A. Definitions of Reductions and Complexity Classes, and N otation List

A.I Reductions

Below, in naming the reductions, "polynomial-time" is implicit unless ot her­ wise noted, e.g., "many-one reduction" means "many-one polynomial-time reduction." This list sorts among t he :Sb first by b, and in the case of identical b's, bya.

:S -y - Gamma reduction. -A :S -y B if there is a nondeterministic polynomial-time machine N such that (a) for each st ring x, N(x) has at least one accepting path, and (b) for each string x and each accepting path p of N(x) it holds that: x E A if and only if the output of N(x) on path p is an element of B. :S ~ - Many-one reduction. -A :S~ B if (3f E FP)(Vx)[x E A {=} f(x) EB] . :S~ o s - (Globally) Positive Turing reduction. -A :S~o s B if there is a deterministic Turing machine Mand a polyno­ mial q such that 1. (VD)[runtimeMD(x):s q(lxl)], B 2. A = L(M ), and 3. (VC, D)[C ç D =} L(MC) ç L(MD)]. :S:08 - Locally positive Turing reduction. -A :S:os B if there is a deterministic polynomial-time Turing machine M such that 1. A = L(MB), 2. (VC)[L(M BUC) :2 L(MB)] , and 3. (VC)[L(M B- C) ç L(MB)] . :ST - Exponential-time Turing reduction. -A :ST B if A E EB. If A E EB via an exponential-time machine, M, having the property that on each input of length n it holds that M makes at most g(n) queries to its orade, we write A :S~ ( n ) - T B. If for some polynomial g(n) it holds that A :S~( n)-T B, we say A :S~olY-T B. :S ~ - Turing reduction. -A :S~ B if A E pB. 116 A. Definitions of Reductions and Complexity Classes, and Notation List

~Tn - Strong nondeterministic 'Iuring reduction. -A ~Tn B if A E NpB ncoNpB.

A.2 Complexity Classes

This list is, with some exceptions for clarity, alphabetically "word-ordered." Also, it orders Greek letters under their Romanized versions (e.g., I;'s are alphabetized as if they were "Sigma"s). co· C - Set-wise complements of complexity class C. A.2 Complexity Classes 117

-A E co . C if A E C. C-:=:;r-complete - Let :=:;r be any reducibility and let C be any complexity class. A set A is said to be C-:=:;r-complete if A E C and A is C-:=:;r-hard. C-:=:;r-hard - Let :=:;r be any reducibility and let C be any complexity class . A set A is said to be C-:=:;r-hard if, for each B E C, B :=:;r A. C-complete - Let C be any complexity class. A set is said to be C-complete if it is C-:=:;~ -complete. C-hard - Let C be any complexity class, A set is said to be C-hard if it is C-:=:;~-hard. C/{F} -A token-based advice class. - Let F be any col1ection of (total) functions from N to N+. Let C be any col1ection of sets. Define

C/{F} = {A I(3g E F)[A E C/{g}J).

CIU} -A token-based advice class . - Let i : N ---. N+. Assume that natural numbers have their standard encoding over binary strings. Let C be any col1ection of sets. Define CIU} = {A I(3B E C)(3h : N ---. N+) [(\ln)[h(n) E {I, .. ., i(n)}] and (\Ix E I;*)[x E A {=} (x, h(lxl)) E B]]}. ClF - A length-based advice class. - Let F be any class of (total) functions mapping from N to No Define

CIF = {A I(3i E F)[A E Clij).

- One common value of F is "poly," which denotes the class of polyno­ mials. CIi - A length-based advice class . - Let i : N ---. N be any (total) function. Let C be any col1ection of sets. Define Cli = {A I (3B E C)(3h : N ---. {a, I }*) [(\ln)[lh(n)I i(n)]/\ (\Ix E I;*)[x E A {=} (x, h(lxl)) EB]]}. coNP - Co-nondeterministic polynomial time. -A E coNP if A E NP. ó~, k ~ 0 - The kth "ó" level of the polynomial hierarchy. _ I\P _ I\p,0 Uk - uk . Óp,A k > 0 k' - - The kth "ó" level of the polynomial hierarchy relativized via oracle A. _ uaI\p,A -.'_ pA 118 A. Definitions of Reductions and Complexity Classes, and Notation List

pA EP.A - For k 2 1, /:),.k' = P k-l . - See, however, the discussion in footnote 6 (on page 43), since the analog of that holds here . Disjunctively self-reducible sets -A set A is said to be disjunctively self-reducible if there is a deterrnin­ istic polynomial-time Turing machine M such that 1. A = L(MA), 2. for each x, MA(x) queries only strings of lengths strictly less than lxi , and 3. the acceptance behavior of M is such that on each input M accepts exactly when either (a) M asks at least one query that is in the oracle set, or (b) M asks no queries and halts in an accepting state. If A is disjunctively self-reducible via a machine M that on each input asks at most two oracle questions, then we say that A is 2-disjunctively self-reducible. DSPACE[f(n)] - Deterministic space f (n). -A E DSPACE[f(n)] if A is accepted by a deterministic Turing machine whose running space is O(J(n)). DTIME[J(n)] - Deterministic time f (n). -A E DTIME[f(n)] if A is accepted by a deterministic Turing machine whose running time is O(J(n)). E- Deterministic exponential time. -E = Uk>o DTIME[2 kn]. EL6.~' k22 - The /:),.1 extended low sets. -A E EL6.~ if /:),.1 ,A ç /:),.1~lEBSAT . EL~p, k > 2 k - - The /:),.1 extended low sets in relativized world W. -A E EL~~ if /:),.1,AEBW ç /:),.1~lEBSATEBW . ELH - The extended low hierarchy. - ELH = Uk~2 ELE~' ELE~' k 2 2 - The E1 extended low sets. -A E ELE~ if E1,A ç E1~lEBSAT. ELe~, k 2 2 - The 81 extended low sets. -A E ELe~ if 81,A ç 81~lEBSAT. Er(C) - The sets :::::>equivalent to some set in C. -A E Er(C) if (:JB E C)[A :::;r B 1\ B :::;r A]. - Note: :::;r must he a defined reduction type. A.2 Complexity Classes 119

EXP - Deterministic "polynomial exponential" time. nkj. - EXP = Uk>O DTIME[2 F-sel - Selectivity via general functions. - Let F be a class of functions. We say a set A is F-selective if there is an f E F such that, for each x and y, 1. set- f(x, y) ç {x, y}, and 2. ifAn{x,y}=f:.0then0=f:.set-f(x,y)ÇA. We say such a function f is an F-selector for A. Let F be a class of functions. F-sel denotes {A IA is F-selective}. FewP - Polynornial-ambiguity (nondeterministic) polynomial time. -B E FewP if (3 polynomial-time 2-ary predicate R)(3 polynomial q) (3 polynomial r)(Vx)[(II{z Ilzl ::; q(lxl) 1\ R(x, z)}11 ::; r(lxl)) 1\ (x E B {::::::::> (3y)[Iyl::; q(lxl) 1\ R(x, y)])j . FP - Deterministic polynomial-time computable functions. - f E FP if f is a (total, single-valued) function computable by a deter­ ministic polynomial-time Turing machine. - All function classes are implicitly partial (unless subscripted with a ''t'' to denote totality) except FP, which by longstanding convention represents the total functions that are computable in deterministic polynomial time. HH - The high hierarchy. - RH = Uk~O HEf:' HEf: ' k ~ 0 - The E~-high sets. -A E HEf: if A E NP and E~,A 2 E~+l' Lc - The C-low sets. -A E Lc if A E NP and CA = C. - Note:C must be a class for which relativization has been defined. L;f - C-low sets in oracle world W. w -A E L;f if A E Np and CAal W = CW . - Note:C must be a class for which relativization has been defined. LH - The low hierarchy. - LH = Uk~O LEf:' NE - Nondeterministic exponential time. - NE = Uk>O NTIME[2kn]. NEXP - Nondeterministic "polynomial exponential" time. nkj - NEXP = Uk>O NTIME[2 . NNT - The implicitly membership-testable sets (also known as the nearly near-testable sets). -A E NNT if (3f E FP)(Vx)[(f(x) = "in" 1\ x E A) V (f(x) = "out" 1\ x 1: A) V (f(x) = "xor" 1\ lI{x,predecessor(x)} n All = 1) V (f(x) = "nxor" 1\ II{x,predecessor(x)} n All == 0 (mod 2))j . NP - Nondeterministic polynomial time. - NP = Uk>O NTIME[nkj. 120 A. Definitions of Reductions and Complexity Classes, and Notation List

