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....out all inconsistent bit strings, arriving at a list L 1 . Having constructed L j , we de ne L j 1 : and let L j 1 contain all consistent bit strings in L j 1 . Let j be a bound on the maximum number of consistent bit strings of length j. As has been shown by di erent authors [10, 11, 20, 14] this number is bounded by S j; f(n) P f(n) 1 i=0 j i j f(n) 1 1 n . As there are m lengths and as checking the consistency of a single bit string takes time at most O n , the runtime of the second stage is at most O mmn O n 2f(n) 1 . In particular, for ....

R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, Stanford, USA, 1987.

....to compute one value of this function. With bounded query complexity we look at the set of functions that can be calculated if we put an upper bound on the number of queries that we allow the computer to ask the oracle. This notion has been extensively studied both in the resource bounded setting [2, 4, 5, 13, 12, 11, 17, 60, 75, 104] and in the recursive setting[15, 16] This notion and its variants has lead to a series of techniques and tools that are used throughout complexity theory. In this chapter we combine some of the bounded query notions with quantum computation. The main goal is to further as was done by Fortnow ....

Richard Beigel. Query-limited Reducibilities. Ph.d. dissertation, Department of Computer Science, Stanford University, 1987. Available on the web via: http://www.eecs.lehigh.edu/beigel/papers/.

....one possibility, does this imply that you can eliminate more, perhaps for higher values of k The next theorem shows how to do this. The proof is similar to Lemma 5.1 of [5] Lemma 19 in [7] or Theorem 4.4.9 in [22] Definition 9.1. Let k, m m. S(m, k) i=0 m . Lemma 9. 2 ([4, 7, 18, 47, 53]) Let k, m m, and let X be such that for any k coordinates, if you project X down to those k coordinates, the resulting set has size 1. Then S(m, k) N, k m, and f : D(ENUM(S(m, k) f ) Proof. Suppose that D(ELIM(f ) t via protocol P . Alice is ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88--1221.

....A 2 NQL that helps SAT via a robust OTM M in quasilinear time. Since A ql m SAT , M can be replaced by a robust OTM M 0 that makes SAT 1 help itself in quasilinear time. The conclusion now follows via Corollary 15. Next we consider the subject of bounded query classes studied in [Bei87b, Bei87a, AG88, BGH89, ABG90, Bei91, BGGO93] In particular, a language L is defined to be P superterse [Bei87a, ABG90] if for all k 1 and all oracle sets B, the function mapping a k tuple of strings x 1 ; x k to the k tuple of answers L(x 1 ) L(x k ) cannot be computed in polynomial ....

....by a robust OTM M 0 that makes SAT 1 help itself in quasilinear time. The conclusion now follows via Corollary 15. Next we consider the subject of bounded query classes studied in [Bei87b, Bei87a, AG88, BGH89, ABG90, Bei91, BGGO93] In particular, a language L is defined to be P superterse [Bei87a, ABG90] if for all k 1 and all oracle sets B, the function mapping a k tuple of strings x 1 ; x k to the k tuple of answers L(x 1 ) L(x k ) cannot be computed in polynomial time while making at most k Gamma 1 queries to B. This notion is relativized to an oracle A as follows: L ....

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R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, 1987.

....the comparator, simply because shortlisting is attempted for all subsets of k bit positions of P a (u i ) The following lemma gives a precise quantitative bound on the number of the elements of F q that will not be excluded by this process. This lemma is a restatement of a lemma due to Beigel [Bei87a, Bei87b] with our parameter settings. Aside: Beigel notes that this lemma also appears in the work of Clarke et al. COS75] also see [BFT97] for an elegant proof using the notion of VC dimension and a lemma proved by Sauer [Sau72] and Shelah [She72] other proofs are possible, for example, by ....

R. Beigel. Query-limited Reducibilities. PhD thesis, Stanford University, 1987.

....out which ones halt by asking n questions. This is the first theorem in the field of Bounded Queries. It was discovered independently by Beigel, Hay, and Owings in the early 1980 s. This observation leads to many other questions of interest. The field of Bounded Queries, founded independently by Beigel [Beigel, 1987] and Gasarch [Gasarch, 1985] raises the following types of questions: 1 2 1 Given a function f and a set X , how many queries to X are needed to compute f 2 Given a set X and an n # 1, are there functions that can be computed with n queries to X but not with n 1 This paper is a survey ....

....versions. For example, FQ X (n, A) is the set of functions that can be computed with n queries to A and an unlimited number of queries to X . 1.3 Enumerability Classes The notion of enumerability is very useful in the study of bounded queries. The concept within computability theory is due to Beigel [Beigel, 1987]. The concept within complexity theory is due independently to Beigel [Beigel, 1987] and Cai Hemachandra [Cai and Hemachandra, 1989] The term enumerability is due to Cai Hemachandra. Definition 1.8 Let f be a function, and let m # 1. We define f # EN(m) f is m enumerable) in two ....

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Beigel, Richard (1987). Query-Limited Reducibilities. PhD thesis, Stanford University. Also available as Report No. STANCS -88--1221.

....this imply that you can eliminate more, perhaps for higher values of k The next theorem shows how to do this. The proof is similar to Lemma 5.1 of [5] Lemma 19 in [7] or Theorem 4.4.9 in [22] Definition 9.1. Let k, m # N such that 1 # k # m. S(m, k) P k 1 i=0 m i . Lemma 9. 2 ([4, 7, 18, 47, 53]) Let k, m # N such that 1 # k # m, and let X # 0, 1 m be such that for any k coordinates, if you project X down to those k coordinates, the resulting set has size # 2 k 1. Then X # S(m, k) Theorem 9.2. Let k, m,n # N, k m, and f : 0, 1 n 0, 1 n # 0, ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88--1221.

