|
|
| Line 1: |
Line 1: |
| In [[logic]], a '''functionally complete''' set of [[logical connective]]s or [[Boolean function|Boolean operators]] is one which can be used to express all possible [[truth table]]s by combining members of the set into a [[Boolean expression]].<ref name="Enderton2001">{{citation | last1=Enderton | first1=Herbert | title=A mathematical introduction to logic | publisher=[[Academic Press]] | location=Boston, MA | edition=2nd | isbn=978-0-12-238452-3 | year=2001}}. ("Complete set of logical connectives").</ref><ref name="Nolt1998">{{citation | last1=Nolt | first1=John | last2=Rohatyn | first2=Dennis | last3=Varzi | first3=Achille | title=Schaum's outline of theory and problems of logic | publisher=[[McGraw–Hill]] | location=New York | edition=2nd | isbn=978-0-07-046649-4 | year=1998}}. ("[F]unctional completeness of [a] set of logical operators").</ref> A well-known complete set of connectives is { AND, NOT }, consisting of binary [[logical conjunction|conjunction]] and [[logical negation|negation]]. The [[singleton (mathematics)|singleton]] sets { [[Sheffer stroke|NAND]] } and { [[Logical NOR|NOR]] } are also functionally complete.
| | Irwin Butts is what my wife enjoys to call me though I don't really like becoming known as like that. [http://www.3Fatchicks.com/5-harmful-types-of-bacteria/ Puerto Rico] is where he's usually been living but she [http://Www.Cosmopolitan.com/sex-love/advice/lube-during-sex requirements] to move because of her family members. One of over [http://c-batang.co.kr/?document_srl=485588&mid=levelup_board std testing at home] testing [http://www.siccus.net/blog/15356 at home std test] home the counter std test ([http://happywedding.lonnieheart.gethompy.com/zbxe/wc_apply/372747 visit the following website]) issues she loves most is to do aerobics and now she is trying to earn money with it. For many years he's been operating as over std home test the counter std test a receptionist. |
| | |
| In a context of [[propositional logic]], functionally complete sets of connectives are also called '''(expressively) adequate'''.<ref name="Smith2003">{{Citation | last1=Smith | first1=Peter | title=An introduction to formal logic | publisher=[[Cambridge University Press]] | isbn=978-0-521-00804-4 | year=2003}}. (Defines "expressively adequate", shortened to "adequate set of connectives" in a section heading.)</ref>
| |
| | |
| From the point of view of [[digital electronics]], functional completeness means that every possible [[logic gate]] can be realized as a network of gates of the types prescribed by the set. In particular, all logic gates can be assembled from either only binary [[NAND gate]]s, or only binary [[NOR gate]]s.
| |
| | |
| ==Formal definition==
| |
| Given the [[Boolean domain]] '''B''' = {0,1}, a set ''F'' of Boolean functions ''ƒ''<sub>i</sub>: '''B'''<sup>''n<sub>i</sub>''</sup> → '''B''' is '''functionally complete''' if the [[clone (algebra)|clone]] on '''B''' generated by the basic functions ''ƒ''<sub>i</sub> contains all functions ''ƒ'': '''B'''<sup>''n''</sup> → '''B''', for all ''strictly positive'' integers {{nowrap|''n'' ≥ 1}}. In other words, the set is functionally complete if every Boolean function that takes at least one variable can be expressed in terms of the functions ''ƒ''<sub>i</sub>. Since every Boolean function of at least one variable can be expressed in terms of binary Boolean functions, ''F'' is functionally complete if and only if every binary Boolean function can be expressed in terms of the functions in ''F''.
| |
| | |
| A more natural condition would be that the clone generated by ''F'' consist of all functions ''ƒ'': '''B'''<sup>''n''</sup> → '''B''', for all integers {{nowrap|''n'' ≥ 0}}. However, the examples given above are not functionally complete in this stronger sense because it is not possible to write a [[arity|nullary]] function, i.e. a constant expression, in terms of ''F'' if ''F'' itself does not contain at least one nullary function. With this stronger definition, the smallest functionally complete sets would have 2 elements.