- BE NP if (3 polynomial-time 2-ary predicate R)(3 polynomial q)(Vx) [x E B ~ (3y)[lyl ~ q(lxl) 1\ R(x ,y)]]. (NP ncoNP)/poly - See entry for C/ F. NP/poly - See entry for C/ F. A Np - Nondeterministic polynomial time relative to orade A. - The class of Ianguages accepted by nondeterministic polynomial-time machines given unit-cost access to orade A. -B E Np A if there is a 2-ary predicate R-computabie in polynomial time relative to A (i.e., being informal about the type of the class pA, R E pA)-and there is a polynomial q such that: (Vx)[x E B ~ (3y) [Iyl ~ q(lxl) 1\ R(x, y)]]. NPMV - The multivalued non deterministie polynomial-time functions. - Each nondeterministic polynomial-time Turing machine is considered to be a function-computing machine as follows. Each path that rejects is considered to have no output. Each path that accepts is considered to output the string of characters stretching, at the moment that path accepts, from the left end of its semi-infinite worktape through (but not induding) the character underneath its worktape head. NPMV denotes the class of all functions f that can be computed (in the sense just stated) by some nondeterministic polynomial-time Turing machine. NPMV-sel - See entries for F-sel and NPMV. NPMVt - The total, multivalued, nondeterministic polynomial-time functions.

- fE NPMVt if fis total (i.e., for all x and y, Ilset-f(x,y)11 > 0) and i « NPMV. NPMVt-sel - See entries for F-sel and NPMVt » NPSV - The single-valued nondeterministic polynomial-time functions. - i e NPSV if fis single-valued (i.e., for all x and y, Ilset-f(x,y)11 ~ 1) and f E NPMV. NPSV-sel - See entries for F-sel and NPSV. NPSVt - The total, single-valued, nondeterministic polynomial-time functions. - t « NPSVt if f is total and fE NPSV. NPSVt-sel - See entries for F-sel and NPSVt. A.2 Complexity Classes 121

NT - The near-testable sets. -A E NT if (~f E FP)(Vx)[ (f(x) = "xor" 1\ 11 {x , predecessor(x)} n All = l)V (f(x) = "nxor" 1\ 11 {x, predecessor(x)} n All == 0 (mod 2))] . NTIME[f(n)] - Nondeterministic time f(n). -A E NTIME[f(n)] if A is accepted by a nondeterministic Turing ma- chine whose nondeterministic running time is O(f(n)). P- Deterministic polynomial time. k - P = Uk>O DTIME[n ]. P-close - The P-close sets. -A EP-close if (~B E P)(~S E SPARSE)[A = B6S], where B6S = (B-S)u(S-B). P-sel - The P-selective sets ; the semi-feasible sets . -A E P-sel if (~f E FP)(Vx, y)[J(x , y) E {x , y} 1\ (f(x, y) nA f 0 ==} f (x, y) E A)]. Such a function f is called a P -selector function for A. P jpoly - See entry for Cj F. pA _ Deterministic polynomial time relative to oracle A. - The class of languages accepted by deterministic polynomial-time ma­ chines given unit-cost access to oracle A. pAffen)] - Deterministic polynomial time relative to oracle A, with a bounded number of queries. - The class of languages accepted by deterministic polynomial-time ma­ chines given unit-cost access to oracle A and allowed on each input x at most f(lxJ) oracle queries. - The obvious generalizations of this from a single oracle to a class of oracles, or from a single function to a class of functions, or both (e.g., pC[F]) are defined and used in the obvious analogous ways. PH - The polynomial hierarchy. p NP NpNP - PH=Uk>oEk =PuNPuNP uNP U· ... - Note: Wesay that the polynomial hierarchy collapses if (~k)[PH = E~] . PP - (Unbounded error) Probabilistic polynomial time. -L E PP if there is a nondeterministic polynomial-time Turing ma­ chine M such that, for every input x, it holds that x E L if and only if more than half of the computation paths of M (x) are accepting paths. PSPACE - Polynomial space. k - PSPACE = Uk>O DSPACE[n ]. qP - Quasipolynomiäl time. - qP = Uk~O DTIME[21og k "l- 122 A. Definitions of Reductions and Complexity Classes, and Notation List

R- Random polynomial time. -B E Rif (3 polynomial-time 2-ary predieate R)(3 polynomial q)(Vx) [(x rf- B ==? 11 {z Ilzl ::; q(lxl) 1\ R(x, z)} 11 = 0) 1\ (x EB==? II{z Ilzl ::;q(lxl) 1\ R(x, z)}11 / II{z Ilzl ::; q(lxl)}11 ~ 1/2)]. Rr(C) - The sets that ::;r-reduee to some set in C. -A E Rr(C) if (3B E C)[A ::;r B]. - Note: ::;r must be a defined reduetion type. S2 - The seeond level of the symmetrie alternation hierarchy. -A E S2 if there exists a set REP and a polynomial q such that, for every x, 1. if x E A then (3y : Iyl ::; q(lxl))(Vz: Izi ::; q(lxl))[(x , y, z) ER], and 2. if x rf- A then (3z : Izi ::; q(lxl))(Vy : Iyl ::; q(lxl))~x , y, z) rf- Rl. - It is known that BPP U t,,~ ç S2 ç ZppNP ç NpN . S~pncoNP - The second level of the symmetrie alternation hierarehy rela­ tivized to NP n coNP. -A E S~pncoNP if there exists a set R E pNPncoNP (equivalently, R E NP ncoNP) and a polynomial q sueh that, for every x , 1. if x E A then (3y : Iyl ::;q(lxl))(Vz: Izi ::; q(lxl))[(x, y, z) ER], and 2. if x rf- A then (3z : [z] ::; q(lxl))(Vy : Iyl ::;q(lxl))[(x, y, z) rf- Rl. NP - It is known that BPP U t"P2-cse2_ SNPncoNP2C -Zpp C - NPNP . Semi-reeursive sets - We say a set A is semi-recursive if (3 reeursive function f)(Vx, y) [(J(x, y) = x V f(x, y) = y) 1\ (J(x, y) n A =I- 0 ==? f(x, y) E A)]. set-f - Outputs of a multivalued function. - Let f be any (possibly partial, possibly multivalued) function. For any strings x and y, set-f(x,y) denotes {z Iz is an output of f(x,y)}. 1:g - The kth level of the arithmetical hierarchy (also known as the Kleene Hierarchy). 1:~, k ~ 0 - The kth "1:" level of the polynomial hierarchy. _ "p _ "p,0 4Jk - ~k . 1:p ,A k > 0 k ' - - The kth "1:" level of the polynomial hierarehy relativized via oracle A. _ "p,A _ pA ~o -. - For k ~ 1, 1:~ ,A = NpEJ: :~\. SPARSE - The sparse sets. -A E SPARSE if (3 polynomial q)(Vn)[IIA=nll ::;q(n)]. TALLY - The tally sets. -A E TALLY if A ç {E, 1, 11, 111, . . .}. A.3 Some Other Notation 123

e~ , k ~ 0 - The kth "8" level of the polynomial hierarchy. p - p - e k - e k,0 . ep,A k > 0 k ' - - The kth "e" level of the polynomial hierarchy relativized via oracle A. _ ep,A _ pA o - . pEt~'\ [O (logn) l - For k >-,1 epk ,A = , where as usual pC!FJ denotes the union over all sets A in C and all functions f in F of the class of languages acceptable by P machines with oracle A that on each input x make at most f(lxl) orade queries . - See, however, the discussion in footnote 6 (on page 43). Turing self-reducible sets - A set A is Turing self-reducible if there is a deterministic polynomial­ time 'Iuring machine M such that A = L(MA) and, for each x, MA(x) queries only strings of lengths strictly less than lxi . UP - Unambiguous (nondeterministic) polynomial time. -B E UP if (3 polynomial-time 2-ary predicate R)(3 polynomial q) (Vx)[(II{z Ilz/ ::; q(lxl) 1\ R(x,z)}11 ::; 1) 1\ (x E B {:::::::} (3y)[ly/::; q(lxl) 1\ R(x, y)])]. - We say a Turing machine Nis unambiguous if and only if, for all x , it holds that N on input x has at most one accepting computation path. A set is in UP exactly if it is accepted by some polynomial-time, unambiguous Turing.. zpp - Expected polynomial time. - A set is in ZPP if there is a probabilistic Turing machine that accepts the set (without error) and whose expected running time is polynomi­ ally bounded in the length of the input. - It is known that ZPP = R n coR.

