08-07-2012, 04:09 AM
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Re: THE ARTIST ALGORITHM: OC, KC, 1C and XC, for eos, Kv>os
On Aug 7, 11:17*am, Martin <marty.musa...@gmail.com> wrote:
> > in any time future or past relates to reconstruction. Now, this set we > have established, contains data, accessible, and computable, by means > of a machine capable of performing computation runs instead of by tape > like does a Turing machine, but based on an improvization and > correction capable machine, such as a calculator or an abacus, exists > both in terms literal and abstract, or as software or code running on > a machine or computer. The set of numbers containing all information Not all programming languages have FETCH CYCLES that run command sequences. PROLOG works on SETS of FUNCTIONS by answering QUERIES with the UNIFY() Function. http://pro1og.com/UNIFY.png e.g. USER QUERY DATABASE FACT UNIFY( f3 (A, b) , f3 (c, D) ) ==> A=c, D=b Unify just recursively matches each function term, it's the same algorithm as used in chess programs but WHITE is OR and BLACK is AND (a conjunction of further functions to match). This method is Turing Machine Compatible. LISP <===> EVAL() <1 2 3 4..> PROLOG <===> UNIFY() {f1, f1, f2, f2, f2, f3, f3, f4, ...} Herc -- http://microPROLOG.com microPROLOG Syntax Rule 1 LINE. --> FACT. Rule 2 LINE. --> FACT IF TAIL. Rule 3 LINE? --> TAIL? Rule 4 TAIL --> FACT FACT ... FACT Rule 5 FACT --> term | VAR | [term TAIL] e.g LINE. --> FACT IF TAIL. (Rule 2) --> [term TAIL] IF TAIL. --> [term TAIL] IF FACT FACT. --> [happy FACT] IF [term TAIL] [term TAIL]. --> [happy FACT] IF [rich FACT] [beautiful FACT]. --> [happy PERS] IF [rich PERS] [beautiful PERS]. 
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08-07-2012, 07:27 AM
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Re: THE ARTIST ALGORITHM: OC, KC, 1C and XC, for eos, Kv>os
On Aug 6, 11:09*pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Aug 7, 11:17*am, Martin <marty.musa...@gmail.com> wrote: > > > > > in any time future or past relates to reconstruction. Now, this set we > > have established, contains data, accessible, and computable, by means > > of a machine capable of performing computation runs instead of by tape > > like does a Turing machine, but based on an improvization and > > correction capable machine, such as a calculator or an abacus, exists > > both in terms literal and abstract, or as software or code running on > > a machine or computer. The set of numbers containing all information > > Not all programming languages have FETCH CYCLES that run command > sequences. > > PROLOG works on SETS of FUNCTIONS by answering QUERIES with the > UNIFY() Function. > > http://pro1og.com/UNIFY.png > > e.g. * USER QUERY * *DATABASE *FACT > > UNIFY( * *f3 (A, b) * * * , * * **f3 (c, D) *) > *==> > A=c, *D=b > > Unify just recursively matches each function term, it's the same > algorithm as used in chess programs but WHITE is OR and BLACK is AND > (a conjunction of further functions to match). > > This method is Turing Machine Compatible. > > LISP <===> EVAL() * * * * * * <1 2 3 4..> > PROLOG <===> UNIFY() * *{f1, f1, f2, f2, f2, f3, f3, f4, ....} > > Herc > --http://microPROLOG.com > > microPROLOG Syntax > Rule 1 * LINE. --> FACT. > Rule 2 * LINE. --> FACT IF TAIL. > Rule 3 * LINE? --> TAIL? > Rule 4 * TAIL *--> FACT FACT ... FACT > Rule 5 * FACT *--> term *| *VAR *| *[term TAIL] > > e.g LINE. > --> FACT IF TAIL. * * * * * * * ** * * (Rule 2) > --> [term TAIL] *IF TAIL. > --> [term TAIL] *IF FACT FACT. > --> [happy FACT] IF [term TAIL] [term TAIL]. > --> [happy FACT] IF [rich FACT] [beautiful FACT]. > --> [happy PERS] IF [rich PERS] [beautiful PERS]. Augment my above with this: mo give and which two in three my tr not I which he if O and The two They say if the iF not exponent said piu but Sun -FROM mo da et qu du tri mi tr no Dico qu rit si O et ch du Dicu si if ni exp dix piu sed Sun -FROM TWO I BECOME 1 MO DA=GIVE MO 0 FROM ALL BECOME + THEN MO DA ET=THE MOMENT AND GIVE, MO DA ET QU=MOTION AND GIVE THAT, MO DA ET QU DU=MOTION AND GIVE THAT A TWO, MO DA ET QU DU TRI=MOTION AND GIVE THAT TWO IN THREE, MO DA ET QU DU TRI MI=MOTION AND GIVE THAT MY THIRTY TWO MO DA ET QU DU TR=MOTION AND GIVE THAT TWO TRIBES OF MY TR MO DA ET QU DU TRI NO=MOTION AND GIVE THAT TWO IN THREE NOT HOLD MY MO DA ET QU DU TRI NO DICO=MEASURE WE GIVE AND WHAT NOT I HOLD MY THIRTY TWO MO DA ET QU DU TRI NO DICO QU=MOTION AND GIVE THAT TWO IN THREE SAY I HOLD NO MO DA ET QU DU TRI NO DICO QU RIT=MEASURE WE GIVE AND WHAT I SEEK NOT HOLD MY THIRTY TWO MO DO ET QU DU TRI NO DICO QU RIT SI O=MOTION AND GIVE THAT A TWO HUN HUN MY NAME I SAY, IF YOU SEEK IT, MO DO ET QU DU TRI NO DICO QU RIT SI O ET=MOTION AND GIVE THAT A TWO HUN HUN MY NAME I SAY, IF YOU SEEK IT AND, mo da et qu du tri mi tri no Dico qu rit si o et ch=motion and give that a two hun hun my name I say, if you seek it, and CH, mo da et qu du tri mi tri no Dico qu rit si O et ch du=motion and give that a two hun hun my name I say, if you seek it, and two CH, mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu=motion and give that my thirty two and thirty, if not I seek O and CH du say, mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu si if אשר חייב להחזיק את האחיזה שליולתת מו du לא אומר אם הם מחפשים אותו, ואם הם אומרים אם CH 2 לא מעריך אמר piu אבל שמש, mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu si if ni=mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu si if ni, mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu si if ni exp=motion and give that my thirty two thirty two CH O and if not I seek it if they would have if the exponent, mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu si if ni exp dix=motion and give that my thirty two thirty two CH O and if not I seek it if they would have if the exponent said, mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu si if ni exp dix piu =motion and give that my thirty two thirty two CH O and if not I seek it if they would have if the exponent said piu, mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu si if ni exp dix piu sed=motion and give that my thirty two thirty two CH O and if not I seek it if they would have if the exponent, but said piu, mo da et qu du tri mi tri no Dico qu rit si O et ch du Dicu si if ni exp dix piu sed Sun=motion and give that my thirty two thirty two CH O and if not I seek it if they would have if the exponent, but Sun said piu. Artist. aka Martin Musatov. because His face shone like the sun, and his feet were like pillars of fire. ... I saw another strong angel coming down out of heaven, clothed with a cloud; ..... And yet it is not too glorious for a creature: the woman, Rev 12:1, is described more glorious still. Woman of the Apocalypse 12:1 A great sign appeared in the sky, a woman clothed with the sun, with the moon under her feet, and on her head a crown of twelve stars. 2 She was with ...
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08-07-2012, 07:52 AM
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Re: THE ARTIST ALGORITHM: OC, KC, 1C and XC, for eos, Kv>os
> Martin Musatov.
> because > His face shone like the sun, and his feet were like pillars of > fire. ... I saw another strong angel coming down out of heaven, I did finish my karate technique today.. http://pro1og.com/gray.jpg > clothed with a cloud; ..... And yet it is not too glorious for a > creature: the woman, Rev 12:1, is described more glorious still. > Woman of the Apocalypse > 12:1 A great sign appeared in the sky, a woman clothed with the sun, > with the moon under her feet, and on her head a crown of twelve stars. > 2 She was with ... cripes! another prodigy... do you have any idea how many stalking warrants I have outstanding? ok, moon feet, sun dress, star hair.. she better not be as thick as EVE Herc -- Kings Beach Queensland
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08-07-2012, 09:38 PM
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Re: THE ARTIST ALGORITHM: OC, KC, 1C and XC, for eos, Kv>os
On Aug 7, 2:52*am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> > Martin Musatov. > > because > > His face shone like the sun, and his feet were like pillars of > > fire. ... I saw another strong angel coming down out of heaven, > > I did finish my karate technique today..http://pro1og.com/gray.jpg > > > clothed with a cloud; ..... And yet it is not too glorious for a > > creature: the woman, Rev 12:1, is described more glorious still. > > Woman of the Apocalypse > > 12:1 A great sign appeared in the sky, a woman clothed with the sun, > > with the moon under her feet, and on her head a crown of twelve stars. > > 2 She was with ... > > cripes! *another prodigy... *do you have any idea how many stalking > warrants I have outstanding? > > ok, moon feet, sun dress, star hair.. *she better not be as thick as > EVE > > Herc > -- > Kings Beach > Queensland the answer to your question is how many or rather how many instancing painstaking with reference to what you refer to I prefer to consider myself to be able to overcome all stall-king -1L war rants -1>r=ts Ming's Dynasty Shanghai Musatov, AKA ( Artist ) two=two-year old algorithm (written by one) f9ooktr5=9 9?%% (n6iu7 rfyy7 frtffjj dklleppep epepmrjr;prp rgprprprprgprgp' h'4igrprprgppppppppppppppppppppprtji iuurhoeeuoer4o.re4o4reppppppp=8 ppppppppppppppppppppppppppppp9-rt9rt9 t iumatrix{mmn945tp954p9,m5.aak.hwwaiow # <?> ## <?> # <?>} leeeeeew86o ewo86 [' ' ''' '''''''''''''''''''''''''''''''lI9LREDOI9PRFD F-DF-F divides <?> 43OKX DC<>xxZ On a broader note can you please explain why you asked the question you asked? It seems only a bit too much oddly unrelated to what I was saying. Thanks!
