text:
Third-party billing
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Third-party billing is a form of billing where an intermediary handles the invoicing and payment between a purchaser and a vendor.
Contents
1 Telecommunications
2 Transportation
3 Cramming
4 References
Telecommunications[edit]
In telecommunications, it can refer to an operator-assisted telephone call, usually in conjunction with the assistance of a live or automated telephone operator.
It can also refer to a service that allows consumers to make purchases through authorized merchants, service providers and telecommunications companies, and have charges placed directly on their local phone bill, as an alternate payment option.[1]
Third party billing serves nearly 12 million households in the United States, and handles hundreds of millions of authorized transactions for consumers and businesses each year.[2] Several businesses, including Fortune 500 companies, choose to have their services included on their phone bills to reduce administrative needs and costs.
Transportation[edit]
In transportation, third party billing is the transference of transportation charges to a party other than the shipper or consignee.
Cramming[edit]
Cramming happens when third party billing occurs on phones without consent of the person being billed.
References[edit]
Jump up ^ http://www.fcc.gov/encyclopedia/understanding-your-telephone-bill/
Jump up ^ http://www.bsgclearing.com/question_about_your_bill/consumer_faqs/
Retrieved from "https://en.wikipedia.org/w/index.php?title=Third-party_billing&oldid=497472590"
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text:
That's What Life Is All About (album)
From Wikipedia, the free encyclopedia
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That’s What Life Is All About
Studio album by Bing Crosby
Released
August 29, 1975
Recorded
October 17, 1974, February, 1975
Genre
Vocal
Label
United Artists (UAG-29730)
Producer
Ken Barnes
Bing Crosby chronology
A Southern Memoir
(1975)A Southern Memoir1975 That's What Life Is All About
(1975) Bingo Viejo
(1975)Bingo Viejo1975
Professional ratings
Review scores
Source
Rating
Allmusic [1]
That’s What Life Is All About is a 1975 vinyl album recorded by Bing Crosby for United Artists at Chappells in London in February 1975.[2] He was accompanied by Pete Moore and his Orchestra.[3]
The songs from the album were included on a 3-CD set called Bing Crosby: The Complete United Artists Sessions issued by EMI Records (7243 59808 2 4) in 1997. This included several previously unreleased alternate takes and studio chat.[4]
Contents
[hide]
1 Background
2 Personnel
3 Reception
4 Track listing
5 References
Background[edit]
In January 1974, Crosby was seriously ill and after two weeks of tests, he underwent three and a half hours of major surgery. Two-fifths of his left lung and an abscess the size of a small orange were removed. The tumor was a rare fungus called nocardia. There were concerns initially that he would not be able to sing again and his recuperation took many months. Record producer Ken Barnes later heard that Crosby was thinking of recording again and he quickly presented his credentials and eventually met Crosby on September 9, 1974 at the singer’s home.[5] Arrangements were soon made for two albums to be made in London. As a prelude, two songs were recorded with Johnny Mercer and a rhythm section at Mercer’s own “Heritage” recording studio at Oak Street in Burbank, California on October 17, 1974. Orchestra was added in February 1975.
Crosby flew to London on February 18 and that afternoon visited the Chappells recording studio. He was shown the vocal booth where he would be singing but said that he did not want to work in that way. He wanted to be with the band and said, “I didn’t come seven thousand miles to sing to a pane of glass.” This worried sound engineer John Timperley who was concerned about separation problems because of the large orchestra which was to be used. Crosby was adamant however and, in the event, he successfully overcame the problems of recording in front of a full orchestra.[6]
Over a period of eight days, Crosby, in six sessions, recorded twenty five tracks. Eleven of the tracks, plus the two Crosby had recorded with Johnny Mercer were issued on this LP. Thirteen, with the addition of another song to be recorded in 1976, were issued on an LP called At My Time of Life and the final recording was included with eleven others made in 1976 and issued on an LP called Beautiful Memories.[7]
Personnel[edit]
Rhythm section - October 17, 1974: Jimmy Rowles (piano); Joe Mondragon (bass) and Alvin Stoller (drums).