A.3 Some Other Notation e The empty string. a E A a is a member of set A. a ~ A a is not a member of set A. =,:;i,::;,~ ,<,> Standard arithmetic relations. = ,Ç,2,~ ,~ ,Ç,~ Standard set relations. o The empty set . An {x (3z}, ... , Zn E A)[x = ZlZ2 '" znJ) . A=n {x X E A l\ lxl = n}. A Sn {x xEAl\ lxl::;n}. A

A6.B (A - B) U (B - A). A The complement of A: E* - A. AuB {XIXEAVXEB}. al\b Logical "and" of the two boolean variables. avb Logical "or" the two boolean variables. --, Logical negation of a boolean value. IIAII The cardinality of set A. A-B AnB. AnB {XIXEAI\XEB}. AE9B {Ox Ix E A} U {ly Iy EB}. N {O, 1, 2, }. N+ {I, 2, 3, l- E The input alphabet, which unless otherwise stated we assume consists of at least two characters, 0 and 1. Unless otherwise stated, sets are subsets of E*. AIR The set of equivalence classes of A with respect to equiv­ alenee relation R. cl(a) The equivalenee class containing a (with respect to a set of equivalence classes that is implicit from the context). ~f When f is a symmetrie P-selector function, we will some­ times use a ~f b to denote f(a,b) = b. -< Used to denote a linear ordering. O(J(n)) Let 9 : N ~ N and f : N ~ N. We say g(n) = O(J(n)) if (3c> O)(3no 2: O)(Vn)[n 2: no ~ g(n) ~ cf(n)]. References

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ti 124 - class of", interpreters 31 11 All 124 - improvement below linear '" for L. 124 recognition of P-selective sets 31 o 123 - length-bounded '" 19 =1-tt 98,99 - linear r-- 24,26,30,31,39 ~ le x 82 - logarithmic '" 39 n 124 - lying '" 33 EB 124 - po lynomial '" 19,39 lal 123 - quadratic '" 30 ~i-tt 74,75,88,90,95 - recursive '" 39 ~~-tt 96,103 - small '" 3 ~~tt 94,116 ~f 21-29,88-91 ,95,99,101,124 - strong '" 35-37 ~î' 74,75, 115 Agrawal 77,113 ~:(n)-T 115 algorithm 16,17,24,26,30,47,52,64, 70,72,86,90,92-94 ~1-T 100 ~1- tt 72,73,98, 116 - '" finding the king of a tournament 63 ~ lex 82 ~~ 2,4,5,8,29,30,46,61,66,73,74, - deterministie polynomial-time '" v, 85,105-107,11 5,117 1,2,5,29,30,64-66,69,72,89 ~~oIY-T 115 - int elligent '" vi ~~o s 8,81,85,86,88,115 - linear-exponential-tirne '" 27 ~ptt 8,85,86, 116 - nondeterministic polynomial-time '" ~r 8,61,85,94,117,118,122 14,90,93 ~T 115 - nondeterministie reeursive '" 92 ~!;. 2,8,27,42,44,67,68,76,99,115 - output of a polynomial-time '" v ~rn 42,44, 116 - P-time deeoding '" for adviee 23 ~ft 8,61,62,65-70,94,108,116 - polynomial-time membership '" 2 124 - reeursive '" v 124 - semi-membership "'s 2 V 124 - smart '" vi ~ 124 - standard brute-foree conversion of a U 124 nondeterministic >- to a deterministic Abadi 40 26 advice 19,20,24,27,28,30-36,39,52, - theory of semi-feasible "'s iii 56,109 Allender vii, viii, 58, 59 - '" for nondeterministically selective ambiguity sets 32 - po lynomial '" 119 - '" for P-selective sets 17,20 Amir 59 - '" interpreter see interpreter, analog advice - complexity-theoretic '" v , 1 134 Index

- complexity-theoretic '" of the low upper "'S on the amount of advice and high hierarchies Erom recursive for P-selective sets 21 function theory 58 upper and lower ",s 49 - nondeterministic '" of P 9 upper and lower "'S on lowness 46 nondeterministic '" to standard de­ Bovet vii terministic results on self-reducibility BPP 122 75 Brauer vin Appel 16 Buhrman vii, viii , 76, 77, 103 approximability 105 Burtschick 39, 76 - bounded '" 113 APT - almost polynomial time 2,16 Cai viii , 39,40,59,76,77, 113 argument 1,4,10,17,23,32,33,63, certificate 33 69,76,84,93,106,108 - '" certifying the guessed answer 12 - counting '" 22 - ad vice '" 34 - divide and conquer '" 54,57 - membership '" 33,34 Arvind 77, 113 - succinct '" 12 assignment Chakaravarthy 40 - collection of "'s 72, 73 Chang 59 - nondeterministie selection of one '" characterization 38 - complete", of the P-selective sets 6 - satisfying '" 72 - complete", of the semi-recursive sets associativity 105,111-113 6 assumption 2,29,30,32,35,37,46,67, Chari 59 74,94,101 circuit 18, 19 - complexity-theoretic "'s 2 - '" for a P-selective set 18 - '" for an arbitrary language 18 Babai 59 - '" for r;* 18 Balcázar vii, 58, 59 '" recognizing a sparse set 17 bAPP 113 - collection of "'s 18 Barrington 16, 39 - encoding of a '" 18 Beigel 59, 76, 77, 103, 113 - exponential-size "'S 18 bit 17,19,23,24,27,29,31,33,66, - families of "'s 17 70-73,82,86,87,91,97-99,116 - polynomial-size "'s 19 - "'s of advice 19,20 - polynomial-sized family of "'s for - input "'s 18 P-selective sets 18 - nondeterministic guess '" 62, 69 - size-bounded "'S 19 bitflip 99 - small "'s v, 3,18,107 bitstring 29,71,97,98, 109, 116 claim 4, 13,24, 25, 45, 48, 52, 63, 68, Book 16,58,59 74,96,103 bound 24,32,46,48,51 - relatively typical optimal lowness '" - absolute lower '" on lowness 46 48 - absolute upper "'s 46 - relativizable '" 13 - adjacent upper and lower "'s 46 class 11,12,41,43,48,105,106,113 - length '" 24 - '" closed under composition with - lower r- 20,30-32,39,46,48,58,59 logspace functions 20 - nontrivial lower '" on lowness 48 '" containing Turing self-reducible - polynomial r- 47 complete sets 75 - query r- 43 - '" known as MAExP 76 - relativized lower "'s on lowness 58 - '" of advice interpreters 31 - time >- 16,26,96 - rv of functions 121 - upper '" 17,20,21,25,28,32,34, - '" of functions computable via 46,49,51,53,55,57-59 G(log n) Turing queries to NP 68 Index 135

