## Algebrization: A new barrier in complexity theory (2007)

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Venue: | MIT Theory of Computing Colloquium |

Citations: | 29 - 2 self |

### BibTeX

@TECHREPORT{Aaronson07algebrization:a,

author = {Scott Aaronson and Avi Wigderson},

title = {Algebrization: A new barrier in complexity theory},

institution = {MIT Theory of Computing Colloquium},

year = {2007}

}

### OpenURL

### Abstract

Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known non-relativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require non-algebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 1

### Citations

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(Show Context)
Citation Context .... [27] and Shamir [37] algebrize: that is, for all A, � A, we have P #PA ⊆ IP e A , and indeed PSPACE A[poly] ⊆ IP e A . Then, in Section 3.3, we sketch an extension to the Babai-Fortnow-Lund theorem =-=[4]-=-, giving us NEXP A[poly] ⊆ MIP e A for all A, � A. The same ideas also yield EXP A[poly] ⊆ MIP e AEXP for all A, � A, where MIPEXP is the subclass of MIP with the provers restricted to lie in EXP. Thi... |

379 | Proofs that Yield Nothing But Their Validity or All languages in N P Have Zero-Knowledge Proof Systems
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Citation Context ... source is cryptography—and more specifically, cryptographic results that exploit the locality of computation. These include the zero-knowledge protocol for NP due to Goldreich, Micali, and Wigderson =-=[16]-=- (henceforth the GMW Theorem), the two-party oblivious circuit evaluation of Yao [45], and potentially many others. Here we focus on the GMW Theorem. As discussed in Section 1, the GMW Theorem is inhe... |

308 | Algebraic methods for interactive proof systems
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Citation Context ...hard. 1.1 The Need for a New Barrier Yet for both of these barriers—relativization and natural proofs—we do know ways to circumvent them. In the early 1990’s, researchers managed to prove IP = PSPACE =-=[27, 37]-=- and other celebrated theorems about interactive protocols, even in the teeth of relativized worlds where these theorems were false. To do so, they created a new technique called arithmetization. The ... |

290 |
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Citation Context ...which computed the Boolean MAJORITY given an input of size 2 n (namely, the truth table of A). But when q = o (n/log n), such a constant-depth circuit violates the celebrated lower bound of Smolensky =-=[38]-=-. Unfortunately, the above argument breaks down when the field size is large compared to n—as it needs to be for most algorithms that would actually exploit oracle access to e A. Therefore, it could b... |

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Citation Context ...A �⊂ PA Ae /poly then we are done, so assume PP ⊂ PA Ae /poly. Then certainly PP ⊂ P e A /poly, so Theorem 3.5 implies that P #PA ⊆ MA e � � A . Therefore ΣP A Ae 2 ⊆ MA as well, since Toda’s Theorem =-=[39]-=- (which relativizes) tells us that ΣP 2 ⊆ P#P and hence � ΣP �A 2 ⊆ P #PA . But Kannan’s Theorem [23] (which also relativizes) tells us that ΣP 2 �⊂ SIZE�nk� for fixed k, and hence � � ΣP A A 2 �⊂ SIZ... |

190 |
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Citation Context ...ful error-correcting properties that were unavailable in the Boolean case. In the case of the natural proofs barrier, a way to circumvent it was actually known since the work of Hartmanis and Stearns =-=[18]-=- in the 1960’s. Any complexity class separation proved via diagonalization—such as P �= EXP or Σ EXP 2 �⊂ P/poly [23]—is inherently non-naturalizing. For diagonalization zeroes in on a specific proper... |

184 |
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Citation Context ...erarchy—or at least separate the polynomial hierarchy from larger classes such as P #P and PSPACE. 17 In the standard oracle setting, these separations were achieved by Furst-Saxe-Sipser [15] and Yao =-=[44]-=- in the 1980’s, whereas in the communication setting they remain notorious open problems. Again, algebraic query complexity provides a natural intermediate case between query complexity and communicat... |

176 | Natural proofs
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Citation Context ... require non-relativizing techniques: techniques that exploit properties of computation that are specific to the real world. The second meta-discovery was natural proofs. In 1993, Razborov and Rudich =-=[35]-=- analyzed the circuit lower bound techniques that had led to some striking successes in the 1980’s, and showed that, if these techniques worked to prove separations like P �= NP, then we could turn th... |