Full details are not given for the recordings in February 1975 but the 3-CD set Bing Crosby: The Complete United Artists Sessions mentions the following:
"The Best Things in Life Are Free" - Kenny Baker, muted trumpet
"Have a Nice Day" - Tommy Reilly, harmonica
Reception[edit]
The album entered the UK album charts on September 20, 1975 and peaked at No. 28 during its 6 weeks in the charts.
The UK magazine The Gramophone reviewed the album saying: "...Crosby’s voice has lost very little power and presence despite his septuagenarian status and the major lung operation he underwent last year, and, estimably aided by the outstanding arrangements of Pete Moore and the first-class orchestra directed by Moore, this album is another landmark in a uniquely long and distinguished career.[8]
Variety commented: “All right, Bing Crosby's voice isn’t as great as it was 20, 30 or whatever years ago. So what? He’s a brilliant stylist and there’s plenty of voice left as he demonstrates here on 13 tracks…”[9]
Track listing[edit]
Side one
No.
Title
Writer(s)
Length
1. "That's What Life Is All About" Les Reed, Peter Dacre, Ken Barnes, Bing Crosby 3:28
2. "Breezin' Along with the Breeze" Richard Whiting, Haven Gillespie, Seymour Simons 2:58
3. "No Time at All" Stephen Schwartz 3:52
4. "I Love to Dance Like They Used to Dance" Billy Goldenberg, Alan and Marilyn Bergman 3:00
5. "Have a Nice Day" Hoyt Axton 3:22
6. "The Pleasure of Your Company" / "Roamin' in the Gloamin'" (with Johnny Mercer) André Previn, Johnny Mercer, Harry Lauder 4:08
Side two
No.
Title
Writer(s)
Length
7. "The Best Things in Life Are Free" Buddy DeSylva, Lew Brown, Ray Henderson 2:26
8. "Some Sunny Day" Irving Berlin 3:11
9. "Bon Vivant" Robert Emmett Dolan, Johnny Mercer 3:49
10. "Good Companions – And Points Beyond" (with Johnny Mercer) André Previn, Johnny Mercer 3:39
11. "Send in the Clowns" Stephen Sondheim 3:16
12. "The Good Old Times" Pete Moore, Ken Barnes, 3:34
13. "That's What Life Is All About" (reprise) Les Reed, Peter Dacre, Ken Barnes, Bing Crosby 1:18
References[edit]
Jump up ^ Allmusic review
Jump up ^ "A Bing Crosby Discography". BING magazine. International Club Crosby. Retrieved May 28, 2017.
Jump up ^ Reynolds, Fred. The Crosby Collection 1926-1977 (Part Five: 1961-1977 ed.). John Joyce. pp. 160–184.
Jump up ^ "Allmusic". Allmusic. Retrieved August 8, 2015.
Jump up ^ Barnes, Ken (1980). The Crosby Years. Elm Tree Books. pp. 35–36. ISBN 0-241-10177-8.
Jump up ^ Barnes, Ken (1980). The Crosby Years. Elm Tree Books. pp. 43–44. ISBN 0-241-10177-8.
Jump up ^ Reynolds, Fred. The Crosby Collection 1926-1977 (Part Five: 1961-1977 ed.). John Joyce. pp. 160–184.
Jump up ^ "The Gramophone". The Gramophone. November 1975.
Jump up ^ "Variety". March 24, 1976.