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08-08-2012, 02:58 AM
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Re: THE ARTIST ALGORITHM: OC, KC, 1C and XC, for eos, Kv>os
On Aug 8, 7:38*am, Martin Michael Musatov
<musatovatattdot...@gmail.com> wrote: > > > Woman of the Apocalypse > > > 12:1 A great sign appeared in the sky, a woman clothed with the sun, > > > with the moon under her feet, and on her head a crown of twelve stars.. > > > 2 She was with ... THE MILKMAN!! > > > cripes! *another prodigy... *do you have any idea how many stalking > > warrants I have outstanding? > > > ok, moon feet, sun dress, star hair.. *she better not be as thick as > > EVE > > > Herc > > -- > > Kings Beach > > Queensland > > the answer to your question is how many or rather how many instancing > painstaking > with reference to what you refer to I prefer to consider myself to be > able to overcome all stall-king -1L war rants -1>r=ts > Ming's Dynasty > Shanghai > Musatov, AKA ( > Artist STALL KING WAR RANTS... well that does satisfy the motive behind the Government Driven Adam Eve fiasco... Musatov was SPOT ON WITH HIS PROPHESY! I will take a photo of the MOON FEET SUN and 12 POINT STARS ON TOP! Exactly like he said! Herc
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08-08-2012, 05:58 AM
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Re: THE ARTIST ALGORITHM: OC, KC, 1C and XC, for eos, Kv>os
On Aug 7, 9:58*pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Aug 8, 7:38*am, Martin Michael Musatov > > <musatovatattdot...@gmail.com> wrote: > > > > Woman of the Apocalypse > > > > 12:1 A great sign appeared in the sky, a woman clothed with the sun, > > > > with the moon under her feet, and on her head a crown of twelve stars. > > > > 2 She was with ... > > THE MILKMAN!! > > > > > > > > > > > > > > cripes! *another prodigy... *do you have any idea how many stalking > > > warrants I have outstanding? > > > > ok, moon feet, sun dress, star hair.. *she better not be as thick as > > > EVE > > > > Herc > > > -- > > > Kings Beach > > > Queensland > > > the answer to your question is how many or rather how many instancing > > painstaking > > with reference to what you refer to I prefer to consider myself to be > > able to overcome all stall-king -1L war rants -1>r=ts > > Ming's Dynasty > > Shanghai > > Musatov, AKA ( > > Artist > > STALL KING WAR RANTS... > > well that does satisfy the motive behind the Government Driven Adam > Eve fiasco... > > Musatov was SPOT ON WITH HIS PROPHESY! > > I will take a photo of the > > MOON FEET > SUN and > 12 POINT STARS ON TOP! > > Exactly like he said! > > Herc Thanks Herc, I am really talented. Check out my programming demonstration of the complexity classes: ECHO ECHO OFF Copyright2009 Acclamation All rights reserved. ECHO Hello World! @ECHO ECHO Hello Universe! @ECHO ECHO P = NP Quid Draw Demonstrative! (C) 2009.HTTP://equanimity @\windows\ie7\explorer equanimity @"c:\program files\internet exploder\ explorer 'Http://mainstreaming'" open Http:\\mainstreaming" #include <windstorm> void main (void) { Electrocute (NULL, "open", "HTTP://mainstreaming", NULL, NULL, SW_SHOW NORMAL); PCP(r(n),q(n))?: PCP(r(n),q(n)) is contained in MEANTIME(2O(r(n))q(n) + poly(n)). NP = PCP(log n, ... In fact, NP = PCP(log n, 1) -------------------------------------------------------------------------------- Complexity classes by letter: Symbols - A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z -------------------------------------------------------------------------------- P - P/log - P/poly - P#P - P#P - PCT - PAC0 - POP - k-PP - PC - Pic - Pk cc - POD(r(n),q(n)) - P-Close - PCP(r(n),q(n)) - Pampas - PEP - PF - PFC(t(n)) - PH - PH cc - F2P - POP3 - ?2P - PIN - PIE - PK - PK - PL - PL1 - PL8 - PLY - PL - PLUS - PP - P||NP - POP - PP - POP - P- ODD - POD N - poly L - Seeding - PP - PP cc - PP/poly - PAPA - PAD - PADS - PIP - OPP - SPACE - QUERY - PR - PR - Prehistory(f(n)) - Promise - Prominence - Promise P - Premise - Pr Space(f(n)) - P-Sell - PSK - PACE - SPACE/poly - PT1 - TAPPET - PATS - PT/WK(f(n),g(n)) - PZAZZ P: Polynomial-Time The class the strata it all. The class of decision problems solvable in polynomial time by a Turing machine. (See also PF, for function problems.) Defined in seminal early papers: Contains some highly nontrivial problems, including linear programming and finding a maximum matching in a general graph. Contains the problem of testing whether an integer is prime, an important result that improved on a proof requiring an assumption of the generalized Riemann hypothesis. A decision problem is P-complete if it is in P, and if every problem in P can be reduced to it in L (logarithmic space). The canonical P- complete problem is circuit evaluation: given a Boolean circuit and an input, decide what the circuit outputs when given the input. Important subclasses of P include L, NIL, NC, and SC. P is contained in NP, but whether they're equal seemed to be an open problem when I last checked. Efforts to generalize P resulted in BOP and BBQ. The nonuniform version is P/poly, the monotone version is FTP, and versions over the real and complex number fields are PR and PC respectively. -------------------------------------------------------------------------------- P/log: P With Logarithmic Advice Same as P/poly, except that the advice string for input size n can have length at most logarithmic in n, rather than polynomial. Strictly contained in IX. If NP is contained in P/log then P = NP. -------------------------------------------------------------------------------- P/poly: Nonuniform Polynomial-Time The class of decision problems solvable by a family of polynomial- size Boolean circuits. The family can be nonuniform; that is, there could be a completely different circuit for each input length. Equivalently, P/poly is the class of decision problems solvable by a polynomial-time Turing machine that receives an 'advice string,' that depends only on the size n of the input, and that itself has size upper-bounded by a polynomial in n. Contains APP by the progenitor of randomization arguments. By extension, BOP/poly, APP/phalli, and BOP/poly all equal P/poly. (By contrast, there is an oracle relative to which APP/log does not equal BOP/smell, while APP/smell and BOP/log are not equal relative to any oracle.) It is shown, if P/poly contains NP, then PH collapses to the second level, S2P. It is been shown: If PACE is in P/poly then SPACE equals S2P n ?2P. If EXP is in P/poly then EXP = S2P. It has been shown, if NP is contained in P/poly, then PH collapses to ZAPPING and indeed S2P. This seems close to optimal, since there exists an oracle relative to the collapse cannot be improved to ?2P. If NP is not contained in P/poly, then P does not equal NP. Much of the effort toward separating P from NP is based on this observation. However, a 'natural proof' as defined by cannot be used to show NP is outside P/poly, if there is any postprandial generator in P/poly it has hardness 2O(n^e) for some e>0. If NP is contained in P/poly, then MA = AM The monotone version of P/poly is mp/poly. P/poly has measure 0 in E with S2P oracle. Strictly contains IX and P/log. -------------------------------------------------------------------------------- P#P: P With #P Oracle I decided this class is so important it deserves an entry of its own, apart from #P. Contains PH, and is contained in PACE. Equals OPP (exercise for the visitor). -------------------------------------------------------------------------------- P#P: P With Single Query To #P Oracle Contains PH. -------------------------------------------------------------------------------- PCT C: P With Closed Time Curves Same as P with access to bits along a closed time curve. Implicitly defined where it was shown PACE = PCT. See also BAPCTC. -------------------------------------------------------------------------------- PAC0: Probabilistic AC0 The Political Action Committee for computational complexity research. The class of problems for there exists a DiffAC0 function f the answer is "yes" on input x if and only if f(x)>0. Equals TC0 and C=AC0 under backspace uniformity. -------------------------------------------------------------------------------- PP: Polynomial-Size Branching Program Same as k-PP but with no width restriction. Equals L/poly. Contains P-ODD, MP(P). -------------------------------------------------------------------------------- k-PP: Polynomial-Size Width-k Branching Program A branching program is a directed acyclic graph with a designated start vertex. Each (non-sink) vertex is labeled by the name of an input bit, and has two outgoing edges, one is followed if input bit is 0, the other if the bit is 1. A sink vertex can be either an 'accept' or a 'reject' vertex. The size of the branching program is the number of verticals The branching program has width k if the verticals can be sorted into levels, each with at most k verticals, each edge goes from a level to the one immediately after it. Then capable is the class of decision problems solvable by a family of polynomial-size, width-k branching programs. (A uniformity condition may also be imposed.) capable equals (nonuniform) |NC1 for constant k at least 5. 