....1 # Thus, #GA cannot take on every value between 1 and n since some of 2 7 these numbers cannot be the order of a subgroup of S n . This leads us to enumerability as a measure of complexity. The concept of enumerability in computational complexity theory was introduced independently by Beigel [Bei87a] and by Cai and Hemachandra [CH89] then later modified by Amir, Beigel, and Gasarch [ABG90] 5 Chicago Journal of Theoretical Computer Science 1999 1 Beals et al. Number of Graph Automorphisms 2 Definition 1 Let b : N # N be polynomially bounded. A function f is Definition 1 1 ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-881221.

....not hold in this form. Our main result states that for any reasonable computational model a selective language is supportive i it is not cheatable. Beigel et al. s result is a corollary of this general theorem since recursively cheatable languages are recursive by Beigel s Nonspeedup Theorem [2]. Our proof is based on a partial information analysis [17, 18] of the involved languages: We establish matching upper and lower bounds for the partial information complexity of the equivalence and reduction closures of selective languages. From this we derive the main results as these bounds ....

....of Fact 5 which holds for both polynomial time and the recursive case. It states that for any reasonable computational model C , a C selective language is parallel C supportive i it is not C cheatable. As recursively cheatable languages are recursive by Beigel s Nonspeedup Theorem [2], Fact 5 becomes a corollary. This paper is organised as follows. Section 1 studies logspace selective languages and intends to motivate why the study of selectivity should not be restricted to P selectivity and semirecursiveness. We transfer Fact 1 to P hard and NL hard languages, see ....

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R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, Stanford, USA, 1987.

....question arose: Are i 1 queries to K more powerful than i queries to K This question can be interpreted in four different ways depending on whether one is computing functions or deciding sets; and whether the queries are serial or parallel. He showed that answer is YES in all these cases [6, 16]. He later extended many of his results to general nonrecursive sets [6] Louise Hay was initially interested in the n r.e. sets, which are a stratification of some of the sets that are T K (we define n r.e. in Section 10) The queries to K hierarchy is another stratification of those same sets. ....

....K This question can be interpreted in four different ways depending on whether one is computing functions or deciding sets; and whether the queries are serial or parallel. He showed that answer is YES in all these cases [6, 16] He later extended many of his results to general nonrecursive sets [6]. Louise Hay was initially interested in the n r.e. sets, which are a stratification of some of the sets that are T K (we define n r.e. in Section 10) The queries to K hierarchy is another stratification of those same sets. Hay was concerned with how these two hierarchies relate to each other. ....

R. Beigel. Query-limited reducibilities. STAN-CS 88-1221, Stanford University, Dept. of Computer Science, July 1988. Ph.D. Thesis.

....So it is not necessary to consider all 4 n possible colors. This also has an in uence on the choice of k in the proof above. However, these are minor details which will not lead to a substantially smaller value of N . On the other hand, Tantau observed that, transferring techniques from Beigel [2], one can show that the union of two languages from (1; n)REG is contained in (1; 2n) 2 )REG (personal communication) We will not give the details of his proof, which would require the introduction of several new notions. Moreover, our construction satis es another nice property. Having read ....

Richard Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, January 1995.

....this imply that you can eliminate more, perhaps for higher values of k The next theorem shows how to do this. The proof is similar to Lemma 5.1 of [5] Lemma 19 in [8] or Theorem 4.4.9 in [17] Definition 10.4 Let k, m # N such that 1 # k # m. S(m, k) P k 1 i=0 m i . Lemma 10.5 ([4, 8, 16, 35, 40]) Let k, m # N such that 1 # k # m, and let X # 0, 1 m be such that for any k coordinates, if you project X down to those k coordinates, the resulting set has size # 2 k 1. Then X # S(m, k) Theorem 10.6 Let k, m, n # N, k m, and f : 0, 1 n 0, 1 n # 0, 1 . ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88--1221.

.... is yes , and hence for the readers familiar with the notation used for complexity classes: P NP[2] tt EQP NP[1] 1) where tt denotes truth table reducibility (non adaptive calls) See Garey and Johnson[7] for an introduction in complexity theory and some of the work by Richard Beigel[2, 3] for an overview of query limited reductions. This question is inspired by the article Two Queries by Harry Buhrman and Lance Fortnow[4] Its answer uses some well known results on Deutsch s problem[6] and its one call, exact solution[5] At the end of the note we will discuss the question if ....

.... efficiently on a classical computer with the use of two non adaptive sat queries, can also be solved by a quantum computer which uses only one sat query (again with polynomial time complexity) For a classical computer with one query at its disposal this is generally believed to be impossible[2, 3, 4]. 3 Conclusion, Question and Reminder We have shown how a quantum computer with one query to an np oracle can solve efficiently all decision problems that a classical computer can calculate in polynomial time with the help of two non adaptive np oracle calls. It is not obvious how to generalise ....

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Richard Beigel, Query-limited Reducibilities, Ph.D. Dissertation, Stanford University, Department of Computer Science (1987). Available at http://www.eecs.lehigh.edu/~beigel/papers/

.... is the corresponding subclass where g(n) 2 O(1) The class P mc(const) coincides with the class of approximable sets that were independently introduced in [BKS94] It is known (see [Bei87b] that P mc(const) further coincides with the class of all nonp superterse sets introduced by Beigel in [Bei87a] As shown in [ABG] any non p superterse set is contained in P=poly. Ogihara [Ogi94] observed that it follows by essentially the same proof that P mc P=poly. He also proved that the classes P mc(k) k 1, and P mc are all closed under polynomialtime one truth table reducibility, and further, ....