[hide]
v
t
e
Bing Crosby
Discography
Filmography
Albums
Music of Hawaii (1939)
Victor Herbert Melodies, Vol. One (1939)
Patriotic Songs for Children (1939)
Cowboy Songs (1939)
Victor Herbert Melodies, Vol. Two (1939)
George Gershwin Songs, Vol. One (1939)
Ballad for Americans (1940)
Favorite Hawaiian Songs (1940)
Christmas Music (1940)
Star Dust (1940)
Hawaii Calls (1941)
Small Fry (1941)
Crosbyana (1941)
Under Western Skies (1941)
Song Hits from Holiday Inn (w/ Fred Astaire) (1942)
Merry Christmas (1945)
Selections from Going My Way (1945)
Selections from The Bells of St. Mary's (1946)
Don't Fence Me In (1946)
The Happy Prince (1946)
Selections from Road to Utopia (1946)
Bing Crosby – Stephen Foster (1946)
What We So Proudly Hail (1946)
Favorite Hawaiian Songs, Vol. One (1946)
Favorite Hawaiian Songs, Vol. Two (1946)
Blue Skies (w/ Fred Astaire) (1946)
Bing Crosby – Jerome Kern (1946)
St. Patrick's Day (1947)
Bing Crosby – Victor Herbert (1947)
Selections from Welcome Stranger (1947)
Our Common Heritage (1947)
El Bingo (1947)
The Small One (1947)
The Man Without a Country (1947)
Drifting and Dreaming (1947)
Blue of the Night (1948)
Selections from Showboat (1948)
The Emperor Waltz (1948)
St. Valentine's Day (1948)
Bing Crosby Sings with Al Jolson, Bob Hope, Dick Haymes and the Andrews Sisters (1948)
Selections from Road to Rio (1948)
Bing Crosby Sings with Judy Garland, Mary Martin, Johnny Mercer (1948)
Bing Crosby Sings with Lionel Hampton, Eddie Heywood, Louis Jordan (1948)
Auld Lang Syne (1948)
A Connecticut Yankee in King Arthur's Court (1949)
Bing Crosby Sings Songs by George Gershwin (1949)
South Pacific (1949)
Christmas Greetings (1949)
Ichabod – The Legend of Sleepy Hollow (1949)
Top o' the Morning / Emperor Waltz (1949)
Songs from Mr. Music (1950)
Go West Young Man (1950)
Le Bing: Song Hits of Paris (1953)
Some Fine Old Chestnuts (1954)
Selections from White Christmas (1954)
Bing: A Musical Autobiography (1954)
High Tor (1956)
A Christmas Sing with Bing Around the World (1956)
High Society (w/ Frank Sinatra, Grace Kelly, and Louis Armstrong) (1956)
Songs I Wish I Had Sung the First Time Around (1956)
Bing Sings Whilst Bregman Swings (1956)
Bing with a Beat (1957)
A Christmas Story (1957)
Ali Baba and the Forty Thieves (1957)
Never Be Afraid (1957)
Jack B. Nimble – A Mother Goose Fantasy (1957)
New Tricks (1957)
Fancy Meeting You Here ( w/ Rosemary Clooney) (1958)
How the West Was Won (1959)
Bing & Satchmo (w/ Louis Armstrong) (1960)
101 Gang Songs (1960)
Holiday in Europe (1960)
The Road to Hong Kong (1962)
On the Happy Side (1962)
On the Sentimental Side (1962)
I Wish You a Merry Christmas (1962)
Reprise Musical Repertory Theatre (1963)
Return to Paradise Islands (1963)
Bing Crosby Sings the Great Country Hits (1963)
America, I Hear You Singing (w/ Frank Sinatra and Fred Waring) (1964)
12 Songs of Christmas (w/ Frank Sinatra and Fred Waring) (1964)
That Travelin' Two-Beat (w/ Rosemary Clooney) (1965)
Bing 'n' Basie (w/ Count Basie) (1972)
A Couple of Song and Dance Men (w/ Fred Astaire) (1975)
Seasons (1977)
Bing Crosby: The Voice of Christmas (1998)
Family
Dixie Lee (first wife)
Gary Crosby (son)
Dennis Crosby (son)
Phillip Crosby (son)
Lindsay Crosby (daughter)
Kathryn Crosby (second wife)
Harry Crosby (son)
Mary Crosby (daughter)
Nathaniel Crosby (son)
Denise Crosby (granddaughter)
Larry Crosby (brother)
Bob Crosby (brother)
Related
Songs Bing Crosby recorded multiple times
A Tribute to Bing Crosby
Retrieved from "https://en.wikipedia.org/w/index.php?title=That%27s_What_Life_Is_All_About_(album)&oldid=802113619"
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text:
Skolem normal form
From Wikipedia, the free encyclopedia
Jump to: navigation, search
In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers.