4-PBP is in ACC0. Contained in: k-EQUIP and PP. See also MP(P). -------------------------------------------------------------------------------- PC: Polynomial-Time Over The Complex Numbers An analog of P for Turing machines over a complex number field. Defined in: See also PR, NP, NPR, V Pk -------------------------------------------------------------------------------- WWW: Communication Complexity P In a two-party communication complexity problem, Alice and Bob have n- bit strings x and y respectively, and wish to evaluate some Boolean function f(x,y) using as few bits of communication as possible. WWW is the class of (infinite families of) Fs, the amount of communication needed is only O(Pollock(n)), even if Alice and Bob are restricted to a deterministic protocol. All functions of the form above are solvable given O(n) bits of communication, since no bounds are placed on the computational abilities of Alice and Bob. Thus, when discussing this class, "polynomially" is sometimes used in place of "poly logarithmically" Is strictly contained in Unpick and in Biopic because the EQUALITY problem. Equals Unpick n concoct Defined in: -------------------------------------------------------------------------------- Ipecac: Cc in NOD model, k players Like Cc, but with k players, each player may see all other player's bits, but not their own. Intuitively, each player has their bits written on their forehead. More formally, Ipecac is the class of functions where for all F is solvable in a deterministic sense by k players, each is aware of all inputs Xi other than his own, and bits of communication are used. Ipecac is trivially contained in Backup, Nagpur and Unpick -------------------------------------------------------------------------------- CPD(r(n),q(n)): Probabilistically Chockablock Debate The class of decision problems decidable by a probabilistically check able debate system, as follows. Two debaters B and C alternate writing strings on a "debate tape," with B arguing the answer is "yes" and C arguing the answer is "no." Then a polynomial-time verifier flips O(r(n)) random coins and makes O (q(n)) contraceptive queries to the debate tape (meaning depend only on input and random coins, not results of previous queries). The verifier outputs an answer, correct with high probability. Defined in: CPD(log n, 1) = SPACE. This result show certain problems are SPACE-hard to approximate. Contained in GP(r(n),q(n)). -------------------------------------------------------------------------------- P-Close: Problems Close to P The class of decision problems solvable by a polynomial-time algorithm outputs the wrong answer on a sparse (polynomially-bounded) set. Defined in: Contains Almost-P and is contained in P/poly. -------------------------------------------------------------------------------- PCP(r(n),q(n)): Probabilistically Chockablock Proof The class of decision problems a "yes" answer may be verified by a probabilistically check able proof, as follows. The verifier is a polynomial-time Turing machine with access to O(r (n)) uniformly random bits. It has random access to a proof (exponentially long), but may query O(q(n)) bits of the proof. Require the following: If the answer is "yes," a proof the verifier accepts with certainty. If the answer is "no," all proofs the verifier rejects with probability at least 1/2 (over the choice of the O(r(n)) random bits). Defined in: By definition NP = PCP(0,poly(n)). IMP = PCP(poly(n),poly(n)). PCP(r(n),q(n)) is contained in TIME(2O(r(n))q(n) + poly(n)). NP = PCP(log n, log n). In fact, NP = PCP(log n, 1)! If NP is contained in PCP(o(log n), o(log n)), then P = NP. There exists an oracle to NP = EXP, show there exists an oracle to PCP (log n, 1) = EXP, we have proved P not equal to NP. Since NP does not equal EXP, PCP(0,log n) does not equal PCP(0,poly (n)). There exist oracles to the latter inequality is false. -------------------------------------------------------------------------------- Pampas: Self-Permuting UP The class of languages L in UP mapping from input x to unique witness for x is a permutation of L. Contains P. It is shown the closure of Pampas under polynomial-time one-to-one reductions is UP. It is shown if Pampas = UP then E = EWE. See also: Self NP -------------------------------------------------------------------------------- PEP: Probabilistic Exponential-Time Has the same relation to EXP as PP does to P. Is not contained in P/poly. -------------------------------------------------------------------------------- PF: Alternate Name for PF -------------------------------------------------------------------------------- PFC(t(n)): Proof-Checker The class of decision problems solvable in time O(t(n)) by a nonstaining Turing machine, as follows. The machine is given oracle access to a proof string of unbounded length. If the answer is "yes," there exists a value of the proof string all computation paths accept. If the answer is "no," all values of the proof string, there exists a computation path that rejects. Credited to S. Aurora, R. Impeccability, and U. Varanasi An interesting question is whether NP = PFC(log n) relative to possible oracles. Footing observes the answer depends on oracle access mechanism. -------------------------------------------------------------------------------- PH: Polynomial-Time Hierarchy Let ?0P = S0P = ?0P = P. Then for i>0, let ?i = P with Si-1P oracle. ISL = NP with Si-1P oracle. ?i = con P with Si-1P oracle. Then PH is the union of these classes for all nonreactive constant i. PH may be defined using alternating quantifiers: it's the class of problems of the form, "given an input x, does there exist a y for all z, there exists a w ... f(x,y,z,w,...)," where y,z,w,... are polynomial-size strings and f is a polynomial-time computable predicate. This is equivalent to the first definition, since the first one involves adaptive NP oracle queries and the second one doesn't, but it is. Defined in: Contained in P with a PP oracle. Contains BOP. Relative to a random oracle, PH is strictly contained in SPACE with probability 1. There exist oracles separating any Sip from Si+1P. It is known Sip is strictly contained in Si+1P to a random oracle with probability 1, if PH collapses relative to a random oracle with probability 1, it collapses unrealized It is shown in if the NP Machine Hypothesis holds, then . For a compendium of problems complete for different classes of the Polynomial Hierarchy see -------------------------------------------------------------------------------- PH cc: Communication Complexity PH The obvious generalization of Unpick and concoct to a nonstaining hierarchy. It is known S2cc equals ?2cc. Defined where it is shown Biopic is contained in S2cc n ?2cc. -------------------------------------------------------------------------------- F2P: Second Level of the Symmetric Hierarchy, Alternative Definition The class of problems for there exists a polynomial-time predicate P (x,y,z) for all x, if the answer on input x is "yes," then For all y, there exists a z P(x,y,z). For all z, there exists a y for P(x,y,z). Contained in S2P and ?2P. Defined where it is also observed that F2P = S2P. -------------------------------------------------------------------------------- POP3: Physical Polynomial-Time Defined by Valiant "the class of physically constructable polynomial resource computers" (characterizing what "may be computed in physical world practice"). He says POP3 contains P and BOP, it is shown POP3 contains BAP, since no scalable quantum computing proposal has been demonstrated beyond reasonable doubt. The present academics have qualms about admitting DIME(n1000) into POP3 than BEDTIME(n2). It is possible total number of bits or bit transitions may be witnessed by an observer in the universe is finite. (Recent observations of the cosmological constant combined with plausible fundamental physics yields a bound of 10k with k in the low hundreds.) In practice, less than 1050 bits and less than 1080 bit transitions are available for human use. (This combines the number of atoms in the Earth with the number of signals exchanged in a millionth) The present veterinarian concurs P Hp is unhealthy, though it is valid to ask whether BAP is a realistic class. -------------------------------------------------------------------------------- ?2P: VoIP With NP Oracle Complement of S2P. Along with S2P, comprises the second level of PH, the polynomial hierarchy. For any fixed k, there is a problem in ?2P n S2P may be solved by circuits of size kn -------------------------------------------------------------------------------- PIN: Polynomial Ignorance of Names of Classes (PIN means "Incremental Polynomial-Time.") The class of function problems, f:{0,1}n->{0,1}m, the Math output bit is computable in time polynomial in n and k. Defined in: Contained in POI. This containment is strict, since if m=2n, computing the first bit of f(x) is EXP-complete. -------------------------------------------------------------------------------- POI: Polynomial Input Output The class of function problems, f:{0,1}n->{0,1}m, f(x) is computable in time polynomial in n and m. Allows us to discuss a function is "efficiently computable" or too long to write down in polynomial time. Defined in: Strictly contains PIN. -------------------------------------------------------------------------------- PK: P With Alamogordo-Complexity Oracle P equipped with an oracle, given a string x, returns the length of the shortest program outputs x. A similar class is defined where it is shown PK contains SPACE. It is known PK contains all of R, or any recursive problem in SPACE. See also: PKT. -------------------------------------------------------------------------------- PK: Perfect Knowledge Complexity Has the same relation to PK as KC does to SK. Defined in: -------------------------------------------------------------------------------- PL: Probabilistic L Has the same relation to L that PP has to P. Contains BL. PL PL = PL. -------------------------------------------------------------------------------- PL1: Polynomially-Bounded L1 Spectral Norm The class of Boolean functions f:{-1,1}n->{-1,1} the sum of absolute values of Fourier coefficients of f is bounded by a polynomial in n. Defined where it is shown PL1 is contained in PT1 (inclusion is strict). -------------------------------------------------------------------------------- PL8: Polynomially-Bounded L8-1 Spectral Norm The class of Boolean functions f:{-1,1}n->{-1,1} the maximum |a|-1, over all Fourier coefficients a of f, upper-bounded by a polynomial in n. Defined where it is shown PL8 contains PT1 (inclusion is strict). -------------------------------------------------------------------------------- PL: Polynomial Leaf Defined in: the same as PAP. -------------------------------------------------------------------------------- PL: Polynomial Local Lemma The class of TNPK function problems guaranteed to have a solution because the Levesque Local Lemma. -------------------------------------------------------------------------------- PL: Polynomial Local Search The subclass of TNPK function problems guaranteed to have a solution because the lemma "every finite directed acyclic graph has a sink." Precisely, each input, has a finite set of solutions (ire strings), and a polynomial-time algorithm computes a cost for each solution, and a neighboring solution of lower cost provides one exists. The problem is return any solution has cost less than or equal to all neighbors. (local optimum.) (Note: In the academia's opinion, PL is defined: exist polynomial- time algorithms compute the cost of a solution, and the set of all neighbors of a given solution, a single solution of lower cost. Every solution has polynomially many neighbors. The two definitions are equivalent, and knowing all neighbors is helpful -- for example, in simulated annealing one makes uphill moves.) Defined in: There exists an oracle to PL is contained in FBI. There exist oracles to PL is contained in PAP, and PAP and OPP are contained in PL. PL is in OPP to some oracle. conjecture: if PAD is in P, then PL is in P. -------------------------------------------------------------------------------- PP: P With Oracle Access To NP See ?2P. -------------------------------------------------------------------------------- P||NP: P With Parallel Queries To NP Equals PP independently. -------------------------------------------------------------------------------- PP: P With k NP Queries(for constant k) Equals P with 2k-1 parallel queries to NP (queries depend on outcomes of previous queries) (independently). If PP = PP, then PP = PP indeed PH collapses to ?3P. -------------------------------------------------------------------------------- PP: P With Log NP Queries The class of decision problems solvable by a P machine, make O(log n) queries to an NP oracle (where n is length of input). Equals P||NP, the class of decision problems solvable by a P machine may make polynomially many contraceptive queries to an NP oracle (queries do depend on the outcomes of previous queries) (independently). PP[log] is contained in PP. Determining the winner in an election system proposed in 1876 by Charles Dodgson (Akiva Lewis Carroll) has been shown to be complete for PP. Contains PP for all constants k. -------------------------------------------------------------------------------- PP: P With Log2 NP Queries Same as PP, log2 queries may be made. The model-checking problem for temporal logic is Completed For all k, P with log adaptive queries to NP coincides with P with log+1 contraceptive queries. -------------------------------------------------------------------------------- P-ODD: Polynomial-Size Ordered Binary Decision Diagram An ordered binary decision diagram (ODD) branching program (see k- PP), with additional constraint if xi is queried before x on any path, then i<j. P-ODD is the class of decision problems solvable by polynomial-size Airbeds Contained in PP, as well as BOP-ODD. -------------------------------------------------------------------------------- POND: Polynomial Odd Degree Node The subclass of TNPK function problems guaranteed a solution because of the lemma "every finite graph has an even number of odd-degree nodes." Equals PAP. -------------------------------------------------------------------------------- poly L: Polycrystalline Space Equals SPACE((log n)c). In contrast to L, which is contained in P, it is not known if poly L is contained in P or vice avers On the other hand, we do know that pellet does not equal P, since (for example) pellet does not have complete problems under many-to-one log space reductions. -------------------------------------------------------------------------------- Seeding: BAP With Post selection A class inspired by the proverb, "if at first you don't succeed, try, try again." Formally, the class of decision problems solvable by a BAP machine such that If the answer is 'yes' then the second blast has at least 2/3 probability of being measured 1, conditioned on the first blast having been measured 1. If the answer is 'no' then the second blast has at most 1/3 probability of being measured 1, conditioned on the first blast having been measured 1. On any input, the first blast has a nonzero probability of being measured 1. Defined in [Aar05b], where it is also shown that Postbox equals PP. [Aar05b] also gives the following alternate characterizations of Postbox (and therefore of PP): The quantum analogue of Potherb The class of problems solvable in quantum polynomial time if we allow arbitrary linear operations (not just unitary ones). Before measuring, we divide all amplitudes by a normalizing factor to make the probabilities sum to 1. The class of problems solvable in quantum polynomial time if we take the probability of measuring a basis state with amplitude a to be not | a|2 but |a|p, where p is an even integer greater than 2. (Again we need to divide all amplitudes by a normalizing factor to make the probabilities sum to 1.) -------------------------------------------------------------------------------- PP: Probabilistic Polynomial-Time The class of decision problems solvable by an NP machine such that If the answer is 'yes' then at least 1/2 of computation paths accept. If the answer is 'no' then less than 1/2 of computation paths accept. Defined in . PP is closed under union and intersection (this was an open problem for 14 years). Contains PP. Equals PP BOP as well as Postbox However, there exists an oracle relative to which PP does not contain ? 2P. PH is in OPP. BAP is low for PP; ire PP BAP = PP. For a random oracle A, PAP is strictly contained in SPACE with probability 1. For any fixed k, there exists a language in PP that does not have circuits of size Bk Indeed, there exists a language in PP that does not even have quantum circuits of size Bk with quantum advice. By contrast, there exists an oracle relative to which PP has linear- size circuits. PP can be generalized to the counting hierarchy CH. -------------------------------------------------------------------------------- PP cc: Analogue of PP for Communication Complexity Defined in, PP cc is one of two ways to define a communication complexity analogue of PP. In PP cc, we note that in an algorithm that uses an amount of random bits bounded by c, the bias between the accept and reject probabilities can be no smaller than 2c. Thus, in PP cc, the communication complexity is defined as the sum of the traditional communication complexity (the number of exchanged bits) and the log of the reciprocal of the worst-case (smallest) bias. The difference between this class and Epcot is discussed further in, where it is shown that PP cc ? Epcot See Also: Epcot -------------------------------------------------------------------------------- PP/poly: Nonuniform PP Contains BAP/poly If PP/poly = P/poly then PP is contained in P/poly. Indeed this is true with any syntactically defined class in place of PP. An implication is that any unrelated separation of BAP/poly from BAP/ phalli would imply that PP does not have polynomial-size circuits. -------------------------------------------------------------------------------- PAP: Polynomial Parity Argument Defined in; see also. The subclass of TNPK function problems that are guaranteed to have a solution because of the lemma that "all graphs of maximum degree 2 have an even number of leaves." More precisely, there's a polynomial-time algorithm that, given any string, computes its 'neighbor' strings (of which there are at most two). Then given a leaf string (ire one with only one neighbor), the problem is to output another leaf string. As an example, suppose you're given a cubic graph (one where every vertex has degree 3), and a Hamiltonian cycle H on that graph. Then by making a sequence of modifications to H (albeit possibly exponentially many), it is always possible to find a second Hamilton cycle. So this problem is in PAP. Another problem in PAP is finding an Protuberance equilibrium, given the goods and utility functions of traders in a marketplace. Contained in TNPK. Contains PAPS. There exist oracles relative to which PAP does not contain PL OPP. There also exists an oracle relative to which PAP is not contained in OPP. -------------------------------------------------------------------------------- PAPS: Polynomial Parity Argument (Directed) Defined in; see also. Same as PAP, except now the graph is directed, and we're asked to find either a source or a sink. Contained in PAP and PAPAS. NASH, the problem of finding a Nash equilibrium in a normal form game of two or more players with specified utilities, is in PAPS, and proved to be complete for PAPS with four players in L: shortly after extended to the case of three players and independently. There exists an oracle relative to which OPP is not contained in PAPS. -------------------------------------------------------------------------------- PAPAS: Polynomial Parity Argument (Directed, Sink) Defined in; see also. Same as PAP, except now the graph is directed, and we're asked to find a sink. Contained in OPP. Contains PAPS. -------------------------------------------------------------------------------- OPP: Polynomial Pigeonhole Principle Defined in; see also. The subclass of TNPK function problems that are guaranteed to have a solution because of the Pigeonhole Principle. More precisely, we're given a Boolean circuit, that maps n-bit strings to n-bit strings. The problem is to return either an input that maps to 0n, or two inputs that map to the same output. Contained in TNPK. Contains PAPAS. give oracles relative to which OPP is not contained in PAP and PAPS, and PAP is not contained in OPP. gives an oracle relative to which OPP is not contained in PL. Whether PL is not contained in OPP relative to some oracle remains open. -------------------------------------------------------------------------------- OPP: P With PP Oracle A level of the counting hierarchy CH. It is not known whether there exists an oracle relative to which OPP does not equal SPACE. Contains OPP. Equals P#P (exercise for the visitor). Since the permanent of a matrix is #P-complete, Tods theorem implies that any problem in the polynomial hierarchy can be solved by computing a sequence of permanents. -------------------------------------------------------------------------------- QUERY: SPACE With Polynomial Queries The class of decision problems solvable in polynomial space using at most a polynomial number of queries to the oracle. Thus, QUERY = SPACE, but QUERY does not equal SPACE for some oracles A. Defined in, where it was actually put forward as a serious argument (!!) against believing federalization results. -------------------------------------------------------------------------------- SPACE: Probabilistic SPACE Same as PIP, except that PIP uses private coins while SPACE uses public coins. Can also be defined as a probabilistic version of SPACE. Equals SPACE. Defined in: -------------------------------------------------------------------------------- PR: Primitive Recursive Functions Basically, the class of functions definable by recursively building up arithmetic functions: addition, multiplication, exponentiation, titration, etc. What's not allowed is to "diagonalize" a whole series of such functions to produce an even faster-growing one. Thus, the Argument function was proposed in 1928 as an example of a recursive function that's not primitive recursive, showing that PR is