....contained in P=poly. Ogihara [Ogi94] observed that it follows by essentially the same proof that P mc P=poly. He also proved that the classes P mc(k) k 1, and P mc are all closed under polynomialtime one truth table reducibility, and further, that the latter class contains all sparse sets. In [Bei87a] it is further shown that P mc(const) is closed under bounded truth table reductions. It has been conjectured by Ogihara that P mc is a proper subclass of P=poly. However, as pointed out in [Ogi94] proving the conjecture is at least as hard as showing that P 6= NP . Prominent and well studied ....

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R. Beigel, Query-limited reducibilities, Ph.D. thesis (Stanford University, 1987).

....x 0 ; Delta Delta Delta ; x k Gamma1 2 Sigma , f(x 0 ; Delta Delta Delta ; x k Gamma1 ) 6= A(x 0 ) Delta Delta Delta A(x k Gamma1 ) A set A is P superterse if and only if A is not P approximable. Note that the above definition of P superterseness is a little different from Beigel s [2] original definition. Ogihara [11] further introduced the following notion of polynomial time membership comparability. Definition 2.6 (Ogihara [11] Let g : N N be a monotonic, nondecreasing, polynomial time computable and polynomial bounded function. 1. A function f is called a ....

....property of P approximable sets which we need latter. If A is P approximable then, for strings x 0 ; Delta Delta Delta ; x s Gamma1 2 Sigma , we can compute in polynomial time a subset of Sigma s which contains A(x 0 ) Delta Delta Delta A(x s Gamma1 ) Proposition 2. 10 (Beigel [2]) If A is P approximable via k 2 N , then there is a polynomial time computable function which computes for any s strings x 0 ; Delta Delta Delta x s Gamma1 a set of at most S(s; k) s 0 s 1 Delta Delta Delta s k Gamma 1 elements from Sigma s which contains ....

R. Beigel. Query-limited Reducibilities. PhD thesis, Stanford University, 1987. 7

....some A 2 NQL that helps SAT via a robust OTM M in quasilinear time. Since A ql m SAT , M can be replaced by a robust OTM M 0 that makes SAT 1 help itself in quasilinear time. The conclusion now follows via Theorem 5.2. Next we consider the subject of bounded query classes studied in [Bei87b, Bei87a, AG88, BGH89, ABG90, Bei91, BGGO93] In particular, a language L is defined to be P superterse [Bei87a, ABG90] if for all k and all oracles B, the function mapping a k tuple of strings x 1 ; x k to the k tuple of answers L(x 1 ) L(x k ) cannot be computed in polynomial time ....

....by a robust OTM M 0 that makes SAT 1 help itself in quasilinear time. The conclusion now follows via Theorem 5.2. Next we consider the subject of bounded query classes studied in [Bei87b, Bei87a, AG88, BGH89, ABG90, Bei91, BGGO93] In particular, a language L is defined to be P superterse [Bei87a, ABG90] if for all k and all oracles B, the function mapping a k tuple of strings x 1 ; x k to the k tuple of answers L(x 1 ) L(x k ) cannot be computed in polynomial time while making at most k Gamma 1 queries to B. Beigel, Kummer, and Stephan [BKS93] proved that SAT is ....

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R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, 1987.

....x 2 ) 2 A. A set A f0; 1g is P multiselective [HJRW] if there exists f 2 FP and a natural constant q 1 such that for all q tuple (x 1 ; x q ) f(x 1 ; x q ) 2 fx 1 ; x q g and (x 1 2 A) x q 2 A) f(x 1 ; x q ) 2 A. A set A f0; 1g is cheatable [Bei87] if there exists f 2 FP and a natural constant q 1 such that for all q tuple (x 1 ; x q ) f(x 1 ; x q ) outputs a set D f0; 1g q of size q which contains A(x 1 ) A(x q ) A set A f0; 1g is easily countable [HN93] if there exists f 2 FP and a natural constant q 1 ....

R. Beigel. Query-limited reducibilities. Technical report, Department of Computer Science, The Johns Hopkins University, Baltimore, MD, 1987. (This is also a Stanford Ph.D. thesis.).

....A 2 NQL that helps SAT via a robust OTM M in quasilinear time. Since A ql m SAT , M can be replaced by a robust OTM M 0 that makes SAT 1 help itself in quasilinear time. The conclusion now follows via Corollary 5.2. Next we consider the subject of bounded query classes studied in [Bei87b, Bei87a, AG88, BGH89, ABG90, Bei91, BGGO93] In particular, a language L is defined to be P superterse [Bei87a, ABG90] if for all k 1 and all oracle sets B, the function mapping a k tuple of strings x 1 ; x k to the k tuple of answers L(x 1 ) L(x k ) cannot be computed in polynomial ....

....by a robust OTM M 0 that makes SAT 1 help itself in quasilinear time. The conclusion now follows via Corollary 5.2. Next we consider the subject of bounded query classes studied in [Bei87b, Bei87a, AG88, BGH89, ABG90, Bei91, BGGO93] In particular, a language L is defined to be P superterse [Bei87a, ABG90] if for all k 1 and all oracle sets B, the function mapping a k tuple of strings x 1 ; x k to the k tuple of answers L(x 1 ) L(x k ) cannot be computed in polynomial time while making at most k Gamma 1 queries to B. This notion is relativized to an oracle A as follows: ....

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R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, 1987.

