Abstract
In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is universal, and this is a hard problem (including the case of whether the NFA accepts all words of a fixed length). On the other hand, it is acceptable to know whether a code is ‘approximately’ maximal, which then boils down to whether a given NFA is ‘approximately’ universal. Here we introduce the notion of a \((1-\varepsilon )\)-universal automaton and present polynomial randomized approximation algorithms to test NFA universality and related hard automata problems, for certain natural probability distributions on the set of words. We also conclude that the randomization aspect is necessary, as approximate universality remains hard for any fixed polynomially computable \(\varepsilon \).
Research supported by NSERC, Canada (Discovery Grants of S.K. and of M.M.) and by CMUP through FCT project UIDB/00144/2021.
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Notes
- 1.
Following the presentation style of [6, pg 193], we refrain from cluttering the notation with the use of a variable for the set of instances.
- 2.
A real \(x\in (0, 1)\) is computable if there is an algorithm that takes as input an integer \(n>0\) and computes the n-th bit of x. It is polynomially computable if the algorithm works in time \(O(n^k)\), for some fixed \(k\in \mathbb {N} _0\), when the input n is given in unary.
- 3.
- 4.
Depending on \(\boldsymbol{t}\), which is a transducer, one can have prefix codes, suffix codes, infix codes, error control codes.
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Konstantinidis, S., Mastnak, M., Moreira, N., Reis, R. (2022). Approximate NFA Universality Motivated by Information Theory. In: Han, YS., Vaszil, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2022. Lecture Notes in Computer Science, vol 13439. Springer, Cham. https://doi.org/10.1007/978-3-031-13257-5_11
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DOI: https://doi.org/10.1007/978-3-031-13257-5_11
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