Every first-order formula may be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization (sometimes spelled Skolemnization). The resulting formula is not necessarily equivalent to the original one, but is equisatisfiable with it: it is satisfiable if and only if the original one is satisfiable.[1]
Reduction to Skolem normal form is a method for removing existential quantifiers from formal logic statements, often performed as the first step in an automated theorem prover.
Contents
[hide]
1 Examples
2 How Skolemization works
3 Uses of Skolemization
4 Skolem theories
5 History
6 See also
7 Notes
8 References
9 External links
Examples[edit]
The simplest form of Skolemization is for existentially quantified variables which are not inside the scope of a universal quantifier. These may be replaced simply by creating new constants. For example, ∃ x P ( x ) {\displaystyle \exists xP(x)} may be changed to P ( c ) {\displaystyle P(c)} , where c {\displaystyle c} is a new constant (does not occur anywhere else in the formula).
More generally, Skolemization is performed by replacing every existentially quantified variable y {\displaystyle y} with a term f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} whose function symbol f {\displaystyle f} is new. The variables of this term are as follows. If the formula is in prenex normal form, x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are the variables that are universally quantified and whose quantifiers precede that of y {\displaystyle y} . In general, they are the variables that are quantified universally [clarification needed] and such that ∃ y {\displaystyle \exists y} occurs in the scope of their quantifiers. The function f {\displaystyle f} introduced in this process is called a Skolem function (or Skolem constant if it is of zero arity) and the term is called a Skolem term.
As an example, the formula ∀ x ∃ y ∀ z . P ( x , y , z ) {\displaystyle \forall x\exists y\forall z.P(x,y,z)} is not in Skolem normal form because it contains the existential quantifier ∃ y {\displaystyle \exists y} . Skolemization replaces y {\displaystyle y} with f ( x ) {\displaystyle f(x)} , where f {\displaystyle f} is a new function symbol, and removes the quantification over y {\displaystyle y} . The resulting formula is ∀ x ∀ z . P ( x , f ( x ) , z ) {\displaystyle \forall x\forall z.P(x,f(x),z)} . The Skolem term f ( x ) {\displaystyle f(x)} contains x {\displaystyle x} , but not z {\displaystyle z} , because the quantifier to be removed ∃ y {\displaystyle \exists y} is in the scope of ∀ x {\displaystyle \forall x} , but not in that of ∀ z {\displaystyle \forall z} ; since this formula is in prenex normal form, this is equivalent to saying that, in the list of quantifiers, x {\displaystyle x} precedes y {\displaystyle y} while z {\displaystyle z} does not. The formula obtained by this transformation is satisfiable if and only if the original formula is.
How Skolemization works[edit]
Skolemization works by applying a second-order equivalence in conjunction to the definition of first-order satisfiability. The equivalence provides a way for "moving" an existential quantifier before a universal one.
∀ x ( R ( g ( x ) ) ∨ ∃ y R ( x , y ) ) ⟺ ∃ f ∀ x ( R ( g ( x ) ) ∨ R ( x , f ( x ) ) ) {\displaystyle \forall x{\Big (}R(g(x))\vee \exists yR(x,y){\Big )}\iff \exists f\forall x{\Big (}R(g(x))\vee R(x,f(x)){\Big )}}
where
f ( x ) {\displaystyle f(x)} is a function that maps x {\displaystyle x} to y {\displaystyle y} .
Intuitively, the sentence "for every x {\displaystyle x} there exists a y {\displaystyle y} such that R ( x , y ) {\displaystyle R(x,y)} " is converted into the equivalent form "there exists a function f {\displaystyle f} mapping every x {\displaystyle x} into a y {\displaystyle y} such that, for every x {\displaystyle x} it holds R ( x , f ( x ) ) {\displaystyle R(x,f(x))} ".