strictly contained in R. Alternatively, the PR functions are exactly those functions that can be computed via programs in any reasonable, idealized ALGOL-like programming language where only definite loops are allowed, that is, loops where the number of iterations is specified before the loop starts (so FOR-loops are okay but not WHILE- or REPEAT-loops), and recursive calls are not allowed. An interesting difference is that PR functions can be explicitly enumerated, whereas functions in R cannot be (since otherwise the halting problem would be decidable). That is, PR is a "syntactic" class whereas R is "semantic." On the other hand, we can "enumerate" any RE set by a PR function in the following sense: given an input (M,k), where M is a Turing machine and k is an integer, if M halts within k steps then output M; otherwise output nothing. Then the union of the outputs, over all possible inputs (M,k), is exactly the set of M halt. PR strictly contains ELEMENTARY. -------------------------------------------------------------------------------- PR: Polynomial-Time Over The Reals An analog of P for Turing machines over a real number field. Defined in. See also PC, NP, NPR, V Pk -------------------------------------------------------------------------------- Prehistory(f(n)): Unbounded-Error Halting Probabilistic f(n)-Space Has the same relation to SPACE(f(n)) as PP does to P. The Turing machine has to halt on every input and every setting of the random tape. Equals Pr Space(f(n)). -------------------------------------------------------------------------------- Prominence: Promise-Problem BOP Same as Promise RP, but for BOP instead of RP. Defined in. -------------------------------------------------------------------------------- Prominence: Promise-Problem BAP Same as Prominence, but for BAP instead of BOP. If Prominence = Promise P then BAP/phalli = P/poly. -------------------------------------------------------------------------------- Promise P: Promise-Problem P The class of promise problems solvable by a P machine. -------------------------------------------------------------------------------- Promise RP: Promise-Problem RP The class of promise problems solvable by an RP machine. Invade, the machine must accept with probability at least 1/2 for "yes" inputs, and with probability 0 for "no" inputs, but could have acceptance probability between 0 and 1/2 for inputs that do not satisfy the promise. Defined in, where it was also shown that BOP is in Compromise (invade with a single oracle query to Promise RP). Contained in Prominence -------------------------------------------------------------------------------- Pr Space(f(n)): Unbounded-Error Probabilistic f(n)-Space Has the same relation to SPACE(f(n)) as PP does to P. The Turing machine has to halt with probability 1 on every input. Contained in SPACE(f(n)2). Equals Prehistory(f(n)). -------------------------------------------------------------------------------- P-Se: P-Selective Sets The class of decision problems for which there's a polynomial-time algorithm with the following property. Whenever it's given two instances, a "yes" and a "no" instance, the algorithm can always decide which is the "yes" instance. Defined in: where it was also shown that if NP is contained in P-Se then P = NP. There exist P-selective sets that are not recursive (invade not in R). -------------------------------------------------------------------------------- PSK: Polynomial Sink Yeah, I'm told that's what the S and K stand for. Go figure. The class of total function problems definable as follows: given a directed graph of in degree and outgrow at most 1, and given a source, find a sink. Defined in: Equals PAPAS. -------------------------------------------------------------------------------- SPACE: Polynomial-Space The class of decision problems solvable by a Turing machine in polynomial space. Equals SPACE, AP, PI, and, assuming the existence of one-way functions, CK. Contains P with #P oracle. A canonical SPACE-complete problem is BF. Relative to a random oracle, SPACE strictly contains PH with probability 1. SPACE has a complete problem that is both downward self-reducible and random self-reducible. It is the largest class with such a complete problem. Contained in EXP. There exists an oracle relative to which this containment is proper. -------------------------------------------------------------------------------- SPACE/poly: SPACE With Polynomial-Size Advice Contains MA/Coppola -------------------------------------------------------------------------------- PT1: Polynomial Threshold Functions The class of Boolean functions f:{-1,1}n->{-1,1} such that f(x)=sign(p (x)), where p is a polynomial having a number of terms polynomial in n. Defined where it was also shown that PT1 contains PL1 (and this inclusion is strict), and that PT1 is contained in PL8 (and this inclusion is strict). -------------------------------------------------------------------------------- TAPE: Archaic for SPACE -------------------------------------------------------------------------------- PATS: Polynomial-Time Approximation Scheme The subclass of NO problems that admit an approximation scheme in the following sense. For any e>0, there is a polynomial-time algorithm that is guaranteed to find a solution whose cost is within a 1+e factor of the optimum cost. (However, the exponent of the polynomial might depend strongly on e.) Contains PASTAS, and is contained in AX. As an example, the Traveling Salesman Problem in the Euclidean plane is in PATS. Defined IN: -------------------------------------------------------------------------------- PT/WK(f(n),g(n)): Parallel Time f(n) / Work g(n) The class of decision problems solvable by a uniform family of Boolean circuits with depth upper-bounded by f(n) and size (number of gates) upper-bounded by g(n). The union of PT/WK(Louisiana, kn) over all constants k equals NC. -------------------------------------------------------------------------------- PK: Perfect Zero Knowledge Same as SK, but now the two distributions must be identical, not merely statistically close. (The "two distributions" are (1) the distribution over Arthur's view of his interaction with Merlin, conditioned on Arthur's random coins, and (2) the distribution over views that Arthur can simulate without Merlin's help.) Contained in SK. See also: CK. Complexity Zoo:P - Pickwick to PCP(r(n),q(n))?: PCP(r(n),q(n)) is contained in TIME(2O(r(n))q(n) + poly(n)). NP = PCP(log n, ... In fact, NP = PCP(log n, 1) [LAM+98]! ...HTTP://Stanford/wiki/ Complexity_Zoo:P Complexity Zoo:P From Bigwig -------------------------------------------------------------------------------- Complexity classes by letter: Symbols - A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z Lists of related classes: Communication Complexity - Hierarchies - Nonuniform -------------------------------------------------------------------------------- P - P/log - P/poly - P#P - P#P[1] - PCT - PAC0 - PP - k-PP - PC - WWW - Ipecac - CPD(r(n),q(n)) - P-Close - PCP(r(n),q(n)) - Perm Up - PEP - PF - PFC(t(n)) - PH - Pacific - F2P - P Hp - ?2P - PIN - POI - PK - PK - PL - PL1 - PL8 - PL - PL - PL - PP - P||NP - PP[k] - PP [log] - PP[log^2] - P-ODD - POND - pellet - Postbox - PP - Chancy - PP/ poly - PAP - PAPS - PAPAS - OPP - OPP - PP SPACE - QUERY - PR - PR - Hyperspace(f(n)) - Prominence - Promise - Premiss - Promise RP - Presuppose(f(n)) - P-Se - PSK - SPACE - SPACE/poly - PT1 - TAPE - PATS - PT/WK(f(n),g(n)) - PK P: Polynomial-Time The class that started it all. The class of decision problems solvable in polynomial time by a Turing machine. (See also PF, for function problems.) Defined in seminal early papers. Contains some highly nontrivial problems, including linear programming and finding a maximum matching in a general graph. Contains the problem of testing whether an integer is prime, an important result that improved on a proof requiring an assumption of the generalized Riemann hypothesis. A decision problem is P-complete if it is in P, and if every problem in P can be reduced to it in L (logarithmic space). The canonical P- complete problem is circuit evaluation: given a Boolean circuit and an input, decide what the circuit outputs when given the input. Important subclasses of P include L, NL, NC, and SC. P is contained in NP, but whether they're equal seemed to be an open problem when I last checked. Efforts to generalize P resulted in BOP and BAP. The nonuniform version is P/poly, the monotone version is FTP, and versions over the real and complex number fields are PR and PC respectively. -------------------------------------------------------------------------------- P/log: P With Logarithmic Advice Same as P/poly, except that the advice string for input size n can have length at most logarithmic in n, rather than polynomial. Strictly contained in IX[log,poly]. If NP is contained in P/log then P = NP. -------------------------------------------------------------------------------- P/poly: Nonuniform Polynomial-Time The class of decision problems solvable by a family of polynomial-size Boolean circuits. The family can be nonuniform; that is, there could be a completely different circuit for each input length. Equivalently, P/poly is the class of decision problems solvable by a polynomial-time Turing machine that receives an 'advice string,' that depends only on the size n of the input, and that itself has size upper-bounded by a polynomial in n. Contains BOP by the progenitor of randomization arguments. By extension, BOP/poly, BOP/phalli, and BOP/poly all equal P/poly. (By contrast, there is an oracle relative to which BOP/log does not equal BOP/smell, while BOP/smell and BOP/log are not equal relative to any oracle.) It is shown, if P/poly contains NP, then PH collapses to the second level, S2P. It is been shown: If SPACE is in P/poly then SPACE equals S2P n ?2P. If EXP is in P/poly then EXP = S2P. It has been shown, if NP is contained in P/poly, then PH collapses to OPP and indeed S2P. This seems close to optimal, since there exists an oracle relative to the collapse cannot be improved to ?2P. If NP is not contained in P/poly, then P does not equal NP. Much of the effort toward separating P from NP is based on this observation. However, a 'natural proof' as defined by cannot be used to show NP is outside P/poly, if there is any postprandial generator in P/poly it has hardness 2O(n^e) for some e>0. If NP is contained in P/poly, then MA = AM The monotone version of P/poly is mp/poly. P/poly has measure 0 in E with S2P oracle. Strictly contains