| |
| | |
| Another natural condition would be that the clone generated by ''F'' together with the two nullary constant functions be functionally complete or, equivalently, functionally complete in the strong sense of the previous paragraph. The example of the Boolean function given by ''S''(''x'', ''y'', ''z'') = ''z'' if ''x'' = ''y'' and ''S''(''x'', ''y'', ''z'') = ''x'' otherwise shows that this condition is strictly weaker than functional completeness.<ref name=Wesselkamper1975a>{{citation
| |
| | title = A sole sufficient operator
| |
| | url = http://projecteuclid.org/euclid.ndjfl/1093891614
| |
| | year = 1975
| |
| | author = Wesselkamper, T.C.
| |
| | journal = Notre Dame Journal of Formal Logic
| |
| | volume = 16
| |
| | pages = 86–88
| |
| | doi = 10.1305/ndjfl/1093891614
| |
| }}</ref><ref name=Massey1975>{{citation
| |
| | title = Concerning an alleged Sheffer function
| |
| | url = http://projecteuclid.org/euclid.ndjfl/1093891898
| |
| | year = 1975
| |
| | author = Massey, G.J.
| |
| | journal = Notre Dame Journal of Formal Logic
| |
| | volume = 16
| |
| | pages = 549–550
| |
| | doi = 10.1305/ndjfl/1093891898
| |
| | issue = 4
| |
| }}</ref><ref name=Wesselkamper1975b>{{citation
| |
| | title = A Correction To My Paper" A. Sole Sufficient Operator
| |
| | url = http://projecteuclid.org/euclid.ndjfl/1093891899
| |
| | year = 1975
| |
| | author = Wesselkamper, T.C.
| |
| | journal = Notre Dame Journal of Formal Logic
| |
| | volume = 16
| |
| | pages = 551
| |
| | doi = 10.1305/ndjfl/1093891899
| |
| | issue = 4
| |
| }}</ref>
| |
| | |
| ==Informal definition==
| |
| Modern texts on logic typically take as primitive some subset of the connectives: [[logical conjunction|conjunction]] (<math>\land</math>), or K''pq''; [[logical disjunction|disjunction]] (<math>\lor</math>), or A''pq''; [[negation]] (<math>\neg</math>), N''p''; or [[material conditional]] (<math>\to</math>), or C''pq''; and possibly the [[Logical biconditional|biconditional]] (<math>\leftrightarrow</math>), or E''pq''. These connectives are functionally complete. However, they do not form a minimal functionally complete set, as the conditional and biconditional may be defined as:
| |
| | |
| :<math>\begin{align}
| |
| A \to B &:= \neg A \lor B\\
| |
| A \leftrightarrow B &:= (A \to B) \land (B \to A).
| |
| \end{align}</math>
| |
| | |
| So <math>\{\neg, \land, \lor\}</math> is also functionally complete. But then, <math>\lor</math> can be defined as
| |
| | |
| :<math>A \lor B := \neg(\neg A \land \neg B).</math>
| |
| | |
| <math>\land</math> can also be defined in terms of <math>\lor</math> in a similar manner.
| |
| | |
| It is also the case that <math> \vee </math> can be defined in terms of <math> \rightarrow </math> as follows:
| |
| | |
| :<math> \ A \vee B := (A \rightarrow B) \rightarrow B. </math>
| |
| | |
| No further simplifications are possible. Hence <math>\neg</math> and one of <math>\{\land, \lor, \rightarrow\}</math> are each minimal functionally complete [[subset]]s of <math>\{\neg, \land, \lor, \to, \leftrightarrow\}</math>.
| |
| | |
| ==Characterization of functional completeness==
| |
| {{further2|[[Post's lattice]]}}
| |
| [[Emil Leon Post|Emil Post]] proved that a set of logical connectives is functionally complete if and only if it is not a subset of any of the following sets of connectives: | |
| | |
| * The [[monotonic]] connectives; changing the truth value of any connected variables from '''F''' to '''T''' without changing any from '''T''' to '''F''' never makes these connectives change their return value from '''T''' to '''F''', e.g. <math>\vee</math>, <math>\wedge</math>, <math>\top</math>, <math>\bot</math>.