This equivalence is useful because the definition of first-order satisfiability implicitly existentially quantifies over the evaluation of function symbols. In particular, a first-order formula Φ {\displaystyle \Phi } is satisfiable if there exists a model M {\displaystyle M} and an evaluation μ {\displaystyle \mu } of the free variables of the formula that evaluate the formula to true. The model contains the evaluation of all function symbols; therefore, Skolem functions are implicitly existentially quantified. In the example above, ∀ x . R ( x , f ( x ) ) {\displaystyle \forall x.R(x,f(x))} is satisfiable if and only if there exists a model M {\displaystyle M} , which contains an evaluation for f {\displaystyle f} , such that ∀ x . R ( x , f ( x ) ) {\displaystyle \forall x.R(x,f(x))} is true for some evaluation of its free variables (none in this case). This may be expressed in second order as ∃ f ∀ x . R ( x , f ( x ) ) {\displaystyle \exists f\forall x.R(x,f(x))} . By the above equivalence, this is the same as the satisfiability of ∀ x ∃ y . R ( x , y ) {\displaystyle \forall x\exists y.R(x,y)} .
At the meta-level, first-order satisfiability of a formula Φ {\displaystyle \Phi } may be written with a little abuse of notation as ∃ M ∃ μ . ( M , μ ⊨ Φ ) {\displaystyle \exists M\exists \mu ~.~(M,\mu \models \Phi )} , where M {\displaystyle M} is a model, μ {\displaystyle \mu } is an evaluation of the free variables, and ⊨ {\displaystyle \models } means that Φ {\displaystyle \Phi } is true in M {\displaystyle M} under μ {\displaystyle \mu } . Since first-order models contain the evaluation of all function symbols, any Skolem function Φ {\displaystyle \Phi } contains is implicitly existentially quantified by ∃ M {\displaystyle \exists M} . As a result, after replacing an existential quantifier over variables into an existential quantifiers over functions at the front of the formula, the formula still may be treated as a first-order one by removing these existential quantifiers. This final step of treating ∃ f ∀ x . R ( x , f ( x ) ) {\displaystyle \exists f\forall x.R(x,f(x))} as ∀ x . R ( x , f ( x ) ) {\displaystyle \forall x.R(x,f(x))} may be completed because functions are implicitly existentially quantified by ∃ M {\displaystyle \exists M} in the definition of first-order satisfiability.
Correctness of Skolemization may be shown on the example formula F 1 = ∀ x 1 … ∀ x n ∃ y R ( x 1 , … , x n , y ) {\displaystyle F_{1}=\forall x_{1}\dots \forall x_{n}\exists yR(x_{1},\dots ,x_{n},y)} as follows. This formula is satisfied by a model M {\displaystyle M} if and only if, for each possible value for x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} in the domain of the model, there exists a value for y {\displaystyle y} in the domain of the model that makes R ( x 1 , … , x n , y ) {\displaystyle R(x_{1},\dots ,x_{n},y)} true. By the axiom of choice, there exists a function f {\displaystyle f} such that y = f ( x 1 , … , x n ) {\displaystyle y=f(x_{1},\dots ,x_{n})} . As a result, the formula F 2 = ∀ x 1 … ∀ x n R ( x 1 , … , x n , f ( x 1 , … , x n ) ) {\displaystyle F_{2}=\forall x_{1}\dots \forall x_{n}R(x_{1},\dots ,x_{n},f(x_{1},\dots ,x_{n}))} is satisfiable, because it has the model obtained by adding the evaluation of f {\displaystyle f} to M {\displaystyle M} . This shows that F 1 {\displaystyle F_{1}} is satisfiable only if F 2 {\displaystyle F_{2}} is satisfiable as well. In the other way around, if F 2 {\displaystyle F_{2}} is satisfiable, then there exists a model M ′ {\displaystyle M'} that satisfies it; this model includes an evaluation for the function f {\displaystyle f} such that, for every value of x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} , the formula R ( x 1 , … , x n , f ( x 1 , … , x n ) ) {\displaystyle R(x_{1},\dots ,x_{n},f(x_{1},\dots ,x_{n}))} holds. As a result, F 1 {\displaystyle F_{1}} is satisfied by the same model because one may choose, for every value of x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , the value y = f ( x 1 , … , x n ) {\displaystyle y=f(x_{1},\dots ,x_{n})} , where f {\displaystyle f} is evaluated according to M ′ {\displaystyle M'} .