152 |
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Citation Context ...ny communication complexity lower bound automatically leads to an algebraic query complexity lower bound. This means, for example, that we can use celebrated lower bounds for the Disjointness problem =-=[33, 22, 25, 34]-=- to show that there exist oracles A, � A relative to which NP A �⊂ BPP eA , and even NP A �⊂ BQP eA and NP A �⊂ coMA e A . For the latter two results, we do not know of any proof by direct constructio... |

152 |
Monotone circuits for connectivity require superlogarithmic depth
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- 1988
(Show Context)
Citation Context ...her hand, they arose directly from the “transfer principle” relating algebrization to communication complexity in Section 4.3. 7.1 Karchmer-Wigderson Revisited Two decades ago, Karchmer and Wigderson =-=[24, 42]-=- noticed that certain communication complexity lower bounds imply circuit lower bounds—or in other words, that one can try to separate complexity classes by thinking only about communication complexit... |

130 | Quantum vs. classical communication and computation
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(Show Context)
Citation Context ... � Am for any m or any m = O (poly (n)) respectively. 12 It is an interesting question whether his lower bound is tight. We know that Disjointness admits a quantum protocol with O( √ N) communication =-=[7, 2]-=-, as well as an MA-protocol with O( √ N log N) communication (see Section 7.2). The question is whether these can be combined somehow to get a QMA-protocol with, say, O(N 1/3 ) communication. 29sNotic... |

123 | Complexity measures and decision tree complexity: a survey
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- 2002
(Show Context)
Citation Context ...s f with probability at least 2/3 on every input. The bounded-error quantum query complexity Q (f) is defined analogously, with quantum algorithms in place of randomized ones. See Buhrman and de Wolf =-=[10]-=- for a survey of these measures. We now define similar measures for algebraic query complexity. In our definition, an important parameter will be the multidegree of the allowed extension (recall that ... |

120 |
Parity, circuits, and the polynomial time hierarchy
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- 1984
(Show Context)
Citation Context ...polynomial hierarchy—or at least separate the polynomial hierarchy from larger classes such as P #P and PSPACE. 17 In the standard oracle setting, these separations were achieved by Furst-Saxe-Sipser =-=[15]-=- and Yao [44] in the 1980’s, whereas in the communication setting they remain notorious open problems. Again, algebraic query complexity provides a natural intermediate case between query complexity a... |

110 | Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses - Klivans, Melkebeek - 2002 |

104 |
On the distributional complexity of disjointness
- Razborov
- 1992
(Show Context)
Citation Context ...ny communication complexity lower bound automatically leads to an algebraic query complexity lower bound. This means, for example, that we can use celebrated lower bounds for the Disjointness problem =-=[33, 22, 25, 34]-=- to show that there exist oracles A, � A relative to which NP A �⊂ BPP eA , and even NP A �⊂ BQP eA and NP A �⊂ coMA e A . For the latter two results, we do not know of any proof by direct constructio... |

90 | Quantum communication complexity of symmetric predicates - Razborov |

81 | Exponential separation of quantum and classical communication complexity
- Raz
- 1999
(Show Context)
Citation Context ... PPcc, which implies PNPA �⊂ PP e A . (iv) Razborov [34] showed that Disjointness has quantum communication complexity Ω( √ N). This implies that NPcc �⊂ BQPcc, and hence NP A �⊂ BQP e A . 11 (v) Raz =-=[30]-=- gave an exponential separation between randomized and quantum communication complexities for a promise problem. This implies that PromiseBQPcc �⊂ PromiseBPPcc, and hence BQP A �⊂ BPP e A (note that w... |

77 | Lower bounds for the size of circuits of bounded depth with basis
- Razborov
- 1987
(Show Context)
Citation Context ...at are not based on arithmetization. Besides the GMW protocol (which we dealt with in Section 8), the following examples have been proposed: (1) Small-depth circuit lower bounds, such as AC 0 �= TC 0 =-=[32]-=-, can be shown to fail relative to suitable oracle gates. (2) Arora, Impagliazzo, and Vazirani [3] argue that even the Cook-Levin Theorem (and by extension, the PCP Theorem) should be considered non-r... |

73 | Circuit-size lower bounds and non-reducibility to sparse sets - Kannan - 1982 |

62 |
Relativized circuit complexity
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- 1985
(Show Context)
Citation Context ... immediately gives A, � A such that NEXP e A ⊂ P A /poly. This collapse is almost the best possible, since Theorem 3.17 implies that there do not exist A, � A such that MA e A EXP ⊂ P A /poly. Wilson =-=[43]-=- gave an oracle A relative to which EXP NPA ⊂ PA /poly. Using similar ideas, one can straightforwardly generalize the construction of Theorem 5.6 to obtain the following: Theorem 5.7 There exist A, � ... |