- '" of functions computable via - incomparable "'es: P-sel and polynornial-time truth-table access weakly-P-rankable 110 to NP 68 - LdP "'es 58 k - "'oflanguages 120,121,123 - length-based advice '" 117 - '" of oracles 121 - low r-es 43 - '" of P-selective sets 81 lowness "'es 43 - '" of sets having interactive proofs - Lr;p "'es 43,58 k 59 - LeP "'es 58 - '" of sets of simpIe organization 43 k - membership in nondeterministic "'es - '" of sets reducible to P-sel 26 80 - '" of the form C/poly 34 - membership in the complexity '" P - <-es in the extended low hierarchy 1 45 - nondeterministic function "'es 9 - "'es of nondeterministically selective - nondeterministic selectivity "'es 9, sets 49 81 - <-es that are not subsets of P /poly - optimal lower bounds for most 76 extended-lowness "'es 59 - "'es that lack hard P-selective sets - reduction '" 97 61 - reduction and equ ivalence "'es 94, - advice '" 19,34 95 - advice upper bounds for reductions - refined advice '" 20 to selectivity "'es 22 - refinement of multivalued nondeter- - advice upper bounds for selectivity ministic function "'es 40 "'es 21 - relationships between nonde­ - arbitrary '" of selector functions 9 terministic selectivity "'es 12, - classic low "'es 41 13 - collapse of "'es 12, see collapse relativizable '" 43 - complement of a '" 11 relativized ~~ '" 43 - complexity '" 3,8,16,19,41,61,62, - selectivity "'es 10 79, 116,117,119 - semi-recursive sets as a '" from - complexity "'es near polynomial time recursive function theory v 2 - separation of reduction "'es from - degree of organizational simplicity of equivalence "'es 96 selectivity "'es 43 - set of equivalen ce "'es 7,124 - ELdJ: "'es 58 - set-wise complements of a complexity - ELr;p "'es 58 k 116 - ELeJ: "'es 58 - structure of polynomial-time - equalit ies and inequalities of re- complexity "'es 48 duetion and equivalence "'es of - 8~ "'es 43 P-selective sets 103 - time-bounded o- 20 - equivalence r- 8,97,100,124 - token-based advice '" 117 - exponential-time complexity "'es 7, classification 16 - '" of sets in NP using the low - first two levels of the high hierarchy hierarchy 42 are well-known "'es 42 clique - function "'es 19, 119, see function, - q-'" 63 class of closure - function analog of <-es 68 - '" properties of P-sel 6 - hardness for complexity "'es 61 - '" under 2-ary connectives of P-sel, - high", 42 NPSVt-sel, and NPMVt-sel 84 - inclusion properties of nondetermin- - '" under bounded-truth-t able istic advice "'es 39 reductions of a class 81 136 Index

- rv under combined self-reducibility - ,....., of the polynomial hierarchy being and 1-truth-table reductions of P-sel a consequence of unique solutions for 103 SAT 38 - rv under complement of NPMV-sel - surprising ,....., of complexity classes 15,82 61 - rv under complement of NPMVt-sel - unexpected ,....., of complexity classes 11,15,34 3 - rv under complement of NPSV-sel collection and NPMVt-sel 81 - ,....., of strings 53, 56, 57 - rv under complement of P-sel 24, commutativity 111, 112 79,81,83 comparability - rv under conjunctive reductions of - membership r- 105, 113 NP 79 - G(logn) membership r- 108 - rv under connectives of P-sel 102 complement - rv under connectives of selectivity - ,....., of a P-selective set 24, 26, 79 classes 83, 84 - ,....., of a set 124 - rv under disjunctive reductions of NP - ,....., of an NPSV-selective set 12 79 complementation - rv under intersect ion of P-sel 82 - ,....., and connectives 84 - rv under k-ary connectives of P-sel , - closure under ,....., of NPMV-sel 82 NPSVt-sel , and NPMVt-sel 84 - closure under rv of NPMVt-sel 11, - rv under many-one reductions of NP 15 79 - closure under ,....., of NPSVt-sel and - rv under many-one reductions of the NPMVt-sel 81 levels of the low hierarchy 46 - closure under ,....., of P-sel 5, 24, 79, 81,83 - rv under nonpositive reductions of completeness 61,117 P-sel 79 - ,....., for NP v - rv under NXOR and XOR of - rv for NP under many-one reductions nondeterministie selectivity classes 2 83 - rv for NP under Turing reductions - rv under positive reductions of P-sel 2 103 - :S ~ - ""'" for UP 66 - rv under positive Turing reductions n - ,....., for NP 42 of P-sel 79,80 - :sr - C-""'" 61,117 - rv under reductions of P-sel 80,85 - C-:S~-,....., 117 - ,....., under Turing reductions of EXP - C-:Sr-rv 61, 117 79 - high hierarchy as a hierarchy of - boolean ,....., of a complexity class 79, generalized ,....., notions 43 81 - NP-rv 42,43 - downward ,....., of P-selective sets 79 complexity 8 - downward ,....., under l-truth-table - rv in terrns of deterministic time v reductions 88 - advice rv 111 - extension of ,....., to NPSVt-selectivity - arbitrary c- 8, 17,39 103 - capture of rv v - reduction r- 85,88 - computational >- vii - relativized world ,....." optimal for - computational rv of a gappy left cut self-reducible P-selective sets 103 31 collapse - computational ,....., of a P-selective set - rv of the boolean hierarchy 76 18 - ,....., of the low hierarchy 48 - computational ,....., of a tally set 17 ,....., of the polynomial hierarchy vi, 2, - concept from computational rv vii 10,38,40,42-44,54,55,57,108,121 - deterministie time rv v Index 137

- left cuts capture the "" of real - transition rv 97,98 numbers v Crescenzi vii - nonuniform "" 17 cut - semi-membership e- 1,3 - computational complexity of a gappy - types of r-- v left rv 31 computability - gappy left rv 31,96,99 - rv in deterministic polynomial time - left rvS capture the complexity of 119 reals v - rv in FPftP 70 - nonempty parts of a gappy left rv - rv in polynomial time 120 31 - deterministie polynomial-time rv - standard left rv of real numbers 3 119 - time-bounded left rvS 16 - easy rv 100 - partial polynomial-time rv 7 decidability 8 degree - polynomial-time e- v, 6,62,88 - recursive rv 31 - rv in NP 8 computation 19,20,33,97, 116, 121 - "" of organizational simplicity of - rv of the value of a circuit 18 selectivity classes 43 - ::;~ _rv 8 - accepting '" 93 - ::;~o s-rv 8 - feasible e- 2 8 - nondeterministic guess of a "" 14 - ::;r-rv - ::;!;._rv 8 - polynomial-time rv 2 - semi-feasible rv v-vii, 1, 2, 16 - Sft-"" 8 - maximum out-e- 63 - world of r-- v - node with maximum out-r- 76 computers - NP rv 8 - making rv smarter VI - recursively enumerable rvS 15 computing - reducibility "" 8 - intuitive rv vi - sets in an NP '" 8 coNE 8 ~b,A 117 conneetion ~~ 61,66,67,122 - structural '" v ~~ 43, 117, 118 connect ive ~~ ,A 45,117,118 - rvS and complementation 84 diagonalization 96,97,99-102 - almost-completely degenerate ""S Diaz vii 80,83 domain - boolean rvS 79,80 - rv of an NPSV function 39 - complementation as a rv 84 DSPACE 118,121 - completely degenerate ""S 80,83,84 DTIME 2,8,31,68,71,82,118,119, - degenerate rvS 84 121 - identity rv 84 Du vii - nondegenerate ""S 83,84 - under which "'s are P-selective sets E 8,22,26,27,109,115,118 closed 79 Ei-T coNP 12-16,20-22,24-26,32-34,37, - ""(P-sel) 95 38,40,42,44,54-56,68,74-76,78,82, El-tt 88,90,116,117,120,122 - rv(P-sel) 95, 97 are nondeterministically selective EATCS i sets hard for "" 73 El;'tt coNPjpoly 21,33,34 - rv(P-sel) 97,100 containment E~_T - nonuniform ""S 67 - rv(P-sel) 97, 100, 103 coR 123 E~_tt count - rv(P- sel) 97,99, 100, 103 138 Index