.... z Universitat Karlsruhe, Institut fur Logik, Komplexitat und Deduktionssysteme, D 76128 Karlsruhe, Germany, email: fstephan ira.uka.de) Supported by the Deutsche Forschungsgemeinschaft (DFG) grant Me 672 4 1 A related notion emerged from the recent work on bounded queries (see [3, 4, 7, 16]) For m 2 n we call a set A strongly (m; n) verbose (in short A 2 V S (m; n) iff there is a recursive function f such that for all x 1 ; x n : jD f(x 1 ; x n ) j m h A (x 1 ) A (x n )i 2 D f(x 1 ; x n ) D i denotes the i th finite set in a canonical indexing of ....

....computes a total function for every oracle, and M X computes F A n : x 1 ; x n : A (x 1 ) A (x n ) by making at most k queries to X. Every set is strongly (2 n ; n) verbose (let D f(x 1 ; x n ) fhb 1 ; b n i : b 1 ; b n ) 2 f0; 1g n g) Beigel [3] (see [4, Theorem 9] proved that every strongly (n; n) verbose set is recursive. This is optimal since every semirecursive set is strongly (n 1; n) verbose. It is easy to see that a set is (1; n) recursive iff it is strongly (2 n Gamma 1; n) verbose. Therefore S n1;1mn Omega Gamma ....

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R. Beigel. Query-limited reducibilities. Ph.D. thesis, Stanford University, Stanford, USA, 1987.

....to a standard left cut, then P 6= PP. We prove that (1) there is a p selective set that is not P tt equivalent to any tally set, while (2) if there is a p selective set that is not P T equivalent to any tally set, then P 6= PP. In a similar vein, we show that if there is a cheatable set [Bei87, Bei91] that is not P m equivalent to any tally set, then P 6= NP, and we obtain with no assumption, a recursive p selective tally set that is not cheatable. Buhrman, van Helden, and Torenvliet [BvHT93] have very recently proved that every p selective self reducible set belongs to P. Their ....

....shown that search nonadaptive reduces to decision for T 0 . By Proposition 1, if T 0 is P tt reducible to a p selective language, then T 0 2 P, which contradicts the assumption that T 0 2 UP Gamma P. 2 5. 2 P selective and P cheatable sets The following definitions are due to Beigel [Bei87, Bei91] For a set A and n 1, Phi A n denotes the function that given x 1 ; Delta Delta Delta ; x n 2 Sigma outputs ( A (x 1 ) Delta Delta Delta ; A (x n ) For a set B and k 1, let PF B k T be the class of functions computed by a polynomial time bounded deterministic oracle ....

R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, 1987.

....n and i. Thus, #GA cannot take on every value between 1 and n since some of these numbers cannot be the order of a subgroup of S n . This leads us to enumerability as a measure of complexity. The concept of enumerability in computational complexity theory was introduced independently by Beigel [5] and by Cai and Hemachandra [8] then later modified by Amir, Beigel and Gasarch [1] Definition 2.2 Let b : N N be polynomially bounded. A function f is b(n) enumerable if there exists a polynomial time computable function g, such that for all x, g(x) outputs a list of at most b(jxj) values, ....

Richard Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

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R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, 1987.

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Richard Beigel. Query-limited Reducibilities. PhD thesis, Department of Computer Science, Stanford University, 1987.

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R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, 1987.

....P = NP, then every 1 cheatable set is 1 tt reducible to a p selective set. Thus showing 1 cheatable sets form a much larger class than p selective sets would be dicult. 1. 1 Previous Work Selman [Sel79] de ned p selective sets as a polynomial time analogue of semirecursive sets [Joc68] Beigel [Bei87a] while studying bounded query classes de ned k membership comparable sets using the terminology of approximable and superterse. A set is k membership comparable if and only if it is k approximable if and only if it is not k 1 superterse. Amir, Beigel, and Gasarch [ABG90] and Ogihara [Ogi95] ....

....[Ogi95] showed that if A is f(n) truth table reducible to a p selective set, then A is ( 1 ) log f(n) membership comparable for any 0. From now, we write k mc sets for k membership comparable sets. Membership comparable sets p selective sets and are studied extensively in the literature [Sel79, Sel82, Ko83, Bei87a, Bei87b, Bei88, ABG90, Tod91, BvHT96, BKS95, Ogi95, AA96, HNOS96]. Ko [Ko83] showed that p selective sets have polynomial size circuits. This result was improved by Amir, Beigel, and Gasarch [ABG90] who showed poly mc sets have polynomial size circuits. Membership comparable sets and p selective sets have played an important role in understanding the structure ....

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R. Beigel. Query-limited reducibilities. PhD thesis, Stanford University, 1987.

....to X in the computation of h. We define a function f 2 PF j tt that returns the answers to the last j queries made to X. This information is sufficient to simulate h in polynomial time without any further oracle queries. 4 The preceding observation is proved in greater detail and generality in [4]. The following observation is quite handy, because it allows us to propagate collapses. i. If PF (k 1) T = PF k T ] ii. If PF (k 1) tt = PF k tt then k tt ] i. Assume that PF (k 1) T . For all t 0, k 1 m) T (k 1) T ffi PF m T by Observation 5(i) k T ffi PF ....

....rounds of queries to an NP set. As in the proof of Observation 5(i) it is easy to see that n 1 tt is the class of problems that can be solved in polynomial time by making n 1 parallel queries to A, followed by n 2 parallel queries to A, followed by n r parallel queries to A. See [4] for formalities and generalizations. The following theorem was proved in [10] for computation without time bound; however the proof also establishes the result for polynomial time. The technique is a generalization of the binary search strategy used in the proof of Observation 13(ii) Theorem ....