59 | Quantum search of spatial regions
- Aaronson, Ambainis
- 2005
(Show Context)
Citation Context ... � Am for any m or any m = O (poly (n)) respectively. 12 It is an interesting question whether his lower bound is tight. We know that Disjointness admits a quantum protocol with O( √ N) communication =-=[7, 2]-=-, as well as an MA-protocol with O( √ N log N) communication (see Section 7.2). The question is whether these can be combined somehow to get a QMA-protocol with, say, O(N 1/3 ) communication. 29sNotic... |

58 | In search of an easy witness: exponential time vs. probabilistic polynomial time
- Impagliazzo, Kabanets, et al.
(Show Context)
Citation Context ...P �⊂ P A /poly • PromiseMA e A �⊂ SIZE A � n k� for all constants k Finally, Section 3.5 discusses some miscellaneous interactive proof results, including that of Impagliazzo, Kabanets, and Wigderson =-=[20]-=- that NEXP ⊂ P/poly =⇒ NEXP = MA, and that of Feige and Kilian [12] that RG = EXP. Throughout the section, we assume some familiarity with the proofs of the results we are algebrizing. 3.1 Self-Correc... |

57 | Pseudorandomness and average-case complexity via uniform reductions
- Trevisan, Vadhan
- 2002
(Show Context)
Citation Context ... we have � , � . NEXP eA[poly] ⊂ P eA /poly =⇒ NEXP A[poly] ⊆ MA eA . 8 Note that Santhanam originally proved his result using a “tight” variant of the IP = PSPACE theorem, due to Trevisan and Vadhan =-=[40]-=-. We instead use a tight variant of the LFKN theorem. However, we certainly expect that the Trevisan-Vadhan theorem, and the proof of Santhanam based on it, would algebrize as well. 16sFeige and Kilia... |

54 |
On time versus space
- Hopcroft, Paul, et al.
- 1977
(Show Context)
Citation Context ...d be considered non-relativizing. (3) Hartmanis et al. [17] cite, as examples of non-relativizing results predating the “interactive proofs revolution,” the 1977 result of Hopcroft, Paul, and Valiant =-=[19]-=- that TIME(f (n)) �= SPACE (f (n)) for any space-constructible f, as well as the 1983 result of Paul et al. [29] that TIME(n) �= NTIME(n). Recent time-space tradeoffs for SAT (see van Melkebeek [28] f... |

51 |
Relativizations of the P=?NP question
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- 1975
(Show Context)
Citation Context ...ue multilinear extension of A. As observed by Babai, Fortnow, and Lund [4], the multilinear extension of any PSPACE language is also in PSPACE. So as in the usual argument of Baker, Gill, and Solovay =-=[5]-=-, we have NP e A = NP PSPACE = PSPACE = P A . The same argument immediately implies that any proof of P �= PSPACE will require nonalgebrizing techniques: Theorem 5.2 There exist A, � A such that PSPAC... |

50 |
Relative to a random oracle A, P A = NP A = coNP A with probability 1
- Bennett, Gill
- 1981
(Show Context)
Citation Context ... A �⊂ BPP e A . Furthermore, the language L that achieves the separation simply corresponds to finding a w ∈ {0,1} n with An (w) = 1. 24sProof. Our proof closely follows the proof of Bennett and Gill =-=[6]-=- that P A �= NP A with probability 1 over A. Similarly to Lemma 4.10, given a Boolean point w and a finite field F, let Dn,w,F be the uniform distribution over all multiquadratic polynomials p : F n →... |

48 |
the polynomial hierarchy
- Perceptrons
- 1994
(Show Context)
Citation Context ... have |St| ≤ |St−1| /2 for all t. Since |S0| ≤ 2 p(n) , it follows that with overwhelming probability one will also have � � �Sp(n) � = 1. 17 Any proof would have to be non-relativizing, since Beigel =-=[6]-=- gave an oracle relative to which P NP �⊂ PP. 18 If the oracle � A only involves a low-degree extension over Fq, for some fixed prime q = o (n/ log n), then we can give A, � A such that PP A �⊂ PH � A... |

44 | Nonrelativizing separations
- Buhrman, Fortnow, et al.
- 1998
(Show Context)
Citation Context ...both barriers, by cleverly combining arithmetization (which is non-relativizing) with diagonalization (which is non-naturalizing). The first such lower bound was due to Buhrman, Fortnow, and Thierauf =-=[8]-=-, who showed that MAEXP, the exponential-time analogue of MA, is not in P/poly. To prove that their result was non-relativizing, Buhrman et al. also gave an oracle A such that MA A EXP ⊂ PA /poly. Usi... |