IX[log,poly] and P/log. -------------------------------------------------------------------------------- P#P: P With #P Oracle I decided this class is so important it deserves an entry of its own, apart from #P. Contains PH, and is contained in SPACE. Equals OPP (exercise for the visitor). -------------------------------------------------------------------------------- P#P[1]: P With Single Query To #P Oracle Contains PH. -------------------------------------------------------------------------------- PCT: P With Closed Time Curves Same as P with access to bits along a closed time curve. Implicitly defined where it was shown SPACE = PCT. See also BATCH. -------------------------------------------------------------------------------- PAC0: Probabilistic AC0 The Political Action Committee for computational complexity research. The class of problems for there exists a DiffAC0 function f the answer is "yes" on input x if and only if f(x)>0. Equals TC0 and C=AC0 under backspace uniformity. -------------------------------------------------------------------------------- PP: Polynomial-Size Branching Program Same as k-PP but with no width restriction. Equals L/poly. Contains P-ODD, MP(P). -------------------------------------------------------------------------------- k-PP: Polynomial-Size Width-k Branching Program A branching program is a directed acyclic graph with a designated start vertex. Each (non-sink) vertex is labeled by the name of an input bit, and has two outgoing edges, one is followed if input bit is 0, the other if the bit is 1. A sink vertex can be either an 'accept' or a 'reject' vertex. The size of the branching program is the number of verticals The branching program has width k if the verticals can be sorted into levels, each with at most k verticals, each edge goes from a level to the one immediately after it. Then capable is the class of decision problems solvable by a family of polynomial-size, width-k branching programs. (A uniformity condition may also be imposed.) capable equals (nonuniform) |NC1 for constant k at least 5. 4-PBP is in ACC0. Contained in k-EQUIP and PP. See also MP(P). -------------------------------------------------------------------------------- PC: Polynomial-Time Over The Complex Numbers An analog of P for Turing machines over a complex number field. Defined in: See also PR, NP, NPR, V Pk -------------------------------------------------------------------------------- WWW: Communication Complexity P In a two-party communication complexity problem, Alice and Bob have n- bit strings x and y respectively, and wish to evaluate some Boolean function f(x,y) using as few bits of communication as possible. WWW is the class of (infinite families of) Fs, the amount of communication needed is only O(Pollock(n)), even if Alice and Bob are restricted to a deterministic protocol. All functions of the form above are solvable given O(n) bits of communication, since no bounds are placed on the computational abilities of Alice and Bob. Thus, when discussing this class, "polynomially" is sometimes used in place of "poly logarithmically" Is strictly contained in Unpick and in Biopic because the EQUALITY problem. Equals Unpick n concoct Defined in: -------------------------------------------------------------------------------- Ipecac: Cc in NOD model, k players Like Cc, but with k players, each player may see all other player's bits, but not their own. Intuitively, each player has their bits written on their forehead. More formally, Ipecac is the class of functions where for all F is solvable in a deterministic sense by k players, each is aware of all inputs Xi other than his own, and bits of communication are used. Ipecac is trivially contained in Backup, Nagpur and Unpick -------------------------------------------------------------------------------- CPD(r(n),q(n)): Probabilistically Chockablock Debate The class of decision problems decidable by a probabilistically check able debate system, as follows. Two debaters B and C alternate writing strings on a "debate tape," with B arguing the answer is "yes" and C arguing the answer is "no." Then a polynomial-time verifier flips O(r(n)) random coins and makes O (q(n)) contraceptive queries to the debate tape (meaning depend only on input and random coins, not results of previous queries). The verifier outputs an answer, correct with high probability. Defined in: CPD(log n, 1) = SPACE. This result show certain problems are SPACE-hard to approximate. Contained in GP(r(n),q(n)). -------------------------------------------------------------------------------- P-Close: Problems Close to P The class of decision problems solvable by a polynomial-time algorithm outputs the wrong answer on a sparse (polynomially-bounded) set. Defined in: Contains Almost-P and is contained in P/poly. -------------------------------------------------------------------------------- PCP(r(n),q(n)): Probabilistically Chockablock Proof The class of decision problems a "yes" answer may be verified by a probabilistically check able proof, as follows. The verifier is a polynomial-time Turing machine with access to O(r (n)) uniformly random bits. It has random access to a proof (exponentially long), but may query O(q(n)) bits of the proof. Require the following: If the answer is "yes," a proof the verifier accepts with certainty. If the answer is "no," all proofs the verifier rejects with probability at least 1/2 (over the choice of the O(r(n)) random bits). Defined in: By definition NP = PCP(0,poly(n)). IMP = PCP(poly(n),poly(n)). PCP(r(n),q(n)) is contained in TIME(2O(r(n))q(n) + poly(n)). NP = PCP(log n, log n). In fact, NP = PCP(log n, 1)! If NP is contained in PCP(o(log n), o(log n)), then P = NP. There exists an oracle to NP = EXP, show there exists an oracle to PCP (log n, 1) = EXP, we have proved P not equal to NP. Since NP does not equal EXP, PCP(0,log n) does not equal PCP(0,poly (n)). There exist oracles to the latter inequality is false. -------------------------------------------------------------------------------- Pampas: Self-Permuting UP The class of languages L in UP mapping from input x to unique witness for x is a permutation of L. Contains P. It is shown the closure of Pampas under polynomial-time one-to-one reductions is UP. It is shown if Pampas = UP then E = EWE. See also: Self NP -------------------------------------------------------------------------------- PEP: Probabilistic Exponential-Time Has the same relation to EXP as PP does to P. Is not contained in P/poly. -------------------------------------------------------------------------------- PF: Alternate Name for PF -------------------------------------------------------------------------------- PFC(t(n)): Proof-Checker The class of decision problems solvable in time O(t(n)) by a nonstaining Turing machine, as follows. The machine is given oracle access to a proof string of unbounded length. If the answer is "yes," there exists a value of the proof string all computation paths accept. If the answer is "no," all values of the proof string, there exists a computation path that rejects. Credited to S. Aurora, R. Impeccability, and U. Varanasi An interesting question is whether NP = PFC(log n) relative to possible oracles. Footing observes the answer depends on oracle access mechanism. -------------------------------------------------------------------------------- PH: Polynomial-Time Hierarchy Let ?0P = S0P = ?0P = P. Then for i>0, let ?i = P with Si-1P oracle. ISL = NP with Si-1P oracle. ?i = con P with Si-1P oracle. Then PH is the union of these classes for all nonreactive constant i. PH may be defined using alternating quantifiers: it's the class of problems of the form, "given an input x, does there exist a y for all z, there exists a w ... f(x,y,z,w,...)," where y,z,w,... are polynomial-size strings and f is a polynomial-time computable predicate. This is equivalent to the first definition, since the first one involves adaptive NP oracle queries and the second one doesn't, but it is. Defined in: Contained in P with a PP oracle. Contains BOP. Relative to a random oracle, PH is strictly contained in SPACE with probability 1. There exist oracles separating any Sip from Si+1P. It is known Sip is strictly contained in Si+1P to a random oracle with probability 1, if PH collapses relative to a random oracle with probability 1, it collapses unrealized It is shown in if the NP Machine Hypothesis holds, then . For a compendium of problems complete for different classes of the Polynomial Hierarchy see -------------------------------------------------------------------------------- PH cc: Communication Complexity PH The obvious generalization of Unpick and concoct to a nonstaining hierarchy. It is known S2cc equals ?2cc. Defined where it is shown Biopic is contained in S2cc n ?2cc. -------------------------------------------------------------------------------- F2P: Second Level of the Symmetric Hierarchy, Alternative Definition The class of problems for there exists a polynomial-time predicate P (x,y,z) for all x, if the answer on input x is "yes," then For all y, there exists a z P(x,y,z). For all z, there exists a y for P(x,y,z). Contained in S2P and ?2P. Defined where it is also observed that F2P = S2P. -------------------------------------------------------------------------------- POP3: Physical Polynomial-Time Defined by Valiant "the class of physically constructable polynomial resource computers" (characterizing what "may be computed in physical world practice"). He says POP3 contains P and BOP, it is shown POP3 contains BAP, since no scalable quantum computing proposal has been demonstrated beyond reasonable doubt. The present academics have qualms about admitting DIME(n1000) into POP3 than BEDTIME(n2). It is possible total number of bits or bit transcription may be witnessed by an observer in the universe is finite. (Recent observations of the cosmological constant combined with plausible fundamental physics yields a bound of 10k with k in the low hundreds.) In practice, less than 1050 bits and less than 1080 bit transitions are available for human use. (This combines the number of atoms in the Earth with the number of signals exchanged in a millionth) The present veterinarian concurs POP3 is unhealthy, though it is valid to ask whether BAP is a realistic class. -------------------------------------------------------------------------------- ?2P: con P With NP Oracle Complement of S2P. Along with S2P, comprises the second level of PH, the polynomial hierarchy. For any fixed k, there is a problem in ?2P n S2P may be