EL~~ 45,46,48,58, 118 - rv computable by a determinist ie polynomial-time Turing machine EL:p 48,118 k 119 ELH 45, 54, 55, 118 - (A,k)-sort rv 113 EL EP 45-50,54,55,58,59,107,118 k - advice rv 19,23 ELe~ 45,46,49, 55, 58, 107, 118 - almost completely degenerate rv 80 van Emde Boas vii, 103 - associative rv 111 enumerability see P-enumerability - boolean rv 79, 83 E 123 - characteristic rv 81,84 equality 43,68 - class of rvs 9,19,117,119 - complete rv versus weak rv 16 - class of rvs computable via G(log n) - notion of rv for partial functions 16 Turing queries to NP 68 equivalence 81,99 - class of rvS computable via truth-table - Turing rv between tally sets and access to NP 68 P-selective sets 17, 30 - collection of rvS 20,117 E~ - completely degenerate rv 80 - rv(P-sel) 97,99,100 - complexity-theoretic study of Ert one-way rvS 16 - rv(P-sel) 97,100 - computable rv 62, 64, 65 example 2,3, 5, 6, 8, 9, 13-15, 17, 18, - degenerate rv 80 21,41,43,46,48,58,61,71,74-76,79, - deterministie polynomial-time 81,82,85,90,97,105,108,109 computable rvs 119 - classic rv of P-selectivity 3 - FPP-selector rv 109 - counter-> 50 - general classes of rvS 16 EXP 27,37,38,66,67,76,79, 119 - logspace rvS 20 - multivalued rv 122 Jê-sel 9,119,120 - multivalued nondeterministic fair-S(k) 106 polynomial-time rvS 120 fair-S(n) 106,107 - multivalued symmetrie rv 10 feasibility - nondegenerate boolean rv 84 - serni-e- vii, 1 - nondeterministic selector rv 32 Feigenbaum 40 - notion of equality for partial rvs 16 FewP 68,69, 119 - NPMV rv 10 FEXP 82,109 - NPMVt rv 10 FEXP-sel 82 - NPSV rv 10 flier - NPSV-selector rv 32 - taking a rv vi - NPSVt rv 10 formula - P-selector rv 1,3-7,21,23,25,26, - boolean rvS 35,36 28,31,32,35,49,63-65,85,88,89,92, - satisfiable rvS vi, 5, 38, 73 94-96,98,99,102,111-113,124 Fortnow viii, 59, 76 - partial rv 16 FP 9,10,12-15,32,68,70,71,77,88, - partial multivalued rv 9,10,122 96,111,112,115,119,121 - polynomial-time computable rv 7, - relativized rv 14 65,107 FpNP[O(log n)] 68,70,71,77 - probabilistic selector rv 105, 109 FPP 109,113 - ranking rv 110 FP~ 70 - selector rv v, 7, 10,32,53,57,81,90, FPftP 68,70,71,77 92,96,98,105-107,109,121 Fpx 96 - selector rv for NPSVt-sel sets 32 fraction - selector rvS sensitive to the order of - dyadic rational rv 3 the arguments 4 function - single-valued rv 10,56, 120 Index 139

- single-valued deterministie Hemaspaandra iii, iv, vii , viii , 15,16, polynomial-time eomputable rv 39,40,58,59,76-78,102,103,112, 9 113, see Hemaehandra - single-valued nondeterministic Hempel vii , 113 polynomial-time rv 120 HH 42,44, 119 - symmetrie seleetor rv 4,22,28,34, hierarchy 96, 101 - arithmetical rv 8, 122 - total rv 117,119 - close conneetion of the extended low - total multivalued nondeterministie rv to the low rv 45 polynomial-time rv 120 - collapse of the boolean rv 76 - total reeursive rv 96 - eollapse of the low rv 48 - total seleetor rv 32 - collapse of the polynomial rv vi , 2, - total single-valued rv 119 10,40,42-44,54,55, 57,108,121 - total single-valued nondeterminist ie - decomposition of NP via the low rvs polynomial-time rv 120 and high rvs 44 - uneomputable rv 18 - extended low rv 118 Furst 39 - high rv 42 - Kleene rv 122 Gabarró vii - low rv 41 Gasareh vii, 59, 113 - lowness rv 43 gate 18 - multiselectivity rv 107 - and rv 18 - polynomial rv and small circuits 35 - exponentially many rvS 18 - relativized polynomial rv 117,122, - not rv 18 123 - or rv 18 - S(k) rv 106 - polynomial number of rvS 18 Hoene vii , 16,39,40,76-78,103,113 Gavaldà 59 Hofmann viii generalization 15, 105-110, 121 Holzwarth viii - rvS of P-seleetivity 105 Homan vii Gil! 113 Homer vii GlaBer 76 HE ~ 42, 44, 119 Goldsmith 15, 113 de Graaf vii immunity guess - C-rv 110 - nondeterministie rv bits 62 - p _rv 110,111 - nondeterministie rv of a eomputation - P-sr-rv 111 path 53 - weakly-Pcrankable-r-- 111 Gundermann 76 incomparability - rv of EHP-sel) and Rl:tt(p-sel) 100 hardness 61,117 - rv of E~(p-sel) and Rit(p-sel) 100 - :::; fcrv 66 - rv of Eit(p-sel) and Rl:tt(p-sel) 100 - eoNP-:::;~-rv 73 inequality 29,42 - NP-rv 61 interpreter - NP-:::;i_tt-rv 74 - advice rv 20,24,31,34,35 - NP-:::;-y-rv 74 - advice rv for SAT 35-37 - NP-:::;ft-rv 68 - class of advice rvS 31 - truth-table rv for NP 67 - nondeterministic advice rv 24, 26, - Turing rv for NP 67 28 Hartmanis 59,76 - NP advice rv 25,26,30 hashing - Padvice rv 30 - half rv 40 - reeursive advice rv 39 Hem aehandra 15,58,59 ,76 ,103 , see intuition vi Hemaspaandra - skating on rv vi 140 Index

Jain 103 loop Jenner 77 - self "-'s 63 Jiang vii, 58,102, 112 Low Jockusch 15,16,103 - "-'(C) 119 Joseph 15,16,113 - "-'(~b) 41 - ,,-,(~n 41 k-walk 97-99 - "-'(~~) 41 - self-avoiding "-' 97 - "-'(~~) 41 Kadin 76 lowness 41-43,45,46,55,58,59 Kämper 40 - "-' of all four types of nondeterminis­ Karloff 59 tically selective sets 58 Karp 20,35, 39, 40 - "-' of nondeterministically selective Karp-Lipton sets 49,58 - "-' Theorem 35 - "-' of NP n P-sel 48 - relativized version of the "-' Theorem - "-' of P-selective sets 46,49 38 - analysis in terms of "-' 58 Kilian 40 - best currently known upper bounds king for "-' of selective sets 51 - "-' of a tournament 24 - best currently known upper bounds Kleene 16 for extended "-' of selective sets 50 - "-' hierarchy 122 - definition of "-' 41 Ko vii, 16,39,58,59,75, 112 - extended "-' 43,46-49, 59 Köbler 16,40, 58, 59 - extended "-' of all four types of Kummer 77,103,113 nondeterministically selective sets 58 Landau 39 - extended-,,-, bounds 50 language see set - extended-r- structure of P-selective - tally "-' see set , tally sets 48,49 Lc,.p 43-45,48,58 - extended-r- upper and lower bounds k 46 p 48 L:k - extended-r- upper bounds 55 Lemma - generalization of "-' 45 - Toda Ordering "-' 76 - more general "-' result 40 - Toda's "-' 76 - nontrivial lower bound on "-' 46, 48 length - refined "-' 43 - advice "-' 34 - upper and lower bounds on "-' 46, - bit-r-- 23 49 - linear "-' 19-22,26,28,30,39, 105, - upper and lower bounds on extended 112 49 - polynomial "-' 19-24,27,32-40,49, Lp 41 54,55,67,68,76,78,105,107-109, LEP 41-45,47-49,51,54,55,57-59, 120,121 k - quadratic "-' 19,20,24,39, 112 107,119 Lep 43-45,50,55,58 LH 41,42,44,46,48,57,58,119 k Lindner 39,76 Lund 59 linear 19 Lipton 20,35,39,40 machine 9,24,27,28,32,47,52,56, list 65,67,69,81,92,95,99,100,102,103, - ~uery "-' 116 116,118 L k 119 - bottleneck "-'s 39 LNP 41 - deterministie f(n)-space Turing "-' logarithm 118 - implicit base of "-' 23 - deterministie f(n)-time Thring "-' Long 16,58,59 118 Index 141