| |
| | |
| * The [[affine transformation|affine]] connectives, such that each connected variable either always or never affects the truth value these connectives return, e.g. <math>\neg</math>, <math>\top</math>, <math>\bot</math>, <math>\leftrightarrow</math>, <math>\not\leftrightarrow</math>.
| |
| | |
| * The '''self-dual''' connectives, which are equal to their own [[de Morgan dual]]; if the truth values of all variables are reversed, so is the truth value these connectives return, e.g. <math>\neg</math>, ''[[majority function|MAJ]]''(''p'',''q'',''r'').
| |
| | |
| * The '''truth-preserving''' connectives; they return the [[truth value]] '''T''' under any interpretation which assigns '''T''' to all variables, e.g. <math>\vee</math>, <math>\wedge</math>, <math>\top</math>, <math>\rightarrow</math>, <math>\leftrightarrow</math>.
| |
| | |
| * The '''falsity-preserving''' connectives; they return the truth value '''F''' under any interpretation which assigns '''F''' to all variables, e.g. <math>\vee</math>, <math>\wedge</math>, <math>\bot</math>, <math>\not\rightarrow</math>, <math>\not\leftrightarrow</math>.
| |
| | |
| In fact, Post gave a complete description of the [[lattice (order)|lattice]] of all [[clone (algebra)|clone]]s (sets of operations closed under composition and containing all projections) on the two-element set {'''T''', '''F'''}, nowadays called [[Post's lattice]], which implies the above result as a simple corollary: the five mentioned sets of connectives are exactly the maximal clones.
| |
| | |
| ==Minimal functionally complete operator sets==
| |
| When a single logical connective or Boolean operator is functionally complete by itself, it is called a '''Sheffer function'''<ref name=Martin1989>The term was originally restricted to ''binary'' operations, but since the end of the 20th century it is used more generally. {{citation
| |
| | title = Systems of logic
| |
| | year = 1989
| |
| | author = Martin, N.M. | |
| | publisher = Cambridge University Press
| |
| | isbn = 978-0-521-36770-7
| |
| | page = 54
| |
| }}.</ref> or sometimes a '''sole sufficient operator'''. There are no [[unary operation|unary]] operators with this property, and the only binary Sheffer functions — [[Sheffer stroke|NAND]] and [[Logical NOR|NOR]] are [[Boolean_algebra#Duality_principle|dual]]. These were discovered but not published by [[Charles Sanders Peirce]] around 1880, and rediscovered independently and published by [[Henry M. Sheffer]] in 1913.<ref name=Scharle1965>{{Citation
| |
| | title = Axiomatization of propositional calculus with Sheffer functors
| |
| | url = http://projecteuclid.org/euclid.ndjfl/1093958259
| |
| | year = 1965
| |
| | author = Scharle, T.W.
| |
| | journal = Notre Dame J. Formal Logic
| |
| | pages = 209–217
| |
| | volume = 6
| |
| | doi = 10.1305/ndjfl/1093958259
| |
| | issue = 3
| |
| }}.</ref>
| |
| In digital electronics terminology, the binary [[NAND gate]] and the binary [[NOR gate]] are the only binary [[universal logic gate]]s.
| |
| | |
| The following are the minimal functionally complete sets of logical connectives with [[arity]] ≤ 2:<ref name="Wernick">Wernick, William (1942) "Complete Sets of Logical Functions," ''Transactions of the American Mathematical Society 51'': 117–32. In his list on the last page of the article, Wernick does not distinguish between ← and →, or between <math>\not\leftarrow</math> and <math>\not\rightarrow</math>.</ref>
| |
| | |
| ;One element: {NAND}, {NOR}.
| |
| ;Two elements: {<math>\vee</math>, ¬}, {<math>\wedge</math>, ¬}, {→, ¬}, {←, ¬}, {→, <math>\bot</math>}, {←, <math>\bot</math>}, {→, <math>\not\leftrightarrow</math>}, {←, <math>\not\leftrightarrow</math>}, {→, <math>\not\to</math>}, {→, <math>\not\leftarrow</math>}, {←, <math>\not\to</math>}, {←, <math>\not\leftarrow</math>}, {<math>\not\to</math>, ¬}, {<math>\not\leftarrow</math>, ¬}, {<math>\not\to</math>, <math>\top</math>}, {<math>\not\leftarrow</math>, <math>\top</math>}, {<math>\not\to</math>, <math>\leftrightarrow</math>}, {<math>\not\leftarrow</math>, <math>\leftrightarrow</math>}.