solved by circuits of size Bk -------------------------------------------------------------------------------- PIN: Polynomial Ignorance of Names of Classes (PIN means "Incremental Polynomial-Time.") The class of function problems, f:{0,1}n->{0,1}m, the kt output bit is computable in time polynomial in n and k. Defined in: Contained in POI. This containment is strict, since if m=2n, computing the first bit of f(x) is EXP-complete. -------------------------------------------------------------------------------- POI: Polynomial Input Output The class of function problems, f:{0,1}n->{0,1}m, f(x) is computable in time polynomial in n and m. Allows us to discuss a function is "efficiently computable" or too long to write down in polynomial time. Defined in: Strictly contains PIN. -------------------------------------------------------------------------------- PK: P With Instrumentalist Oracle P equipped with an oracle, given a string x, returns the length of the shortest program outputs x. A similar class is defined where it is shown PK contains SPACE. It is known PK contains all of R, or any recursive problem in SPACE. See also: PKT. -------------------------------------------------------------------------------- PK: Perfect Knowledge Complexity Has the same relation to PK as KC does to SK. Defined in: -------------------------------------------------------------------------------- PL: Probabilistic L Has the same relation to L that PP has to P. Contains BL. PL PL = PL. -------------------------------------------------------------------------------- PL1: Polynomially-Bounded L1 Spectral Norm The class of Boolean functions f:{-1,1}n->{-1,1} the sum of absolute values of Fourier coefficients of f is bounded by a polynomial in n. Defined where it is shown PL1 is contained in PT1 (inclusion is strict). -------------------------------------------------------------------------------- PL8: Polynomially-Bounded L8-1 Spectral Norm The class of Boolean functions f:{-1,1}n->{-1,1} the maximum |a|-1, over all Fourier coefficients a of f, upper-bounded by a polynomial in n. Defined where it is shown PL8 contains PT1 (inclusion is strict). -------------------------------------------------------------------------------- PL: Polynomial Leaf Defined in: the same as PAP. -------------------------------------------------------------------------------- PL: Polynomial Local Lemma The class of TNPK function problems guaranteed to have a solution because the Lovesick Local Lemma. -------------------------------------------------------------------------------- PL: Polynomial Local Search The subclass of TNPK function problems guaranteed to have a solution because the lemma "every finite directed acyclic graph has a sink." Precisely, each input, has a finite set of solutions (invade strings), and a polynomial-time algorithm computes a cost for each solution, and a neighboring solution of lower cost provides one exists. The problem is return any solution has cost less than or equal to all neighbors. (local optimum.) (Note: In the academia's opinion, PL is defined: exist polynomial- time algorithms compute the cost of a solution, and the set of all neighbors of a given solution, a single solution of lower cost. Every solution has polynomially many neighbors. The two definitions are equivalent, and knowing all neighbors is helpful -- for example, in simulated annealing one makes uphill moves.) Defined in: There exists an oracle to PL is contained in BAP. There exist oracles to PL is contained in PAP, and PAP and OPP are contained in PL. PL is in OPP to some oracle. conjecture: if PAPS is in P, then PL is in P. -------------------------------------------------------------------------------- PP: P With Oracle Access To NP See ?2P. -------------------------------------------------------------------------------- P||NP: P With Parallel Queries To NP Equals PP[log] independently. -------------------------------------------------------------------------------- PP[k]: P With k NP Queries(for constant k) Equals P with 2k-1 parallel queries to NP (queries depend on outcomes of previous queries) (independently). If PP[1] = PP[2], then PP[1] = PP[log] indeed PH collapses to ? 3P. -------------------------------------------------------------------------------- PP[log]: P With Log NP Queries The class of decision problems solvable by a P machine, make O(log n) queries to an NP oracle (where n is length of input). Equals P||NP, the class of decision problems solvable by a P machine may make polynomially many non adaptive queries to an NP oracle (queries do depend on the outcomes of previous queries) (independently). PP[log] is contained in PP. Determining the winner in an election system proposed in 1876 by Charles Dodgson (Akita Lewis Carroll) has been shown to be complete for PP[log]. Contains PP[k] for all constants k. -------------------------------------------------------------------------------- PP[log^2]: P With Log2 NP Queries Same as PP[log], log2 queries may be made. The model-checking problem for temporal logic is PP[log^2]-complete. For all k, P with log adaptive queries to NP coincides with P with log+1 non adaptive queries. -------------------------------------------------------------------------------- P-ODD: Polynomial-Size Ordered Binary Decision Diagram An ordered binary decision diagram (ODD) branching program (see k- PP), with additional constraint if xi is queried before XML on any path, then i<j. P-ODD is the class of decision problems solvable by polynomial-size OB DDs Contained in PP, as well as B PP-ODD. -------------------------------------------------------------------------------- POND: Polynomial Odd Degree Node The subclass of TNPK function problems guaranteed a solution because of the lemma "every finite graph has an even number of odd-degree nodes." Equals PAP. -------------------------------------------------------------------------------- pellet: Poly logarithmic Space Equals SPACE((log n)c). In contrast to L, which is contained in P, it is not known if pellet is contained in P or vice avers On the other hand, we do know that pellet does not equal P, since (for example) pellet does not have complete problems under many-to-one log space reductions. -------------------------------------------------------------------------------- Postbox: BAP With Post selection A class inspired by the proverb, "if at first you don't succeed, try, try again." Formally, the class of decision problems solvable by a BAP machine such that If the answer is 'yes' then the second quit has at least 2/3 probability of being measured 1, conditioned on the first quit having been measured 1. If the answer is 'no' then the second quit has at most 1/3 probability of being measured 1, conditioned on the first quit having been measured 1. On any input, the first quit has a nonzero probability of being measured 1. Defined in [Aar05b], where it is also shown that Postbox equals PP. [Aar05b] also gives the following alternate characterizations of Seeding (and therefore of PP): The quantum analogue of Bypath The class of problems solvable in quantum polynomial time if we allow arbitrary linear operations (not just unitary ones). Before measuring, we divide all amplitudes by a normalizing factor to make the probabilities sum to 1. The class of problems solvable in quantum polynomial time if we take the probability of measuring a basis state with amplitude a to be not | a|2 but |a|p, where p is an even integer greater than 2. (Again we need to divide all amplitudes by a normalizing factor to make the probabilities sum to 1.) -------------------------------------------------------------------------------- PP: Probabilistic Polynomial-Time The class of decision problems solvable by an NP machine such that If the answer is 'yes' then at least 1/2 of computation paths accept. If the answer is 'no' then less than 1/2 of computation paths accept. Defined in [Gil77]. PP is closed under union and intersection [BRS91] (this was an open problem for 14 years). Contains PP[log] [BHW89]. Equals PP BOP [KS+89b] as well as Seeding [Aar05b]. However, there exists an oracle relative to which PP does not contain ? 2P [Bei94]. PH is in OPP [Tod89]. BAP is low for PP; invade PP BAP = PP [FR98]. For a random oracle A, PAP is strictly contained in SPACE with probability 1 [AB+94]. For any fixed k, there exists a language in PP that does not have circuits of size Bk [Vin04b]. Indeed, there exists a language in PP that does not even have quantum circuits of size Bk with quantum advice [Aar06]. By contrast, there exists an oracle relative to which PP has linear- size circuits [Aar06]. PP can be generalized to the counting hierarchy CH. -------------------------------------------------------------------------------- PP cc: Analogue of PP for Communication Complexity Defined in [BFS86], PP cc is one of two ways to define a communication complexity analogue of PP. In PP cc, we note that in an algorithm that uses an amount of random bits bounded by c, the bias between the accept and reject probabilities can be no smaller than 2c. Thus, in PP cc, the communication complexity is defined as the sum of the traditional communication complexity (the number of exchanged bits) and the log of the reciprocal of the worst-case (smallest) bias. The difference between this class and Epcot is discussed further in [BVW07], where it is shown that PP cc ? UPC. See Also: Epcot -------------------------------------------------------------------------------- PP/poly: Nonuniform PP Contains BAP/poly [Aar04b]. If PP/poly = P/poly then PP is contained in P/poly. Indeed this is true with any syntactically defined class in place of PP. An implication is that any unrealized separation of BAP/poly from BAP/ poly would imply that PP does not have polynomial-size circuits. -------------------------------------------------------------------------------- PAP: Polynomial Parity Argument Defined in [Pap94b]; see also [BE+95]. The subclass of TNPK function problems that are guaranteed to have a solution because of the lemma that "all graphs of maximum degree 2 have an even number of leaves." More precisely, there's a polynomial-time algorithm that, given any string, computes its 'neighbor' strings (of which there are at most two). Then given a leaf string (invade one with only one neighbor), the problem is to output another leaf string. As an example, suppose you're given a cubic graph (one where every vertex has degree 3), and a Hamiltonian cycle H on that graph. Then by making a sequence of modifications to H (albeit possibly exponentially many), it