Formulas Resilient to Short-Circuit Errors

We show how to efficiently convert any boolean formula F into a boolean formula E that is resilient to short-circuit errors (as introduced by Kleitman et al. [KLM94]). A gate has a short-circuit error when the value it computes is replaced by the value of one of its inputs. We guarantee that E computes the same function as F, as long as at most (1/10 - ε) of the gates on each path from the output to an input have been corrupted in E. The corruptions may be chosen adversarially, and may depend on the formula E and even on the input. We obtain our result by extending the Karchmer-Wigderson connection between formulas and communication protocols to the setting of adversarial error. This enables us to obtain error-resilient formulas from error-resilient communication protocols.

[1]  Bruce E. Hajek,et al.  On the maximum tolerable noise for reliable computation by formulas , 1991, IEEE Trans. Inf. Theory.

[2]  Eli Upfal,et al.  Computing with unreliable information , 1990, STOC '90.

[3]  Silvio Micali,et al.  Algorithmic Tamper-Proof (ATP) Security: Theoretical Foundations for Security against Hardware Tampering , 2004, TCC.

[4]  Frank Thomson Leighton,et al.  On the design of reliable Boolean circuits that contain partially unreliable gates , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[5]  Péter Gács,et al.  Lower bounds for the complexity of reliable Boolean circuits with noisy gates , 1994, IEEE Trans. Inf. Theory.

[6]  Nicholas Pippenger,et al.  Reliable computation by formulas in the presence of noise , 1988, IEEE Trans. Inf. Theory.

[7]  Anna Gál,et al.  Lower bounds for the complexity of reliable Boolean circuits with noisy gates , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[8]  Falk Unger.  Noise Threshold for Universality of Two-Input Gates , 2007, IEEE Transactions on Information Theory.

[9]  Anna Gál,et al.  Fault tolerant circuits and probabilistically checkable proofs , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[10]  Ronald de Wolf,et al.  Upper bounds on the noise threshold for fault-tolerant quantum computing , 2008, Quantum Inf. Comput..

[11]  Andrew Chi-Chih Yao,et al.  On Fault-Tolerant Networks for Sorting , 1985, SIAM J. Comput..

[12]  Amit Sahai,et al.  Efficient and Explicit Coding for Interactive Communication , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[13]  Tomás Feder,et al.  Reliable computation by networks in the presence of noise , 1989, IEEE Trans. Inf. Theory.

[14]  Nicholas Pippenger,et al.  On networks of noisy gates , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[15]  Mark Braverman,et al.  Towards deterministic tree code constructions , 2012, ITCS '12.

[16]  Yuval Ishai,et al.  Private Circuits II: Keeping Secrets in Tamperable Circuits , 2006, EUROCRYPT.

[17]  Yael Tauman Kalai,et al.  Cryptography with Tamperable and Leaky Memory , 2011, CRYPTO.

[18]  J. von Neumann,et al.  Probabilistic Logic and the Synthesis of Reliable Organisms from Unreliable Components , 1956 .

[19]  Mark Braverman,et al.  Toward Coding for Maximum Errors in Interactive Communication , 2011, IEEE Transactions on Information Theory.

[20]  C. Greg Plaxton,et al.  Breaking the Theta (n log² n) Barrier for Sorting with Faults , 1997, J. Comput. Syst. Sci..

[21]  Eli Upfal,et al.  Fault tolerant sorting network , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[22]  Leonard J. Schulman,et al.  Deterministic coding for interactive communication , 1993, STOC.

[23]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[24]  Leonard J. Schulman,et al.  Signal propagation and noisy circuits , 1999, IEEE Trans. Inf. Theory.

[25]  Leonard J. Schulman,et al.  On the maximum tolerable noise of k-input gates for reliable computation by formulas , 2003, IEEE Trans. Inf. Theory.

[26]  Eli Upfal,et al.  Computing with Unreliable Information (Preliminary Version) , 1990, STOC 1990.