| |
| ;Three elements: {<math>\lor</math>, <math>\leftrightarrow</math>, <math>\bot</math>}, {<math>\lor</math>, <math>\leftrightarrow</math>, <math>\not\leftrightarrow</math>}, {<math>\lor</math>, <math>\not\leftrightarrow</math>, <math>\top</math>}, {<math>\land</math>, <math>\leftrightarrow</math>, <math>\bot</math>}, {<math>\land</math>, <math>\leftrightarrow</math>, <math>\not\leftrightarrow</math>}, {<math>\land</math>, <math>\not\leftrightarrow</math>, <math>\top</math>}.
| |
| | |
| There are no minimal functionally complete sets of more than three at most binary logical connectives.<ref name="Wernick" /> Constant unary or binary connectives and binary connectives that depend only on one of the arguments have been suppressed to keep the list readable. E.g. the set consisting of binary <math>\vee</math> and the binary connective given by negation of the first argument (ignoring the second) is another minimal functionally complete set.
| |
| | |
| == Examples ==
| |
| | |
| * Examples of using the <code>NAND</code> completeness. As illustrated by,<ref>"NAND Gate Operations" at http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nand.html</ref>
| |
| ** ¬A = A <code>NAND</code> A
| |
| ** A ∧ B = ¬(A <code>NAND</code> B) = (A <code>NAND</code> B) <code>NAND</code> (A <code>NAND</code> B)
| |
| ** A ∨ B = (A <code>NAND</code> A) <code>NAND</code> (B <code>NAND</code> B)
| |
| | |
| * Examples of using the <code>NOR</code> completeness. As illustrated by,<ref>"NOR Gate Operations" at http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nor.html</ref>
| |
| ** ¬A = A <code>NOR</code> A
| |
| ** A ∧ B = (A <code>NOR</code> A) <code>NOR</code> (B <code>NOR</code> B)
| |
| ** A ∨ B = (A <code>NOR</code> B) <code>NOR</code> (A <code>NOR</code> B)
| |
| | |
| Note that, an electronic circuit or a software function is optimized by the reuse, that reduce the number of gates. For instance, the "A ∧ B" operation, when expressed by <code>NAND</code> gates, is implemented with the reuse of "A <code>NAND</code> B",
| |
| : X = (A <code>NAND</code> B); A ∧ B = X <code>NAND</code> X
| |
| | |
| == In other domains ==
| |
| Apart from logical connectives (Boolean operators), functional completeness can be introduced in other domains. For example, a set of [[reversible computation|reversible]] gates is called functionally complete, if it can express every reversible operator.
| |
| | |
| The 3-input [[Fredkin gate]] is functionally complete reversible gate by itself – a sole sufficient operator. There are many other [[three-input universal logic gate]]s, such as the [[Toffoli gate]].
| |
| | |
| == Set theory ==
| |
| There is an [[isomorphism]] between the [[Algebra of sets]] and the [[Boolean algebra]], that is, they have the same [[Boolean algebra (structure)|structure]]. Then, if we map boolean operators into set operators, the "translated" above text are valid also for sets: there are many "minimal complete set of set-theory operators" that can generate any other set relations. The more popular "Minimal complete operator sets" are {¬, ∩} and {¬, ∪}.
| |
| | |
| ==See also==
| |
| * [[Algebra of sets]]
| |
| * [[Boolean algebra]]
| |
| * Neither [[Completeness (order theory)]] nor [[Total order]] are in the domain of discourse for this article.
| |
| | |
| ==References==
| |
| {{Reflist|30em}}
| |
| | |
| [[Category:Boolean algebra]]
| |
| [[Category:Propositional calculus]]
| |
| [[Category:Logic in computer science]]
| |