our paper ``Tableaux, Path Dissolution, and Decomposable Negation Normal Form for Knowledge Compilation,'' Proceedings of the International Conference TABLEAUX 2003 - Analytic Tableaux and Related Methods, Rome, Italy, September 2003.
	www.cs.albany.edu /~nvm/res_ints.html (408 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Converts a term already in negation normal form into conjunctive normal form.
		When applied to a term already in negation normal form (see NNF_CONV), meaning that all other propositional connectives have been eliminated in favour of conjunction, disjunction and negation, and negation is only applied to atomic formulas, WEAK_CNF_CONV puts the term into an equivalent conjunctive normal form, which is a conjunction of disjunctions.
		The ordering and associativity of the resulting form are not guaranteed, and it may contain duplicates.
	www.cl.cam.ac.uk /users/jrh13/hol-light/HTML/WEAK_CNF_CONV.html (168 words)

	The ctlsp package
		This function tests each LTL formula in negation normal form to see if it is a propositional formula.
		This function tests an LTL formula in negation normal form to see if it is a propositional formula.
		A literal is an atomic proposition or the negation of an atomic popposition.
	vlsi.colorado.edu /~vis/doc/html/ctlspExtDet.html (2074 words)

	What's New in TPS and ETPS
		NNF, NNF-EXPAND: Converts a line to or from negation normal form.
		When the user sets a mode, all important flags are now set to their default values unless explicitly set by the mode.
		There is a new version of NAT-ETREE for translating from a normal natural deduction proof to an expansion proof.
	gtps.math.cmu.edu /whatsnew.html (2107 words)

	Philosophy 112
		Use Double Negation, DeMorgan's Laws, Idempotence, and Communitivity to simplify the sentences as much as possible and put it into negation normal form.
		Thus, the negation of a logically true sentence will be false on all non-spurious rows.
		And the negation of a logically false sentence will be true on all non-spurious rows.
	www.philosophy.ilstu.edu /horvath/PHI112/Exam1PS.htm (716 words)

	CIS 301 Captain's log
		Using DeMorgan, and double negation, a sentence can be converted to Negation Normal Form (NNF).
		Disjunction elimination corresponds to "proof by cases"; negation introduction corresponds to "proof by contradiction".
		In addition to expressing the four Aristotelian forms, we address how to translate numeric claims such as "exactly one" or "at most one".
	www.cis.ksu.edu /~tamtoft/CIS301/Fall05/log.html (2068 words)

	Amplifying Electrochemical Indicators
		In the first step of a two-step process, the system description is converted into a logical expression, in what is known as a conjunctive normal form (CNF) that is based on the effects and preconditions of actions in an n-step plan.
		In the second step, the aforementioned research results are used to convert the CNF representation into a decomposable negation normal form (DNNF) representation.
		It turns out that the computation time needed to evaluate a DNNF expression to compute an optimal n-step plan increases only linearly with the DNNF representation size.
	www.nasatech.com /Briefs/Jan05/NPO40296.html (677 words)

	Conjunctive Normal Form
		First, we define a literal to be either a propositional letter or its negation, e.g.
		Suppose that after removing the implication operators and pushing negation in as far as possible, we have a disjunction:
		This is the property that we need for the resolution procedure (see below) to operate correctly.
	www.mathcs.duq.edu /simon/Summer01/notes-6-20/node2.html (266 words)

	Journal of the ACM Bibliography (Site not responding. Last check: 2007-10-10)
		Path dissolution, a rule of inference that operates on formulas in negation normal form and that employs a representation called semantic graphs, is introduced.
		Furthermore, one need not (and cannot) restrict attention to conjunctive normal form (CNF) when employing dissolution: A single application (even to a CNF formula) generally produces a non-CNF formula that is more compact than any of its CNF equivalents.
		Path dissolution is a global rule; as such, it is employed at the first order level differently from the way locally-oriented techniques such as resolution are.
	theory.lcs.mit.edu /~jacm/References/murrayr1993:504.html (254 words)

	CS206: Formal Methods in Computer Science
		Normal forms of propositional logic formulae: negation normal form (NNF), conjunctive normal form (CNF), disjunctive normal form (DNF).
		Proof that any formula can be converted to a semantically equivalent CNF formula.
		Prenex normal forms and use of quantifier equivalences in obtaining prenex normal forms.
	www.cse.iitb.ac.in /~supratik/courses/cs206 (1308 words)

	redlog User Manual - Miscellaneous Normal Forms (Site not responding. Last check: 2007-10-10)
		Go to the first, previous, next, last section, table of contents.
		Note that in all contexts, we use languages such that all negations can be encoded by relations (see section Format and Handling of Formulas).
		The number of quantifier changes in the result is minimal among all prenex normal forms that can be obtained from
	www.fmi.uni-passau.de /~redlog/htmldoc/redlog_33.html (205 words)

	Simplification of Many-Valued Logic Formulas Using Anti-Links -- BECKERT et al. 8 (4): 569 -- Journal of Logic and ...
		form of a given formula; rather, one removes certain forms of
		main advantage is that no clausal normal form has to be computed
		logic formulas called signed NNF and it is shown that all interesting
	logcom.oxfordjournals.org /cgi/content/abstract/8/4/569 (246 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Converts a term already in negation normal form into disjunctive normal form.
		When applied to a term already in negation normal form (see NNF_CONV), meaning that all other propositional connectives have been eliminated in favour of disjunction, disjunction and negation, and negation is only applied to atomic formulas, WEAK_DNF_CONV puts the term into an equivalent disjunctive normal form, which is a disjunction of conjunctions.
		See DNF_CONV for a stronger (but somewhat slower) variant where this is important.
	www.cl.cam.ac.uk /~jrh13/hol-light/HTML/WEAK_DNF_CONV.html (148 words)

	c2d (Site not responding. Last check: 2007-10-10)
		c2d is a system that compiles CNF into d-DNNF (deterministic, decomposable negation normal form).
		d-DNNF is a tractable logical form that allows operations such as model counting, clausal entailment, and model enumeration and minimization to be performed in linear time.
		Send questions and comments to darwiche at cs.ucla.edu.
	reasoning.cs.ucla.edu /c2d (85 words)

	DBLP: Erik Rosenthal (Site not responding. Last check: 2007-10-10)
		Reiner Hähnle, Neil V. Murray, Erik Rosenthal: Normal Forms for Knowledge Compilation.
		Neil V. Murray, Erik Rosenthal: Tableaux, Path Dissolution, and Decomposable Negation Normal Form for Knowledge Compilation.
		Reiner Hähnle, Neil V. Murray, Erik Rosenthal: Completeness for Linear Regular Negation Normal Form Inference Systems.
	www.acm.org /turing/sigmod/dblp/db/indices/a-tree/r/Rosenthal:Erik.html (544 words)

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