Abstract
This paper develops a free energy theory from physics including the variational principles for automata and languages and also provides algorithms to compute the energy as well as efficient algorithms for estimating the nondeterminism in a nondeterministic finite automaton. This theory is then used as a foundation to define a semantic similarity metric for automata and languages. Since automata are a fundamental model for all modern programs while languages are a fundamental model for the programs’ behaviors, we believe that the theory and the metric developed in this paper can be further used for real-word programs as well.
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- 1.
The factor 2, intuitively, comes from the fact that we “stretch”, by a factor of 2, a run in finite automaton M to correspond it to a walk in graph \({\hat{M}}\). A somewhat more efficient way to compute \({{{\mathcal {E}}}}(M_V)\) is to construct an \(m\times m\) Gurevich matrix \(\mathbf{M}'\) where m is the number of states in M such that \(\mathbf{M}'_{ij}=0\) if there is no transition from \(p_i\) to \(p_j\) in M, else \(\mathbf{M}'_{ij}=\sum _{a:(p_i,a,p_j)\in T} e^{V(p_i,a,p_j)}\). Herein, \(p_1,\cdots , p_m\) are all states in M. One can show that \({{{\mathcal {E}}}}(M_V)=\ln \lambda '\) where \(\lambda '\) is the Perron-Frobenius eigenvalue of \(\mathbf{M}'\). We omit the details.
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Acknowledgements
We would like to thank Jean-Charles Delvenne, David Koslicki, Daniel J. Thompson, Eric Wang, William J. Hutton III, and Ali Saberi for discussions. We would also like to thank the seven referees for suggestions and comments that have improved the presentation of our results.
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Cui, C., Dang, Z. (2016). A Free Energy Foundation of Semantic Similarity in Automata and Languages. In: Amsaleg, L., Houle, M., Schubert, E. (eds) Similarity Search and Applications. SISAP 2016. Lecture Notes in Computer Science(), vol 9939. Springer, Cham. https://doi.org/10.1007/978-3-319-46759-7_3
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