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. <p <p <p <p -tt, -ptt' -btt' -k-tt - Various types of truth-table reductions. - Truth-table reductions are often defined in terms of separate "query generation" and "answer interpretation" machines. In this book, we will more commonly use a more intuitive, though equivalent, formu­ lation. We say that A ~rt B if there is a 'Iuring machine, having a designated query tape, that on each input x will enter a special query state at most once . When it does, our model is that the content of the query tape is interpreted as a collection of strings (say, each separated by the character "#"), and in one time step the content of the query tape is replaced by the bitstring bsbz:': bz , where bi is 1 if the ith string on the tape is an element of Band is 0 otherwise, and where z is the number of strings that were on the query tape. The Turing machine in this same step has its state moved to a special query-was­ just-answered state. (It is henceforth not allowed to query.) It then continues on , eventually accepting or rejecting. Note in particular that the query tape is readabie, and so our Turing machine may read the bitstring b1b2 ... bz , and these bits may affect the acceptance. For each natural number k, if A ~rt B via a Turing machine M of the form described above then we say that A ~~-tt B if on no input does the machine ever enter the query state with more than k - 1 "#" characters on its query tape. If there is a natural number k such that A ~~-tt B, then we say that A ~~tt B. The reducibility ~~tt is referred to as bounded-truth-table reducibility. Suppose that A ~rt B via a Turing machine M of the form described in the above definition of ~rt . We will, when this use is clear from content (i.e., when the machine is a truth-table machine), use MB to denote the computation of M when B is used to answer the query list written to the oracle tape. We say that A ~~tt B (A positive-truth­ table reduces to B) if there exists a polynomial-time Turing machine M (that functions as a truth-table machine) such that A ~rt B via M, and (VC, D)[L(Mc ) ç L(Mc UD)]. 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.