Uses of Skolemization[edit]
One of the uses of Skolemization is automated theorem proving. For example, in the method of analytic tableaux, whenever a formula whose leading quantifier is existential occurs, the formula obtained by removing that quantifier via Skolemization may be generated. For example, if ∃ x . Φ ( x , y 1 , … , y n ) {\displaystyle \exists x.\Phi (x,y_{1},\ldots ,y_{n})} occurs in a tableau, where x , y 1 , … , y n {\displaystyle x,y_{1},\ldots ,y_{n}} are the free variables of Φ ( x , y 1 , … , y n ) {\displaystyle \Phi (x,y_{1},\ldots ,y_{n})} , then Φ ( f ( y 1 , … , y n ) , y 1 , … , y n ) {\displaystyle \Phi (f(y_{1},\ldots ,y_{n}),y_{1},\ldots ,y_{n})} may be added to the same branch of the tableau. This addition does not alter the satisfiability of the tableau: every model of the old formula may be extended, by adding a suitable evaluation of f {\displaystyle f} , to a model of the new formula.
This form of Skolemization is an improvement over "classical" Skolemization in that only variables that are free in the formula are placed in the Skolem term. This is an improvement because the semantics of tableau may implicitly place the formula in the scope of some universally quantified variables that are not in the formula itself; these variables are not in the Skolem term, while they would be there according to the original definition of Skolemization. Another improvement that may be used is applying the same Skolem function symbol for formulae that are identical up to variable renaming.[2]
Another use is in the resolution method for first order logic, where formulas are represented as sets of clauses understood to be universally quantified. (For an example see drinker paradox.)
Skolem theories[edit]
In general, if T {\displaystyle T} is a theory and for each formula F {\displaystyle F} with free variables x 1 , … , x n , y {\displaystyle x_{1},\dots ,x_{n},y} there is a Skolem function, then T {\displaystyle T} is called a Skolem theory.[3] For example, by the above, arithmetic with the Axiom of Choice is a Skolem theory.
Every Skolem theory is model complete, i.e. every substructure of a model is an elementary substructure. Given a model M of a Skolem theory T, the smallest substructure containing a certain set A is called the Skolem hull of A. The Skolem hull of A is an atomic prime model over A.
History[edit]
SNF is named after Thoralf Skolem.
See also[edit]
Herbrandization, the dual of Skolemization
Predicate functor logic
Notes[edit]
Jump up ^ "Normal Forms and Skolemization" (PDF). max planck institut informatik. Retrieved 15 December 2012.
Jump up ^ R. Hähnle. Tableaux and related methods. Handbook of Automated Reasoning.
Jump up ^ "Sets, Models and Proofs" (3.3) by I. Moerdijk and J. van Oosten
References[edit]
Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Skolem function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Skolemization on PlanetMath.org
Skolemization by Hector Zenil, The Wolfram Demonstrations Project.
Weisstein, Eric W. "SkolemizedForm". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Skolem_normal_form&oldid=798013127"
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values: If the formula is not in prenex form, also some existentially quantified variables may belong to the x, e.g. the formula ¬∃x ¬∃y. p(x,y) should be skolemized to is equivalent to ¬∃x. ¬ p(x,f(x)). (August 2013)

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values: \exists M \exists \mu ~.~ ( M,\mu \models \Phi)

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