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R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....unless NP=poly = co NP=poly [59] 3 Note that if NP=poly = co NP=poly then the polynomial time hierarchy collapses. Thus for all of the preceding separations it suffices to assume that the PH is proper. Other work on polynomial time computations with a bounded number of queries includes [3, 4, 13, 14, 7, 8, 9, 11, 22, 23, 24, 25, 32, 33, 34, 35, 40, 44, 45, 51, 58]. Related work in recursion theory includes [10, 15, 16, 17, 18, 19, 47] When 2 q(n) log n, the preceding results show that, assuming the PH is proper, q(n) 1) T 6 PF (q(n) 1) T 6 P (q(n) 1) tt 6 P (q(n) 1) tt P q(n) T : In view of the power of oracle queries ....

....to A, followed by n r parallel queries to A. Similarly, PF ffi (PF n 1 tt ) is the class of problems that can be solved in polynomial time by making n 1 parallel queries to A, simultaneous with n 2 parallel queries to A, simultaneous with n r parallel queries to A. See [7] for formalities and generalizations. These observations lead to a neat tradeoff. Lemma 45 (Synchronization) k T k Delta Delta Delta k PF Proof: The left hand side is the class of all sets than can be computed by j polynomial time Turing machines, working in parallel, that each ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....The arithmetical hierarchy is the recursion theoretic analogue of the more modern polynomial hierarchy. All levels of the arithmetical hierarchy are distinct [17, 18] i.e. it contains no gaps. ffl Delta 1 ae Sigma 1 ae Delta 2 ae Sigma 2 ae Delta Delta Delta Bounded Query Hierarchies [1] Let A be a language. FP m T is the class of functions computed by polynomial time oracle Turing machines that make at most m queries to A on each input. FP m tt is the class of functions computed by polynomial time oracle Turing machines that make at most m nonadaptive queries to A on each ....

....paper we investigate the hierarchy of languages decided by a polynomial time Turing machine that makes a bounded number of parallel queries to a fixed language A. In light of the three results listed above, one might expect that m tt : 1) A simple divide and conquer argument is given in [1], from which one can easily deduce a weak form of (1) m tt : 2) 2) can be understood informally as follows: Consider an unknown assignment ff to a set of Boolean variables. Suppose that we are given a black box that takes a 2m ary Boolean formula f(x 1 ; x 2m ) and produces an ....

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R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford Univ., 1987. Available as Report No. STAN-CS-881221.

....X 2 that separates C 2 such that jX 2 j k 2 Gamma 1. Let X = X 1 [ X 2 [ fxg. Then X separates C and jXj k 1 Gamma 1 k 2 Gamma 1 1 = k 1 k 2 Gamma 1 = k Gamma 1. We also present a different proof of Lemma 5.2. 2, which is based on our original proof of the Weak Nonspeedup Theorem in [Bei86]. In our proof we first show that there is a finite set Y of m points that separates C. We construct X by taking two cases: If there is one point that is necessary in order to separate two of the sets in C then we put the necessary point in X. If none of the points is necessary in order to ....

Richard Beigel. Query-limited reducibilities. Working draft, May 1986.

....use Turing degree, since this notion is not quantitative. Hence we use a notion of complexity that is quantitative like time or space, yet captures difficulty within recursion theory like Turing degree. It is for these reasons that the theory of bounded queries in recursion theory was developed [1, 2, 6]. Given that a function f is recursive in a set X , our main concern will be the number of queries to X required to compute f . This will be our measure of complexity. Several natural functions have been classified this way (see [2, 3, 7, 11, 12] In this paper we investigate the complexity of ....

....possibilities (unless we eventually get n possibilities) If f 2 SEN(n) then, given x, we can list out n possibilities for f(x) one of which is correct; moreover, we do have all the possibilities and we know we have them all. This concept first appeared in a recursion theoretic framework in [1]. Some very general theorems about EN(n) were proved in [12] The term enumerable is from [4] where it was defined in a polynomial time bounded framework. Definition 3. Let B be a set. B 2 Q(n; A) if B 2 FQ(n; A) B 2 Q jj (n; A) if B 2 FQ jj (n; A) B 2 QC(n; A) if B 2 FQC(n;A) Definition ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....This general question admits several variants, depending on whether we are bounding the number of queries made in a truth table reduction or in a Turing reduction and on whether we are computing sets or computing functions. 2. Preliminaries Much of the material in this paper is also derived in [Bei87a, Bei87b, Bei87d, BG87], sometimes by different techniques. Various kinds of reducibilities have been defined in the literature [Rog67, Soa87] We present some of the well studied reducibilities, along with some that have received attention only recently. Throughout this paper, we will consider a Turing machine with a ....

....be computed by m parallel processes that each make at most n serial queries to A and then pass their output to a recursive function. If we force the processes to synchronize before each query, then we can make all the queries in n rounds of m parallel queries. We use the following theorem from [Bei87b]. A complexity theoretic variant of the theorem also appears in [Bei87c] 14 Theorem 26 If A is not superterse then for every n there exists a set C n tt A such that n 2 FQ k (O(log n) C) The following theorem provides a nice relation between supportiveness and superterseness. Theorem 27 ....

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Richard Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS88 -1221.

....If 0 i joej then oe(i) denotes the i bit of oe, where we begin counting at 0. All logarithms in this paper are base 2. Applications of our work to recursive graph theory appear in [5,6] Questions concerning Q(n; A) the class of sets B such that B = feg for some e, are considered in [3,4,7]. Similar questions have been considered in a polynomial time framework. See [1] for a list of references. 2. Verbose Sets An important question is whether all nonrecursive sets are terse. They are not: in fact the halting set is verbose [7] In this section we show that every tt degree contains ....

