Dmitry Gavinsky
ABSTRACT
One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P ⊈ NC1). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.
Karchmer, Raz, and Wigderson [21] suggested to approach this problem by proving the following conjecture: given two boolean functions f and g, the depth complexity of the composed function g o f is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that P ⊈ NC1.
As a starting point for studying the composition of functions, they introduced a relation called "the universal relation", and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [12]. An alternative proof was given later by Håstad and Wigderson [18]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.
In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it.
Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations -- communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.
Supplemental Material
- A. E. Andreev. On a method for obtaining more than quadratic effective lower bounds for the complexity of π-schemes. Moscow University Mathematics Bulletin, 42(1):24--29, 1987.Google Scholar
- Z. Bar-Yossef, T. S. Jayram, R. Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci., 68(4):702--732, 2004. Google ScholarDigital Library
- B. Barak, M. Braverman, X. Chen, and A. Rao. How to compress interactive communication. In STOC, pages 67--76, 2010. Google ScholarDigital Library
- M. L. Bonet and S. R. Buss. Size-depth tradeoffs for boolean fomulae. Inf. Process. Lett., 49(3):151--55, 1994. Google ScholarDigital Library
- M. Braverman. Interactive information complexity. In STOC, pages 505--524, 2012. Google ScholarDigital Library
- M. Braverman and A. Rao. Information equals amortized communication. In FOCS, pages 748--757, 2011. Google ScholarDigital Library
- M. Braverman and O. Weinstein. A discrepancy lower bound for information complexity. In APPROX-RANDOM, pages 459--470, 2012.Google ScholarCross Ref
- R. P. Brent. The parallel evaluation of general arithmetic expressions. J. ACM, 21(2):201--206, 1974. Google ScholarDigital Library
- A. Chakrabarti, Y. Shi, A. Wirth, and A. C.-C. Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In FOCS, pages 270--278, 2001. Google ScholarDigital Library
- T. M. Cover and J. A. Thomas. Elements of information theory. Wiley-Interscience, 1991. Google ScholarDigital Library
- M. Dietzfelbinger and H. Wunderlich. A characterization of average case communication complexity. Inf. Process. Lett., 101(6):245--249, 2007. Google ScholarDigital Library
- J. Edmonds, R. Impagliazzo, S. Rudich, and J. Sgall. Communication complexity towards lower bounds on circuit depth. Computational Complexity, 10(3):210--246, 2001.Google ScholarCross Ref
- P. Frankl and N. Tokushige. The Erdős-ko-rado theorem for integer sequences. Combinatorica, 19(1):55--63, 1999.Google ScholarCross Ref
- D. Gavinsky, O. Meir, O. Weinstein, and A. Wigderson. Toward better formula lower bounds: An information complexity approach to the krw composition conjecture. Electronic Colloquium on Computational Complexity (ECCC), (190), 2013.Google Scholar
- M. Grigni and M. Sipser. Monotone separation of logspace from nc. In Structure in Complexity Theory Conference, pages 294--298, 1991.Google Scholar
- P. Harsha, R. Jain, D. A. McAllester, and J. Radhakrishnan. The communication complexity of correlation. IEEE Transactions on Information Theory, 56(1):438--449, 2010. Google ScholarDigital Library
- J. Håstad. The shrinkage exponent of de morgan formulas is 2. SIAM J. Comput., 27(1):48--64, 1998. Google ScholarDigital Library
- J. Håstad and A. Wigderson. Composition of the universal relation. In Advances in computational complexity theory, AMS-DIMACS, 1993.Google ScholarCross Ref
- R. Jain, J. Radhakrishnan, and P. Sen. A direct sum theorem in communication complexity via message compression. In ICALP, pages 300--315, 2003. Google ScholarDigital Library
- T. S. Jayram, R. Kumar, and D. Sivakumar. Two applications of information complexity. In STOC, pages 673--682, 2003. Google ScholarDigital Library
- M. Karchmer, R. Raz, and A. Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5(3/4):191--204, 1995.Google ScholarCross Ref
- M. Karchmer and A. Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM J. Discrete Math., 3(2):255--265, 1990.Google ScholarCross Ref
- I. Kerenidis, S. Laplante, V. Lerays, J. Roland, and D. Xiao. Lower bounds on information complexity via zero-communication protocols and applications. Electronic Colloquium on Computational Complexity (ECCC), 19:38, 2012.Google Scholar
- V. M. Khrapchenko. A method of obtaining lower bounds for the complexity of π-schemes. Mathematical Notes Academy of Sciences USSR, 10:474--479, 1972.Google Scholar
- R. Raz and A. Wigderson. Monotone circuits for matching require linear depth. J. ACM, 39(3):736--744, 1992. Google ScholarDigital Library
- A. A. Razborov and S. Rudich. Natural proofs. J. Comput. Syst. Sci., 55(1):24--35, 1997. Google ScholarDigital Library
- P. M. Spira. On time-hardware complexity tradeoffs for boolean functions. In Proceedings of the Fourth Hawaii International Symposium on System Sciences, pages 525--527, 1971.Google Scholar
- I. Wegener. The complexity of Boolean functions. Wiley-Teubner, 1987. Google ScholarDigital Library
- A. C.-C. Yao. Some complexity questions related to distributive computing (preliminary report). In STOC, pages 209--213, 1979. Google ScholarDigital Library
Index Terms
- Toward better formula lower bounds: an information complexity approach to the KRW composition conjecture
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