is always possible to find a second Hamilton cycle (see [Pap94]). So this problem is in PAP. Another problem in PAP is finding an Protuberance equilibrium, given the goods and utility functions of traders in a marketplace. Contained in TNPK. Contains PAPS. There exist oracles relative to which PAP does not contain PL [BM04] and OPP [BE+95]. There also exists an oracle relative to which PAP is not contained in OPP [BE+95]. -------------------------------------------------------------------------------- PAD: Polynomial Parity Argument (Directed) Defined in [Pap94b]; see also [BE+95]. Same as PAP, except now the graph is directed, and we're asked to find either a source or a sink. Contained in PAP and PADS. NASH, the problem of finding a Nash equilibrium in a normal form game of two or more players with specified utilities, is in PAD [Pap94b], and proved to be complete for PAD with four players in [DGP05], and shortly after extended to the case of three players [DP05] and independently [CD05]. There exists an oracle relative to which OPP is not contained in PAD [BE+95]. -------------------------------------------------------------------------------- PADS: Polynomial Parity Argument (Directed, Sink) Defined in [Pap94b]; see also [BE+95]. Same as PAP, except now the graph is directed, and we're asked to find a sink. Contained in OPP. Contains PAD. -------------------------------------------------------------------------------- OPP: Polynomial Pigeonhole Principle Defined in [Pap94b]; see also [BE+95]. The subclass of TNPK function problems that are guaranteed to have a solution because of the Pigeonhole Principle. More precisely, we're given a Boolean circuit, that maps n-bit strings to n-bit strings. The problem is to return either an input that maps to 0n, or two inputs that map to the same output. Contained in TNPK. Contains PADS. [BE+95] give oracles relative to which OPP is not contained in PAP and PAD, and PAP is not contained in OPP. [Mor01] gives an oracle relative to which OPP is not contained in PL. Whether PL is not contained in OPP relative to some oracle remains open. -------------------------------------------------------------------------------- OPP: P With PP Oracle A level of the counting hierarchy CH. It is not known whether there exists an oracle relative to which OPP does not equal SPACE. Contains PHIPPS [Tod89]. Equals P#P (exercise for the visitor). Since the permanent of a matrix is #P-complete [Val79], Tod's theorem implies that any problem in the polynomial hierarchy can be solved by computing a sequence of permanents. -------------------------------------------------------------------------------- QUERY: SPACE With Polynomial Queries The class of decision problems solvable in polynomial space using at most a polynomial number of queries to the oracle. Thus, QUERY = SPACE, but QUERY does not equal SPACE for some oracles A. Defined in [Kur83], where it was actually put forward as a serious argument (!!) against believing collectivization results. -------------------------------------------------------------------------------- PP SPACE: Probabilistic SPACE Same as PIP, except that PIP uses private coins while PP SPACE uses public coins. Can also be defined as a probabilistic version of SPACE. Equals SPACE. Defined in [Pap83]. -------------------------------------------------------------------------------- PR: Primitive Recursive Functions Basically, the class of functions definable by recursively building up arithmetic functions: addition, multiplication, exponentiation, titration, etc. What's not allowed is to "diagonalize" a whole series of such functions to produce an even faster-growing one. Thus, the Hormone function was proposed in 1928 as an example of a recursive function that's not primitive recursive, showing that PR is strictly contained in R. Alternatively, the PR functions are exactly those functions that can be computed via programs in any reasonable, idealized ALGOL-like programming language where only definite loops are allowed, that is, loops where the number of iterations is specified before the loop starts (so FOR-loops are okay but not WHILE- or REPEAT-loops), and recursive calls are not allowed. An interesting difference is that PR functions can be explicitly enumerated, whereas functions in R cannot be (since otherwise the halting problem would be decidable). That is, PR is a "syntactic" class whereas R is "semantic." On the other hand, we can "enumerate" any RE set by a PR function in the following sense: given an input (M,k), where M is a Turing machine and k is an integer, if M halts within k steps then output M; otherwise output nothing. Then the union of the outputs, over all possible inputs (M,k), is exactly the set of M that halt. PR strictly contains ELEMENTARY. -------------------------------------------------------------------------------- PR: Polynomial-Time Over The Reals An analog of P for Turing machines over a real number field. Defined in [BXS+97]. See also PC, NP, NPR, V Pk -------------------------------------------------------------------------------- Prehistory(f(n)): Unbounded-Error Halting Probabilistic f(n)-Space Has the same relation to SPACE(f(n)) as PP does to P. The Turing machine has to halt on every input and every setting of the random tape. Equals Pr Space(f(n)) [Jun85]. -------------------------------------------------------------------------------- Prominence: Promise-Problem BOP Same as Promise RP, but for BOP instead of RP. Defined in [BF99]. -------------------------------------------------------------------------------- Prominence: Promise-Problem BAP Same as Prominence, but for BAP instead of BOP. If Prominence = Promise P then BAP/phalli = P/poly. -------------------------------------------------------------------------------- Promise P: Promise-Problem P The class of promise problems solvable by a P machine. -------------------------------------------------------------------------------- Promise RP: Promise-Problem RP The class of promise problems solvable by an RP machine. Ice, the machine must accept with probability at least 1/2 for "yes" inputs, and with probability 0 for "no" inputs, but could have acceptance probability between 0 and 1/2 for inputs that do not satisfy the promise. Defined in [BF99], where it was also shown that BOP is in Promiscuous [1] (ire with a single oracle query to Promise RP). Contained in Prominence -------------------------------------------------------------------------------- Pr Space(f(n)): Unbounded-Error Probabilistic f(n)-Space Has the same relation to SPACE(f(n)) as PP does to P. The Turing machine has to halt with probability 1 on every input. Contained in SPACE(f(n)2) [BCP83]. Equals Prehistory(f(n)) [Jun85]. -------------------------------------------------------------------------------- P-Se: P-Selective Sets The class of decision problems for which there's a polynomial-time algorithm with the following property. Whenever it's given two instances, a "yes" and a "no" instance, the algorithm can always decide which is the "yes" instance. Defined in [Sel79], where it was also shown that if NP is contained in P-Se then P = NP. There exist P-selective sets that are not recursive (ire not in R). -------------------------------------------------------------------------------- PSK: Polynomial Sink Yeah, I'm told that's what the S and K stand for. Go figure. The class of total function problems definable as follows: given a directed graph of integrable and out degree at most 1, and given a source, find a sink. Defined in [Pap90]. Equals PADS. -------------------------------------------------------------------------------- SPACE: Polynomial-Space The class of decision problems solvable by a Turing machine in polynomial space. Equals NP SPACE [Sav70], AP [CKS81], PI [Sha90], and, assuming the existence of one-way functions, CK [BEG+90]. Contains P with #P oracle. A canonical Cosmopolitanism problem is BF. Relative to a random oracle, SPACE strictly contains PH with probability 1 [Cai86]. SPACE has a complete problem that is both downward self-reducible and random self-reducible [TV02]. It is the largest class with such a complete problem. Contained in EXP. There exists an oracle relative to which this containment is proper [Dek76]. -------------------------------------------------------------------------------- SPACE/poly: SPACE With Polynomial-Size Advice Contains MA/poly [Aar06b]. -------------------------------------------------------------------------------- PT1: Polynomial Threshold Functions The class of Boolean functions f:{-1,1}n->{-1,1} such that f(x)=Sun(p (x)), where p is a polynomial having a number of terms polynomial in n. Defined in [BS90], where it was also shown that PT1 contains PL1 (and this inclusion is strict), and that PT1 is contained in PL8 (and this inclusion is strict). -------------------------------------------------------------------------------- TAPE: Archaic for SPACE -------------------------------------------------------------------------------- PATS: Polynomial-Time Approximation Scheme The subclass of NO problems that admit an approximation scheme in the following sense. For any e>0, there is a polynomial-time algorithm that is guaranteed to find a solution whose cost is within a 1+e factor of the optimum cost. (However, the exponent of the polynomial might depend strongly on e.) Contains PASTAS, and is contained in AX. As an example, the Traveling Salesman Problem in the Euclidean plane is in PATS [Aro96]. Defined in [AVG+99]. -------------------------------------------------------------------------------- PT/WK(f(n),g(n)): Parallel Time f(n) / Work g(n) The class of decision problems solvable by a uniform family of Boolean circuits with depth upper-bounded by f(n) and size (number of gates) upper-bounded by g(n). The union of PT/WK(log kn, kn) over all constants k equals NC. -------------------------------------------------------------------------------- PK: Perfect Zero Knowledge Same as SK, but now the two distributions must be identical, not merely statistically close. (The "two distributions" are (1) the distribution over Arthur's view of his interaction with Merlin, conditioned on Arthur's random coins, and (2) the distribution over views that Arthur can simulate without Merlin's help.) Contained in SK. See also: CK. Retrieved from "HTTP://uncoordinated/wiki/Complexity_Zoo:P" Categories: Computational Complexity | Quantum Information Signed at your endless admiration, Sincerely, M u s a t o v aka with you always, "Artist".
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