....show that this result is tight. As a corollary, we show that all Kolmogorov random sets are superterse, and hence that almost all sets are superterse. Let Gamma n Delta denote the binomial coefficient n choose i and let S(n; k) 0ik Gamma1 The following lemma has appeared in [4]. A complexity theoretic variant with essentially the same proof has appeared in [2] Lemma 19. 4] If F Gamma 1 partial recursive functions then, for all n, F n is computable by a set of S(n; k) partial recursive functions. We note that the set of S(n; k) functions obtained in [4] is in ....

[Article contains additional citation context not shown here]

BEIGEL, R. Query Limited Reducibilities. Ph.D thesis, Stanford University (1987).

....7 Lemma 4.6 FQ k (n; K) FQ(dlog (n 1)e; K) Proof: By Lemma 4.5(iii,ii) and the transitivity lemma. Corollary 4.7 FQ k (2 Gamma 1; K) FQ(n; K) Thus in reductions to K, 2 Gamma 1 parallel queries can be replaced by n serial queries. The reverse inclusion was originally proved in [Bei87] Since the proof uses substantially different techniques, it is deferred to Section 8. For our purposes, it is sufficient to prove the reverse only for total functions; this will be done in the next section. 5 Limiting Recursive Functions The notion of bounded mind changes in recursive ....

Richard Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....was motivated by the recursion theoretic Nonspeedup Theorem [10] which states that all recursively cheatable sets are recursive; by analogy the Nonspeedup Theorem suggests that cheatable sets might be easy in some standard complexitytheoretic sense. Amir and Gasarch [2] and (later) ourselves [8] have shown that 1 cheatable sets cannot be bi immune for P, so membership in a 1cheatable set can be decided in polynomial time infinitely often. In contrast, we show that 2 cheatable sets can be bi immune for P. This is a qualitative distinction between 1 cheatable sets and the general class of ....

....if A is k cheatable for some k. Although we defined two different brands of k query p terseness (super and regular) we do not define both analogous brands of k cheatability because they lead to equivalent definitions of cheatability. We will need the following obvious result, which is proved in [8]. 5 (k 1) k tt A then i. 8n) F k tt A) ii. A is k cheatable. See [10, 11] for some results on bounded query reductions in recursion theory. 3. An Oracle Free Definition of Cheatability The concepts of k query p superterseness and k cheatability can be described in a purely ....

[Article contains additional citation context not shown here]

R. Beigel. Query-limited reducibilities. STAN-CS 88-1221, Stanford University, Dept. of Computer Science, July 1988. Ph.D. Thesis.

....1) ary search is not optimal. In Section 9 we examine using parallel queries and an auxiliary (weaker) oracle. Section 10 contains a summary of our results and some open questions. Other work on bounded queries in a recursion theoretic context has been done by Beigel, Gasarch, Gill, Hay and Owings [6, 8, 5, 12, 13, 10, 29]. In a polynomial framework, work on bounded queries has been done by Amir, Beigel, and Gasarch [2, 1, 6, 7, 9, 20] and Goldsmith, Joseph and Young [21] Krentel [25] Rosier and Yen [31] Wagner and Wechsung [37, 38] Other work on recursive graph theory has been done by Bean [3, 4] Burr [14] ....

....contains a summary of our results and some open questions. Other work on bounded queries in a recursion theoretic context has been done by Beigel, Gasarch, Gill, Hay and Owings [6, 8, 5, 12, 13, 10, 29] In a polynomial framework, work on bounded queries has been done by Amir, Beigel, and Gasarch [2, 1, 6, 7, 9, 20] and Goldsmith, Joseph and Young [21] Krentel [25] Rosier and Yen [31] Wagner and Wechsung [37, 38] Other work on recursive graph theory has been done by Bean [3, 4] Burr [14] Carstens and Pappinghaus [15, 18, 16, 17] Kierstead [22, 23, 24] Manaster and Rosenstein [27, 28] Schmerl [32, ....

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R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88--1221.

....setting where determinism and nondeterminism are not equal. In particular we look at deterministic and nondeterministic oracle machines with a bound on the number of calls to the oracle. Other work on bounding the number of calls to an oracle, in the context of recursion theory, can be found in [4, 3, 6, 7, 8, 9, 10]. Bounded queries in the context of complexity theory (i.e. polynomial time computations) can be found in [2, 1, 4, 5, 13, 14, 15, 16, 18, 19, 20] If A # N (the natural numbers) and n # 1, let F A n (x 1 , x n ) ##A (x 1 ) #A (x n )#. where #A is the characteristic ....

....machines with a bound on the number of calls to the oracle. Other work on bounding the number of calls to an oracle, in the context of recursion theory, can be found in [4, 3, 6, 7, 8, 9, 10] Bounded queries in the context of complexity theory (i.e. polynomial time computations) can be found in [2, 1, 4, 5, 13, 14, 15, 16, 18, 19, 20]. If A # N (the natural numbers) and n # 1, let F A n (x 1 , x n ) ##A (x 1 ) #A (x n )#. where #A is the characteristic function of A ( i.e. x # A # #A (x) 1, x # A # #A (x) 0) Obviously a Turing Machine having A as an oracle can compute F A n in n ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88--1221.

No context found.

Richard Beigel. Query-Limited Reducibilities. PhD thesis, Stanford Univ., 1987. Available as Report No. STAN-CS-88-1221.

....The arithmetical hierarchy is the recursion theoretic analogue of the more modern polynomial hierarchy. All levels of the arithmetical hierarchy are distinct [17, 18] i.e. it contains no gaps. ffl Delta 1 ae Sigma 1 ae Delta 2 ae Sigma 2 ae Delta Delta Delta Bounded Query Hierarchies [1] Let A be a language. FP A m T is the class of functions computed by polynomial time oracle Turing machines that make at most m queries to A on each input. FP A m tt is the class of functions computed by polynomial time oracle Turing machines that make at most m nonadaptive queries to A on ....

