# Crytographic limitations on learning Boolean formulae and finite automata

@inproceedings{Kearns1989CrytographicLO, title={Crytographic limitations on learning Boolean formulae and finite automata}, author={Michael Kearns and Leslie G. Valiant}, booktitle={Symposium on the Theory of Computing}, year={1989} }

In this paper we consider the problem of learning from examples classes of functions when there are no restrictions on the allowed hypotheses other than that they are polynomial time evaluatable. We prove that for Boolean formulae, finite automata, and constant depth threshold circuits (simplified neural nets), this problem is computationally as difficult as the quadratic residue problem, inverting the RSA function and factoring Blum integers (composite number p q where p and q are both primes…

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## References

SHOWING 1-10 OF 89 REFERENCES

### An O(n0.4)-approximation algorithm for 3-coloring

- PhysicsSTOC '89
- 1989

A polynomial-time algorithm to color any 3-colorable n-node graph with O(n) colors improves the best previously known bound of O (√n/√log) by reducing the number of colors needed to color a 3- colorable graph.

### Fast probabilistic algorithms for hamiltonian circuits and matchings

- Computer ScienceSTOC '77
- 1977

It is shown that for each problem there is an algorithm that is extremely fast, and which with probability tending to one finds a solution in randomly chosen graphs of sufficient density, and the results contrast with the known NP-completeness of the first two problems.

### Complexity of Automaton Identification from Given Data

- Computer Science, MathematicsInf. Control.
- 1978

### Log Depth Circuits for Division and Related Problems

- Computer ScienceSIAM J. Comput.
- 1984

This work presents optimal depth Boolean circuits for integer division, powering, and multiple products and describes an algorithm for testing divisibility that is optimal for both depth and space.

### Digital signatures and public key functions as intractable as factoring

- M.I.T. Laboratory for Computer Science, technical report number TM-212
- 1979

### An d(n(] ‘)-approximation algorlthm for 3-coloring

- Proceedings of the 21sr ACM Symposuon on the Theow of Computmg
- 1989

### Lecture notes on the complexity of some problems in number theory

- Lecture notes on the complexity of some problems in number theory
- 1982

### Log depth cu’cults for dwlslon and related problems

- SIAM J. Comput. 15, 4 (1986), 994-1003.
- 1986

### A theory of the learnable

- Computer ScienceSTOC '84
- 1984

This paper regards learning as the phenomenon of knowledge acquisition in the absence of explicit programming, and gives a precise methodology for studying this phenomenon from a computational viewpoint.

### On the Markov Chain Simulation Method for Uniform Combinatorial Distributions and Simulated Annealing

- MathematicsProbability in the Engineering and Informational Sciences
- 1987

Uniform distributions on complicated combinatorial sets can be simulated by the Markov chain method. A condition is given for the simulations to be accurate in polynomial time. Similar analysis of…