42 | The role of relativization in complexity theory
- Fortnow
- 1994
(Show Context)
Citation Context ...communication complexity to construct A, � A such that NEXP e A ⊂ P A /poly and NP e A ⊂ SIZE A (n). 1.4 Related Work In a survey article on “The Role of Relativization in Complexity Theory,” Fortnow =-=[13]-=- defined a class of oracles O relative to which IP = PSPACE. His proof that IP A = PSPACE A for all A ∈ O was similar to our proof, in Section 3.2, that IP = PSPACE algebrizes. However, because he wan... |

41 |
How to generate and exchange secrets (extended abstract
- Yao
- 1986
(Show Context)
Citation Context ... locality of computation. These include the zero-knowledge protocol for NP due to Goldreich, Micali, and Wigderson [16] (henceforth the GMW Theorem), the two-party oblivious circuit evaluation of Yao =-=[45]-=-, and potentially many others. Here we focus on the GMW Theorem. As discussed in Section 1, the GMW Theorem is inherently non-black-box, since it uses the structure of an NP-complete problem (namely 3... |

40 |
On determinism versus nondeterminism and related problems
- Paul, Pippenger, et al.
- 1983
(Show Context)
Citation Context ...ing the “interactive proofs revolution,” the 1977 result of Hopcroft, Paul, and Valiant [19] that TIME(f (n)) �= SPACE (f (n)) for any space-constructible f, as well as the 1983 result of Paul et al. =-=[29]-=- that TIME(n) �= NTIME(n). Recent time-space tradeoffs for SAT (see van Melkebeek [28] for a survey) have a similar flavor. There are two points we can make regarding these examples. Firstly, the smal... |

36 | On computation and communication with small bias
- Buhrman, Vereshchagin, et al.
- 2007
(Show Context)
Citation Context ...wer bound of [33, 22] to show that Disjointness has MA communication complexity Ω( √ N). From this it follows that coNPcc �⊂ MAcc, and hence coNP A �⊂ MA eA . (iii) Buhrman, Vereshchagin, and de Wolf =-=[9]-=- showed that PNP cc �⊂ PPcc, which implies PNPA �⊂ PP e A . (iv) Razborov [34] showed that Disjointness has quantum communication complexity Ω( √ N). This implies that NPcc �⊂ BQPcc, and hence NP A �⊂... |

19 | Rectangle size bounds and threshold covers in communication complexity
- Klauck
- 2003
(Show Context)
Citation Context ...ny communication complexity lower bound automatically leads to an algebraic query complexity lower bound. This means, for example, that we can use celebrated lower bounds for the Disjointness problem =-=[33, 22, 25, 34]-=- to show that there exist oracles A, � A relative to which NP A �⊂ BPP eA , and even NP A �⊂ BQP eA and NP A �⊂ coMA e A . For the latter two results, we do not know of any proof by direct constructio... |

14 |
Making games short
- Feige, Kilian
- 1997
(Show Context)
Citation Context ... Finally, Section 3.5 discusses some miscellaneous interactive proof results, including that of Impagliazzo, Kabanets, and Wigderson [20] that NEXP ⊂ P/poly =⇒ NEXP = MA, and that of Feige and Kilian =-=[12]-=- that RG = EXP. Throughout the section, we assume some familiarity with the proofs of the results we are algebrizing. 3.1 Self-Correction for #P: Algebrizing In this subsection we examine some non-rel... |

14 | A survey of lower bounds for satisfiability and related problems
- Melkebeek
(Show Context)
Citation Context ...t [19] that TIME(f (n)) �= SPACE (f (n)) for any space-constructible f, as well as the 1983 result of Paul et al. [29] that TIME(n) �= NTIME(n). Recent time-space tradeoffs for SAT (see van Melkebeek =-=[28]-=- for a survey) have a similar flavor. There are two points we can make regarding these examples. Firstly, the small-depth circuit lower bounds are already “well covered” by the natural proofs barrier.... |

12 | Relativizing versus nonrelativizing techniques: the role of local checkability
- Arora, Impagliazzo, et al.
- 1992
(Show Context)
Citation Context ...he following examples have been proposed: (1) Small-depth circuit lower bounds, such as AC 0 �= TC 0 [32], can be shown to fail relative to suitable oracle gates. (2) Arora, Impagliazzo, and Vazirani =-=[3]-=- argue that even the Cook-Levin Theorem (and by extension, the PCP Theorem) should be considered non-relativizing. (3) Hartmanis et al. [17] cite, as examples of non-relativizing results predating the... |