- deterministie polynomial-time Turing nonelosure 2,47,88,106,118,119,123 - "-' of ELE~ under many-one - enumeration of partial reeursive "-'S reduetions 46 31 - "-' of the extended low hierarchy 46 - exponential-time e- 27 - "-' under a funetion of P-sel 84 - FewP,,-, 68 - "-' under intersection for all selectivity - FP,,-, 12 classes 82 - FPft "-' 71 - "-' under k-ary connectives of - funetion-eomputing Turing "-' 10, NPSV-sel and NPMV-sel 84 120 - "-' under nondegenerate connectives - nondeterministie polynomial-time of NPSV-sel and NPMV-sel 84 funetion-eomputing Turing "-' 10 - nondeterministic polynomial-time - "-' under reduetions of P-sel 85 Turing "-' 10,11,14,29,38,47,50, - "-' under union of selective sets 82 55,62,65,69,90,109,115,120,121 - simultaneous capture of "-' under - NP "-' 53,56,67 intersection for all versions of - oracle "-' 47,86 selectivity 82 - p A EI1 SAT "-' 47 nondeterminism vi, 39 - polynomial-time oracle "-' 29 - linear amount of "-' 39 - polynomial-time Turing "-' 27,69 - understanding of "-' 9 - probabilistic polynomial-time Turing notation 123 123 notion - query-cloeked "-' 100 - "-' of being "easily k-countable" 113 - simulating "-' 27 - "-' of equality for partial functions - Turing "-' 5,116 16 - Turing reduetion "-' 94 - "-'s closely related to P-selectivity - unambiguous polynomial-time Turing 109 67,123 - "-'S related to membership compara- - unambiguous Turing "-' 67 bility 108 Magklis viii - advice "-' 35 Mayer viii - complexity-theoretic "-' vii MeLaughlin 16 - refinements of the "-' of membership measure comparability 108 - "-' of resource 19 NP v, 2, 3, 5, 8, 9,11-16,20-22,25,26, - complexity "-'s 107 29,30,32-59,61,66-72,74-79,81,82, measurement 88,90,91,94,103,105,107,108,111, - fine-grained "-' of advice 20 112,116,117,119-122 Meyer 16 - complete for "-' v MinimumPath 62,65-67,69,70 - completeness for "-' under many-one f\j 124 reductions 2 f\j+ 124 - completeness for "-' under Turing Naik vii , 16,39,40,59,76-78,103 reductions 2 Nasipak vii , 39 - relativized "-' 42,43,47,48, 119, 120 nature NPjpoly 21,33,34 - nondeterministic "-' of gamma NPMV 10,11,15,16,34,39,54,74,91, reductions 90 92,120 NE 8,119 NPMV-sel 10,11,13,21,34,41,50,51, NEXP 119 54,55,57,58,74,75,82-84,120 Nickeisen vii , viii, 113 NPMVt 10-12,15,16,34, 54,74,75, Nisan 59 81,90,91,120 NNT 15 NPMVt-sel 11-13,21,34,41,50,51, - implicitly membership-testable sets 54,55,57-59,74,75,81-84,90,120 2,119 NpNP 55,57,121 142 Index

NPSV 9, 10, 12, 15, 16,32,34,38,39, - El-tt(--") 95,97 49,50,52,55,56,74,120 - E~tt("') 97,100 NPSV-sel 10-13,21,32-34,41,49-51, - Et-T("') 97,100,103 54,55,57-59,68,74,83,84,120 - Et-tt(",) 97,99,100,103 NPSVt 10,12-16,74,81,88,90,103, - E~( "') 97,99, 100 120 - Eft("') 97,100 NPSVt-sel 11-14,21,32,41,50,51,54, - Ri-T("') 95 57,58,68,74,81,83,84,120 - Rl-tt("') 95, 96 NT - R~-tt ('") 96 - near-testable sets 2, 121 - R~tt("') 97,100 NTIME 119, 121 - relativized Rt-tt ("') 103 - Rt-T("') 95,97,99 Ogihara vii, viii, 16,39,40,59,77,78, - Rt-tt(",) 95,97,99,103 103, 113, see Ogiwara - R~O(1)_T("') 28 Ogiwara 76, 103, see Ogihara - R~(logn)-T("') 95 optimality R~(nk)-T("') - relativized '" 48 - 27 oracle 12,14,27-29,41,43,47,52-54, - RH"') 35,97,100,102 56,57,59,80,81,88,98-103,115,118, - Rft("') 95,97,99,100,102,108 120,121,123 P-sr 110,111,113 - '" query 5 - the polynomial-time semi-rankable - low sets as "'s 41 sets 110 - NP", 71 P jpoly 21-24,35-37,39,78,105,107, - open r- questions 59 108 ordering Papadimitriou vii - lexicographical e- 71,108,110 Parkins vii, 39 - linear '" 7, 124 Pasanen 16 - linear '" of {1}* 6 Paterson 16 - linear '" of ~. 6 path 11,12,14,53,65,81,92,93 - linear '" of ~. 6 - accepting '" 11,29,56,62,66-70, - partial '" 7 90,94,120,121,123 - partial '" 7 - accepting e- of a FewP machine 68 output - accepting '" of a function-computing - '" of a polynomial-time algorithm v machine 10, 120 - linearly bounded '" of an advice - computation '" 14,33, 56, 62, 66, function 19 69,70,87,109,121 - quadratically bounded '" of an advice - directed '" 63,64,76 function 19 - directed '" in a tournament 64 Owings 113 - guess bits of an accepting '" 70 - guessed '" 14 P v , 1-9, 12-16, 19-24, 26, 28-30,32, - guessed computation '" 11 33,35-39,41-49,54,55,58,61,62, - minimum accepting '" of a nondeter- 64-68,76-79,82,83,85,88,94,95, ministic Turing machine 62,65-67, 103,105-112,115,117,118,120-123 69, 70 P-close 2,49, 121 - nondeterministic e- 53,54,57 P-enumerability 77 - nondeterministic guess of a P-mc 108 computation '" 11, 53 - "'(const) 108 - nondownward '" 20 - R~tt ("'(const)) 108 - rejecting '" of a function-computing P-sel 1-5,8,9,11-13,18,20-27,30,31, machine 10, 120 35,39,41,43,46-51,66-68,70,72,76, - rejecting "'s 66 77,79,81-85,88,94-96,102,103,105, - short "'s in a tournament 63 106,110-112,121 - simulated '" 53 - Ei-T("') 95 PH Index 143

