Abstract
The basic problem of feedback coding is vividly described by Rényi [23, p. 47] as a problem of fault-tolerant adaptive search with errors, as follows: […] I made up the following version, which I called “Bar-kochba with lies”. Assume that the number of questions which can be asked to figure out the “something” being thought of is fixed and the one who answers is allowed to lie a certain number of times. The questioner, of course, doesn’t know which answer is true and which is not. Moreover the one answering is not required to lie as many times as is allowed. For example, when only two things can be thought of and only one lie is allowed, then 3 questions are needed […] If there are four things to choose from and one lie is allowed, then five questions are needed. If two or more lies are allowed, then the calculation of the minimum number of questions is quite complicated […] It does seem to be a very profound problem […]
* The first author was supported by the Sofja Kovaleskvaja Award of the Alexander von Humboldt Foundation. The second author was partially supported by Cofin-2004 Project on Manyvalued Logic.
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Notes
- 1.
The reader may provide the necessary reformulation of “answer” for the cooperative model where Carole knows Paul’s searching strategy.
- 2.
Here \(\dot -\) denotes subtraction truncated to 0.
- 3.
The dependence of the S i ’s on b is understood.
- 4.
This latter paper gives the state of the art on the asymmetric Rényi-Ulam game.
- 5.
Note the asymmetric effect of noise.
- 6.
- 7.
Definition 8.1 is a generalization of the present one, but only applies to the symmetric channel.
- 8.
Since search on a symmetric channel is harder than on the Z-channel, a fortiori Paul can successfully use Gilbert packing for this part of his strategy on the asymmetric channel.
- 9.
Of course, answers sent on channel j may still contain relevant information about any \(w\not=z.\) In accordance with our aims in this section, we shall not be interested in defining the various types of optimization problems concerning search in a multichannel game: suffice to say that communication on a noisy channel is cheaper than on a low-noise channel. Thus Paul’s minimum cost search of x secret will require a careful analysis of which channels should be used when: generally speaking, an expensive channel is to be sparingly used, after the search space has been greatly reduced by preliminary extensive use of noisy channels.
- 10.
In the co-operative model where Carole knows Paul’s strategy and errors are due to distortion, Carole sends honest bits \(b=0,1\) to signify her negative or positive answers. It is expected that up to e j of these answers may be erroneous/mendacious.
- 11.
Even if Carole’s answers sent on a noisy channel \(d>1\) may suggest that z is not a possible candidate for x secret , Carole’s further answers on a more reliable channel \(d'<d\) may well have the contrary effect. When this happens we are led to revise the error parameter e d , or to conclude that an exceptionally large number of errors has affected channel d during this particular session. The essential role of each e d is to tentatively fix an upper bound for our counting of distorted bits sent through channel d: thus, \(e_{d}+1, e_{d}+2,\ldots\) wrong bits are counted as e d . While it is true that only the first channel can give a definitive verdict about which x’s should be excluded as possible candidates for x secret , also the other (less expensive) channels are useful in the search of x secret : in most concrete applications they can be used to substantially reduce the size of the search space.
- 12.
This is a generalization of the notion of state given in the first part of this paper.
- 13.
Compare with [5, p. 247].
- 14.
For any linearly ordered BL-algebra A, the Wajsberg hoops W i such that \(A\cong \bigoplus W_{i}\) will be called the Wajsberg components of A.
- 15.
In fact also the converse is true.
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Acknowledgements
We are grateful to Franco Montagna for his valuable assistance in the writing of the proof of (iii) ⇒ (iv) in Theorem 8.1. We also thank Manuela Busaniche for her further simplification of this proof.
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Cicalese*, F., Mundici, D. (2011). Recent Developments of Feedback Coding and Its Relations with Many-Valued Logic. In: van Benthem, J., Gupta, A., Parikh, R. (eds) Proof, Computation and Agency. Synthese Library, vol 352. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0080-2_8
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