12 | Circuit lower bounds for Merlin-Arthur classes
- Santhanam
- 2007
(Show Context)
Citation Context ...d k, the class PP does not have circuits of size n k ; and Aaronson [1] showed that Vinodchandran’s result was non-relativizing, by giving an oracle A such that PP A ⊂ SIZE A (n). Recently, Santhanam =-=[36]-=- gave a striking improvement of Vinodchandran’s result, by showing that for every fixed k, the class PromiseMA does not have circuits of size n k . As Santhanam [36] stressed, these results raise an i... |

8 | On the power of quantum proofs
- Raz, Shpilka
- 2004
(Show Context)
Citation Context ...omise problem. This implies that PromiseBQPcc �⊂ PromiseBPPcc, and hence BQP A �⊂ BPP e A (note that we can remove the promise by simply choosing oracles A, � A that satisfy it). (vi) Raz and Shpilka =-=[31]-=- showed that PromiseQMAcc �⊂ PromiseMAcc. As in (iv), this implies that QMA A �⊂ MA e A . We end by mentioning, without details, two other algebraic oracle separations that can be proved using the con... |

6 | Oracles are subtle but not malicious
- Aaronson
- 2006
(Show Context)
Citation Context ...hrman et al. also gave an oracle A such that MA A EXP ⊂ PA /poly. Using similar ideas, Vinodchandran [41] showed that for every fixed k, the class PP does not have circuits of size n k ; and Aaronson =-=[1]-=- showed that Vinodchandran’s result was non-relativizing, by giving an oracle A such that PP A ⊂ SIZE A (n). Recently, Santhanam [36] gave a striking improvement of Vinodchandran’s result, by showing ... |

6 |
Efficient learning algorithms yield circuit lower bounds
- Fortnow, Klivans
(Show Context)
Citation Context ...ere exist A, � A such that BPEXP e A ⊂ P A /poly. We omit the details of the above two constructions. However, we would like to mention one interesting implication of Theorem 5.8. Fortnow and Klivans =-=[14]-=- recently showed the following: Theorem 5.9 ([14]) If the class of polynomial-size circuits is exactly learnable by a BPP machine from membership and equivalence queries, or is PAC-learnable by a BPP ... |

5 |
A note on the circuit complexity of PP. ECCC
- Vinodchandran
- 2004
(Show Context)
Citation Context ...onential-time analogue of MA, is not in P/poly. To prove that their result was non-relativizing, Buhrman et al. also gave an oracle A such that MA A EXP ⊂ PA /poly. Using similar ideas, Vinodchandran =-=[41]-=- showed that for every fixed k, the class PP does not have circuits of size n k ; and Aaronson [1] showed that Vinodchandran’s result was non-relativizing, by giving an oracle A such that PP A ⊂ SIZE ... |

4 |
Relativization: a revisionistic perspective
- Hartmanis, Chang, et al.
- 1992
(Show Context)
Citation Context ... suitable oracle gates. (2) Arora, Impagliazzo, and Vazirani [3] argue that even the Cook-Levin Theorem (and by extension, the PCP Theorem) should be considered non-relativizing. (3) Hartmanis et al. =-=[17]-=- cite, as examples of non-relativizing results predating the “interactive proofs revolution,” the 1977 result of Hopcroft, Paul, and Valiant [19] that TIME(f (n)) �= SPACE (f (n)) for any space-constr... |

3 | The black-box query complexity of polynomial summation. Preliminary version at www.cs.sfu.ca/ kabanets/Research/polysum.html
- Juma, Kabanets, et al.
- 2007
(Show Context)
Citation Context ...ent proof techniques for solving open problems in complexity theory, then a non-recursive definition like ours seems essential. Recently (and independently of us), Juma, Kabanets, Rackoff and Shpilka =-=[21]-=- studied an algebraic query complexity model closely related to ours, and proved lower bounds in this model. In our terminology, they “almost” constructed an oracle A, and a multiquadratic extension �... |

2 |
Information theoretic reasons for computational difficulty
- Wigderson
- 1990
(Show Context)
Citation Context ...her hand, they arose directly from the “transfer principle” relating algebrization to communication complexity in Section 4.3. 7.1 Karchmer-Wigderson Revisited Two decades ago, Karchmer and Wigderson =-=[24, 42]-=- noticed that certain communication complexity lower bounds imply circuit lower bounds—or in other words, that one can try to separate complexity classes by thinking only about communication complexit... |