- polynomial hierarchy 15,35,37-40, query 12,14,26-28,47,52-54,56,57, 42,48,58,67,68,74-76,78,108,113, 65,66,68-71,73,81,86,89-102,107, 121 108,115,116,118,121,123 IIl; 37,38 - answers to ""s 69, 70 poly 19 - answers to rvS on the MinimumPath Popeye 41 70 - "" the Sailor Man 41 - linear limit to number of ""s 27 - cotton candy is low for "" 41 - linear number of rvS by a Turing - spinach is not low for "" 41 reduction 26 power 41,49,94 - membership rv 70 - ""s of two 20 - nonadaptive rv 102 - distinguishing the "" of reductions - oracle r- 5 v - polynomial number of ""s to a - relative "" 2 P-selective set 76 - separating the "" of reducibilities 2, - possible answers to rvS 69 15 - set of answers to rv 71 PP 28-30,46-49,55,75,109, 121 - truth-table rv 70 PP/poly 109 predecessor R 94 - lexicographical "" 2 - random polynomial time 68,122, preorder 7 123 procedure Ri-T - rv(P-sel) 95 - nondeterministic polynomial-time rv 91 Ri-tt - ",,(P-sel) 95, 96 program ~-tt - Selman's "" 16 - rv(P-sel) 96 - Selman's structural "" v Ramachandran 59 pronouncement 77 range 101 proof - "" of natural senses v - nonrelativizable "" 59 Ranjan 59 - relativizable "" 46 rank 110 property rankability - closure "" 84 - Pvsemi-e- 110 - closure ""s of P-sel 6, 79 rational - closure rvS of P-sel, NPSVt-sel, and - dyadic e- 3 NPMVt-sel 84 R~tt PSPACE 37,58,59,66,67,75,88, 105, - rv(P-mc(const)) 108 109,121 - rv(P-sel) 97, 100 PW-sei 48 realization - rv of an FP~P function 77 qP recursiveness - the quasipolynomial time sets 2, 16, - semi-e- 2,6,82,83,85 121 reducibility 8, 117 quadratic 19 - l-truth-table rv to a P-selective set quantification 88 - universal rv 33 - 2-disjunctive self-r- 5, 118 quantifier - rv degree 8 - alternating rvS 43 - disjunctive self->- 5,38, 118 - number of rvS needed to remove - disjunctive self->- of SAT 35 aset's ability to provide useful - self-r-- vi, 16,74,75,79,80,88,90, information 43 103 - polynomially bounded rv 9 - Turing >- 26 - unbounded "" 9 - Turing self-rv 5,74,75,88-92,123 144 Index

- Turing self-e- class es containing - standard set "'s 123 complete sets with that property 75 - structural '" 7 reduction - transitive '" 7 - advice upper bounds for "'s to relativization 13,41 ,43,49,55,88,90, selectivity classes 22 119 - comparison of polynomial-time "'S - '" of a proof 32 2 - '" on a per set basis 38 - completeness for NP under many-one - '" remains a useful approach 59 "'s 2 - positive '" 59 - completeness for NP under Turing - survey of open "" questions 59 "'S 2 requirement 20 - conjunctive "'S 81 research v-vii, 2, 3, 5 - disjunctive "'S 81 - unification of semi-feasibility '" vi - exponential-time Turing '" 115 result 5,7,12-16,20,23,34,37,39,40, - gamma r- 90, 115 45-49,55,61,66-68,73-80,85,88,90, - linear Turing '" 26 91,100,102,105,106,108,110-113 - locally pos itive Turing '" 103, 115 - ""S distinguishing reducibility notions - many-one "-'S 81 2 - parity "-'S 81 - "'s on topics beyend selectivity 3 - polynomial-time Turing "-' 95,99 - advice "'S 54 - positive Turing '" 29,79,81,85,86, - class ic r-- of Hartmanis and Stearns 115 76 - positive-truth-table "-' 87 - classic ""S 3 - streng nondeterministic Turing '" - complexity '" 107 116 - extended-lowness '" 47,48,54 - truth-table "'s 116 - hardness '" 62 - Turing '" 26,29,94, 100, 115 - immediate corollaries of lowness "'s - variants of positive "'s 80 46 refinement 16,39,40,110 - links between oracle "'s and "'s in - "" of P-selectivity 110,111 the real world 59 - '" of the amount of advice 20 - lowness ""S 43,46,48, 55 - meaning and weight of relativization - '" of the P-select ive sets 105 - '" of the semi-feasible sets v "'s 59 - NPSV", 39 - nonrelativized "'S 76 - relativizable "'s 58 Regan vii - relativized "'s 76 relation 7,29 - relativizing ""S 90 - '" between nondeterministic - value of relativized "'S 49 selectivity classes 12 R~_T - '" between the multiselectivity - ""(P-sel) 95,97,99 hierarchy and the extended low R~_tt hierarchy 107 - ",(P-sel) 95,97,99,103 - '" between truth-table equivalence RP'x classes and Turing equivalence classes k -tt 100 - ",(P-sel) 103 R~ - close e- between P-selective sets and standard left cuts 39 - ",(S(k» 106 - equivalence '" 7,105,124 R~O (l)_T - equivalence '" 7 - ",(P-sel) 28 RP - equivalence '" on E" 7 O(J ogn)-T - preorder '" 7 - ",(P-sel) 95 - refiexive '" 7 R~(nk)_T - refiexive and transitive '" 7 - ",(P-sel) 27 - standard arithmetic "'S 123 Rogers 40 Index 145

Rohatgi 59 - NPSV-rv 12,15,32,38 Rothe vii, 16,58, 112 - NP8Vt-rv 12,14-16,88,90,103 Royer 40 - other types of rv than p _cv 102 Rozenberg viii - P_rv 1,3-7,9,12,14-16,31,49,62, Rr(C) 122 64,68,74,82,88,102,103,105,109, R~ 110 - rv(P-sel) 35, 97, 100, 102 - P_rv 1 - rv(SPARSE) 35 - probabilistic rv 105, 109 Rft - relationships between nondeterminis- - rv(P-sel) 95,97,99, 100, 102, 108 tic rv classes 12 - rv(TALLY) 35 - study of nondeterministic rv 15 - understanding of cv 9 S(JTï) 106 - weak rv 105,112 S(2) 107 selector 52,64,81,97,99 S(k) 106 - cv function see function, selector R~(rv) - 106 - associative cv 111-112 8(logn) 106 - commutative cv 111-112 8(n) 106 - F-cv 9,119 82 40,76,78,113,122 S~pncoNP 40, 76, 78, 122 - more powerful cvs 109 - NPMV-rv 11,15,74,92 Salomaa viii SAT 5,9,14,15,35-39,45,47,54,68, - NPMVt-rv 11,12,81,90 71-73,75,76,82,108,118 - NP8V-rv 32,34,38,50,52,55,56 8chnorr 16 - NPSVt-rv 14,15 8chöning 15,16,58,59 - P_rv 7,92 segment - symmetrie rv 4, 86 - initial rv of a linear ordering 6 - symmetrie P_rv 100 - initial rv of a polynomial-time 8elman v, vii, 1,2,4,5,15,16,39,40, computable linear ordering 6 59,76-78,103 - initial rv of a recursive linear ordering sequence 6 - characteristic cv 102 selectivity 9, 16,79, 106 set - rv and self-reducible sets 88 - cvs $r-equivalent to some set in C - rv via general functions 119 118 - (A , k)-rv 113 - (a, b)p-recursive cvs 109 - (i,j)-rv 106 - advice complexity of the P-selective - associative rv 111-112 cvS 111 - broadening of rv 9 - advice for P-selective cvS 20 - commutative rv 111-112 - best currently known upper bounds - deterministic rv 90 for extended lowness of selective rvS - F-rv 9, 119 50 - FEXP-rv 109 - best currently known upper bounds - forms of rv 105 for lowness of selective rvS 51 - four types of nondeterministie cv 15 - cheatable rvS 109, 113 - FPP-cv 109, 113 - class containing Turing self-reducible - FPf-rv 88 complete rvS 75 - generalizations of cv 105 - closure properties of the P-selective - multi-> 105, 106, 112, 113 rvS 6 - nondeterministic rv 9, 80 - closure under complement of - NP-rv 38 NPMV-selective cvS 15 - NP2V-cv 39 - closure under complement of NPMVt - NPMV-cv 11,15,16,34,74,91 selective cvs 15 - NPMVt-cv 11,12,15,16,74,75,90, - collection of cvS 19,20,79,81,117 91 - complement of a cv 11 146 Index