....languages decided by a polynomial time Turing machine that makes a bounded number of parallel queries to a fixed language A. In light of the three results listed above, one might expect that P A (m 1) tt = P A m tt ) P A btt = P A m tt : 1) A simple divide and conquer argument is given in [1], from which one can easily deduce a weak form of (1) P A 2m tt = P A m tt ) P A btt = P A m tt : 2) 2) can be understood informally as follows: Consider an unknown assignment ff to a set of Boolean variables. Suppose that we are given a black box that takes a 2m ary Boolean formula f(x ....

[Article contains additional citation context not shown here]

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford Univ., 1987. Available as Report No. STAN-CS-881221.

....was motivated by the recursion theoretic Nonspeedup Theorem [10] which states that all recursively cheatable sets are recursive; by analogy the Nonspeedup Theorem suggests that cheatable sets might be easy in some standard complexitytheoretic sense. Amir and Gasarch [2] and (later) ourselves [8] have shown that 1 cheatable sets cannot be bi immune for P, so membership in a 1cheatable set can be decided in polynomial time infinitely often. In contrast, we show that 2 cheatable sets can be bi immune for P. This is a qualitative distinction between 1 cheatable sets and the general class of ....

....if A is k cheatable for some k. Although we defined two different brands of k query p terseness (super and regular) we do not define both analogous brands of k cheatability because they lead to equivalent definitions of cheatability. We will need the following obvious result, which is proved in [8]. Theorem 4 If F A (k 1) P k tt A then i. 8n) F A n P k tt A) ii. A is k cheatable. See [10, 11] for some results on bounded query reductions in recursion theory. 3. An Oracle Free Definition of Cheatability The concepts of k query p superterseness and k cheatability can be ....

[Article contains additional citation context not shown here]

R. Beigel. Query-limited reducibilities. STAN-CS 88-1221, Stanford University, Dept. of Computer Science, July 1988. Ph.D. Thesis.

....the less possible answers. Definition 2.2 Let a 2 N and f be any total function. f is a enumerable if there exists a recursive function g such that, for all x, jW g(x) j a and f(x) 2 W g(x) We denote this by f 2 EN(a) This concept first appeared in a recursion theoretic framework in [3]. The name enumerable is from [7] where it was defined in a polynomial bounded framework. If f is a enumerable then, given x, we can find g(x) and try to enumerate W g(x) looking for possibilities for f(x) While doing this we do not know when W g(x) will have stopped generating ....

....set. Hence we can obtain all the possibilities and know we have them all. Definition 2.3 Let a 2 N and f be any total function. f is strongly a enumerable if there exists a recursive function g such that, for all x, jD g(x) j a and f(x) 2 D g(x) We denote this by f 2 SEN(a) Lemma 2. 4 ([3, 5]) Let a 2 N and let f be any function. 1. 9X) f 2 FQ(a; X) iff f 2 EN(2 a ) 2. 9X) f 2 FQC(a;X) iff f 2 SEN(2 a ) In this paper we will prove upper and lower bounds in terms of enumerability (or strong enumerability) Using Lemma 2.4 the reader can obtain corollaries about upper and ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....Lemma 4.6 FQ k (n; K) FQ(dlog (n 1)e; K) Proof: By Lemma 4.5(iii,ii) and the transitivity lemma. Corollary 4.7 FQ k (2 n Gamma 1; K) FQ(n; K) Thus in reductions to K, 2 n Gamma 1 parallel queries can be replaced by n serial queries. The reverse inclusion was originally proved in [Bei87] Since the proof uses substantially different techniques, it is deferred to Section 8. For our purposes, it is sufficient to prove the reverse only for total functions; this will be done in the next section. 5 Limiting Recursive Functions The notion of bounded mind changes in recursive ....

Richard Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....unless NP=poly = co NP=poly [59] Note that if NP=poly = co NP=poly then the polynomial time hierarchy collapses. Thus for all of the preceding separations it suffices to assume that the PH is proper. Other work on polynomial time computations with a bounded number of queries includes [3, 4, 13, 14, 7, 8, 9, 11, 22, 23, 24, 25, 32, 33, 34, 35, 40, 44, 45, 51, 58]. Related work in recursion theory includes [10, 15, 16, 17, 18, 19, 47] When 2 q(n) 1 2 log n, the preceding results show that, assuming the PH is proper, PF NP (q(n) 1) T 6 PF NP q(n) T PF NP (q(n) 1) tt 6 PF NP q(n) tt P NP (q(n) 1) T 6 P NP q(n) T P NP (q(n) 1) tt 6 P ....

....queries to A. Similarly, PF ffi (PF A nr tt k Delta Delta Delta k PF A n 1 tt ) is the class of problems that can be solved in polynomial time by making n 1 parallel queries to A, simultaneous with n 2 parallel queries to A, simultaneous with n r parallel queries to A. See [7] for formalities and generalizations. These observations lead to a neat tradeoff. Lemma 45 (Synchronization) PF A k T k Delta Delta Delta k PF A k T z j PF A j tt ffi Delta Delta Delta ffi PF A j tt z k Proof: The left hand side is the class of all sets than can ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....Arithmetical Hierarchy The arithmetical hierarchy is the recursion theoretic analogue of the more modern polynomial hierarchy. All levels of the arithmetical hierarchy are distinct [17, 18] i.e. it contains no gaps. ffl D 1 ae S 1 ae D 2 ae S 2 ae Delta Delta Delta Bounded Query Hierarchies [1] Let A be a language. FP A m T is the class of functions computed by polynomial time oracle Turing machines that make at most m queries to A on each input. FP A m tt is the class of functions computed by polynomial time oracle Turing machines that make at most m nonadaptive queries to A on ....

....of languages decided by a polynomial time Turing machine that makes a bounded number of parallel queries to a fixed language A. In light of the three results listed above, one might expect that P A (m 1) tt = P A m tt ) P A btt = P A m tt : 1) One of the seminal works on bounded queries [1] gives a simple divide and conquer argument from which one can easily deduce a weak form of (1) P A 2m tt = P A m tt ) P A btt = P A m tt : 2) 2) can be understood informally as follows: Consider an unknown assignment a to a set of Boolean variables. Suppose that we are given a black box ....

[Article contains additional citation context not shown here]

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Available as Report No. STAN-CS-88-1221.

....X 2 that separates C 2 such that jX 2 j k 2 Gamma 1. Let X = X 1 [ X 2 [ fxg. Then X separates C and jXj k 1 Gamma 1 k 2 Gamma 1 1 = k 1 k 2 Gamma 1 = k Gamma 1. We also present a different proof of Lemma 5.2. 2, which is based on our original proof of the Weak Nonspeedup Theorem in [Bei86]. In our proof we first show that there is a finite set Y of m points that separates C. We construct X by taking two cases: If there is one point that is necessary in order to separate two of the sets in C then we put the necessary point in X. If none of the points is necessary in order to ....

Richard Beigel. Query-limited reducibilities. Working draft, May 1986.

....to X in the computation of h. We define a function f 2 PF X j tt that returns the answers to the last j queries made to X. This information is sufficient to simulate h in polynomial time without any further oracle queries. The preceding observation is proved in greater detail and generality in [4]. The following observation is quite handy, because it allows us to propagate collapses. Observation 6 i. If PF B (k 1) T = PF B k T then (8j k) PF B j T = PF B k T ] ii. If PF B (k 1) tt = PF B k tt then (8j k) PF B j tt = PF B k tt ] Proof: i. Assume that PF B k T = PF B ....

....proof of Observation 5(i) it is easy to see that PF A nr tt ffi Delta Delta Delta ffi PF A n 1 tt is the class of problems that can be solved in polynomial time by making n 1 parallel queries to A, followed by n 2 parallel queries to A, followed by n r parallel queries to A. See [4] for formalities and generalizations. The following theorem was proved in [10] for computation without time bound; however the proof also establishes the result for polynomial time. The technique is a generalization of the binary search strategy used in the proof of Observation 13(ii) Theorem ....

[Article contains additional citation context not shown here]

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....section we show that if a set is self reducible and cheatable then it is in P. 2 We use this to show that, under a suitable hypothesis, certain sets A are not cheatable. We then prove a lemma that extends this to any set B such that A p T B. 2 This result was first stated (without proof) in [10], crediting the current authors. We subsequently learned that [36] obtained the result independently. Many natural sets, e.g. all NP complete sets in [31] are self reducible, as defined by Schnorr in [62] We use an alternative definition which is more general and is implicit in the ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

....i joej then oe(i) denotes the i th bit of oe, where we begin counting at 0. All logarithms in this paper are base 2. Applications of our work to recursive graph theory appear in [5,6] Questions concerning Q(n; A) the class of sets B such that B = feg A(n) for some e, are considered in [3,4,7]. Similar questions have been considered in a polynomial time framework. See [1] for a list of references. 2. Verbose Sets An important question is whether all nonrecursive sets are terse. They are not: in fact the halting set is verbose [7] In this section we show that every tt degree contains ....

....that this result is tight. As a corollary, we show that all Kolmogorov random sets are superterse, and hence that almost all sets are superterse. Let Gamma n i Delta denote the binomial coefficient n choose i and let S(n; k) X 0ik Gamma1 n i : The following lemma has appeared in [4]. A complexity theoretic variant with essentially the same proof has appeared in [2] Lemma 19. 4] If F A k is computable by a set of 2 k Gamma 1 partial recursive functions then, for all n, F A n is computable by a set of S(n; k) partial recursive functions. We note that the set of S(n; k) ....

[Article contains additional citation context not shown here]

BEIGEL, R. Query Limited Reducibilities. Ph.D thesis, Stanford University (1987).

....use Turing degree, since this notion is not quantitative. Hence we use a notion of complexity that is quantitative like time or space, yet captures difficulty within recursion theory like Turing degree. It is for these reasons that the theory of bounded queries in recursion theory was developed [1, 2, 6]. Given that a function f is recursive in a set X, our main concern will be the number of queries to X required to compute f . This will be our measure of complexity. Several natural functions have been classified this way (see [2, 3, 7, 11, 12] In this paper we investigate the complexity of ....

....possibilities (unless we eventually get n possibilities) If f 2 SEN(n) then, given x, we can list out n possibilities for f(x) one of which is correct; moreover, we do have all the possibilities and we know we have them all. This concept first appeared in a recursion theoretic framework in [1]. Some very general theorems about EN(n) were proved in [12] The term enumerable is from [4] where it was defined in a polynomial time bounded framework. Definition3. Let B be a set. B 2 Q(n; A) if B 2 FQ(n; A) B 2 Q jj (n; A) if B 2 FQ jj (n; A) B 2 QC(n; A) if B 2 FQC(n;A) ....

R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987. Also available as Report No. STAN-CS-88-1221.

No context found.

R. Beigel, Query-Limited Reducibilities, Stanford Ph.D. thesis, Palo Alto, CA.

No context found.

R. Beigel. Query-limited Reducibilities. Ph.D. thesis, Stanford University, Stanford, USA, 1987.

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