- complement of an NPSV-selective '" - P-superterse "'s 109 12 - Pvverbose e-s 109 - complete ",s 13, 38 - polynomial-time semi-rankable ,.....,s - complete characterization of 110 P-selective "'s 6 - polynomial-time Turing equivalent - complete characterization of ,.....,s 8 semi-recursive ",s 6 - query r--s 69 - computationally simple '" in a - refinement of the semi-feasible e-s v natural sense v - self-reducible e-s 79,88 - ~i extended low "'s 118 - self-reducible Psselective e- 88 - disjunctively self-reducible "'S 103, - semi-feasible o-s v, vi, 1, 121 105,118 - serni-recursive r-s v, 6, 15,83,96, - easily-countable "'S 108, 109 109, 122 - FPP-selective,....., 109 - ~i extended low r-s 118 - implicitly membership-testable ,.....,s - sparse r-s 2, 122 2,15,119 - standard left cut", see cut

- left cut r- 28, see cut - superterse r-s 113 - lowness for nondeterministically - tally >- 8, 17 selective e-s 49 - ei extended low r-s 118 - lowness of Pvselectlve e-s 46 - Turing self-reducible "'s 123 - membership comparable i--s 107, - verbose e-s 108 108 - weakly Psrankable e-s 110 - membership complexity of ,.....,s 1 - weakly P-selective,.....,s 105 - near-testable o-s 2, 15, 108, 109, 121 - weakly-Pvrankable e-e 110 - nearly near-testable r-s 2, 119 - weakly-P-selective "'s 105 - nondeterministic version of semi- set-f 9-12,14,32-34,38,39,50,52, feasible r- vi 55,56,74,90,92,93,119,120,122 - nondeterministically selective ,.....,s setting 10,61,67,73,90 - satisfiability r-s 73 - nonempty finite sub-v of a P-selective Sewelson 76 set 21,24,25 SR 106,107 - nonempty finite sub-v of an Shamir 59 NPMV-selective set 34 Sheu 58,59 - nonrecursive r- v ~. 1,4-7,12,17-20,23,28,29,39,53, - NP,.....,s 2 56,77,80,85,89,91,95,108,109,111, - NP-~ft-hard,....., 68 117,124 - NP-~~-complete""'" 68 ~b 41,122 - NP-complete""'" 5, 8, 14, 15,42,61 ~i 41 - Nf'Mv-selectlve e-s 34,54 ~~ 35-38,40,41,47,67,75,111 - NPMVt-selective ,.....,s 16,34,54 ~~ 36,41 - NPSV-selective,.....,s 16,49 ~i 37,38,42,43,45,54,58,118,119, - NPSVt-selective,.....,s 14,16,88 121,122 - oracle r- 5 ~i,A 42,45, 118, 119, 122, 123 - Pvclose e-s 2,15, 121 simplicity 8 - Pvenumerable e- 77 - lowness is a notion of organizational - P-selective,....., v , 1-9, 17, 18,20, 43 21,23-32,39,46-49,61-68,70,73, - organizational ,....., of selectivity classes 76,79-83,85,86,88,94-96,98-103, 43 105-111,113,121 - structural e- 8, 9 - P-selective "'s whose intersection is simulation 12, 27, 53 not semi-recursive 82 Sipser vii - P-selective nonrecursive r-s v Sivakumar vii, 76, 77 - Pvsemi-renkable e-s 105, 110 sorting Index 147

rv of queries according to aselector - rv of semi-feasible algorithms 11l 64,65,92,94-96,98,99,101,102 - rv of semi-feasible computation v, (A, k)-rv 113 vii space 19 - advice rv 43 - deterministic J(n) rv 118 - basic lowness rv 41 - physical rv taken by a circuit 19 - coding rv 97 - polynomial rv 121 - complexity rv 1,3,16,41,46,103 SPARSE 46,48,50,107,121,122 - computational complexity rv vii - R~(rv) 35 - extended-lowness rv 46 spinach 41 - lowness rv 41,43,46 - rv is not low for Popeye 41 - P-selectivity rv 68 Stearns 76 - recursive funetion rv v, 2, 58,103 step 26,31,32,51,53,56,57,62,64, - selectivity rv 3, 105 86,87,89,91,93,98,99,116 - tournament rv 63 - computation rvS 62 81:,A 123 Stephan 77, 103, 113 e~ 43, 118, 123 Stol vii 8~,A 45, 118, 123 Stricker vii i Thierauf 16,39,40,76-78,103 string thresholds 106 - advice rv 20,25,33 time - col1ection of rvs 21,57,116 - almost polynomial rv (APT) 2,16 - concatenation of rvs 17 - co-nondeterministic polynomial rv - easily decodable advice rv 23 117 - evil rv 52 - deterministic "polynomial exponen- study 9 tial" rv 119 - rv of associative selectivity 112 - deterministic exponential rv 118 - rv of nondeterministic selectivity 9, - deterministic J(n) rv 118 15 - deterministic polynomial rv 121 - rv of P-selectivity 1,7 - deterministic polynomial rv relative - complexity-theoretic rv of one-way to an orade 121 functions 16 - deterministie polynomial rv relative - complexity-theoretic rv of semi­ to an oracle with a bounded number membership complexity 1 of queries 121 - exponential rv vi, 26 TALLY 6,30,47,122 - linear rv 98, 100 - Rft(rv) 35 - linear exponential rv 27 Tantau vii , 102, 103 - nondeterministic "polynomial tape 116 exponential" rv 119 - oracle rv 116 - nondeterministic exponential rv 119 technique - nondeterministic J(n) rv 121 - minimum path rv 61,62,68,75 - nondeterministic polynomial rv 24, - parallel census rv 68 119,120 test 1,49 - nondeterministic running rv 121 - dassic simplicity rv 3 - probabilistic polynomial rv 121 testability - quasipolynomial rv (qP) 2, 16, 121 - near-r- 109, 113 - random polynomial rv 122 Thakur vii - unambiguous nondeterministic Theorem polynomial rv 123 - Karp-Lipton rv 35 Toda 75,76 - relativized version of the Karp- - rv Ordering Lemma 76 Lipton rv 38 - rv's Lemma 76 theory token - rv of positive relativization 59 - advice rv 24-27,31,32,34,39,117 148 Index

Torán 77 Wagner 76 Torenvliet iii, iv, viii, 39, 77,103 Wang vii, 16,39,40,76-78,103,113 tournament 63,64, 76 Watanabe vii, 40, 58, 112 - king of a '" 24 weakly-FpELrankable 111 tree 26 weakly-P-rankable 110 - '" of possible queries 100 - the weakly P-rankable sets 110 - self-r eduction '" 90,91,93 Wechsung 40, 76, 77 West 76 un ion 7,65,96,98,99, 102, 105, 123 worktape 10,120 UP 66, 67, 123 - semi-infinite '" 10,120 worid - real r-- 49 vari abie 5,9,72,79,84 - relativized '" 46,48,49,76, 103, - logical and of "'s 124 108, 118, 119 - logical or of "'s 124 Wössner viii Veltman viii Verbeek vii Young 15,16, 113 verboseness 113 Vereshchagin 58 Zaki vii, 16, 113 verification Zimand vii, 16, 59, 113 - polynomial-time '" of a certificate ZP P 37-40,67,68,74,78, 108, 122, 33 123 Vyskoë 59 Zp p NP 40,67,68,78,122 Monographs in Theoretical Computer Science . An EATCS Series

K. lensen A. de Luca, S. Varricchio Coloured Petri Nets Finiteness and Regularity Basic Concepts, Analysis Methods in Semigroups and Formal Languages and Practical Use, Vol. 1 E. Best, R. Devillers , M. Koutny 2nded. Petri Net Algebra K. lensen Coloured Petri Nets S.P. Dem ri, E. S. Orlowska Incomplete Information: Basic Concepts, Analysis Methods Structure, Inference, Complexity and Practical Use, Vol. 2 J.C.M. Baeten, C.A. Middelburg K. Jensen Process Algebra with Timing Coloured Petri Nets Basic Concepts, Analysis Methods L.A. Hern aspaandra, L.Torenvliet and Practical Use, Vol. 3 Theory of Semi-Feasible Algorithms A. Nait Abda llah The Logic of Partial Information Z. Pülöp, H. Vogler Syntax-Directed Semantics Form al Models Based on Tree Transducers

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