Abstract
In the last few years, several publications discussed a new method of matrix completion, called matrix abduction, which is particularly useful in the context of Talmudic logic and legal ruling systems in general. Given a matrix \(\mathbf{A}\) with entries that are either in \(\{0,1\}\) or blank, the method allows us to decide whether each blank entry should be 0 or 1, or remain undecided. Unlike existing matrix completion methods, which are based on notions of analogy using distance (analogy to nearest neighbors), the new method takes a different approach and completes the matrix using only topological criteria; as a result, the outcome of the process is often significantly different. This chapter will focus specifically on one of these criteria, involving the representation of the finite partially ordered set as minimally generated multisets (ordered by inclusion) for which there is no known exact polynomial-time evaluation algorithm. We examine the existing efficient greedy algorithm for this criteria, its connection the similar known concepts dimension and two dimension of a poset (partial ordered set) then derive new useful properties and an algorithm. We start by proving that the exact evaluation of the criteria is NP-complete both in the general case and in the limited context of matrix abduction. We then discuss one-point removal properties of the problematic criteria, which may aid in solving specific cases and be utilized by new algorithms. Finally, we present a new greedy algorithm that offers a significant improvement over the existing one, and discuss the possibility of reusing existing algorithms that approximate 2-dimension of a poset.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The only exception to this is when elements are shared (contracted), but the desired limitation still holds.
References
Abraham, M., Gabbay, D.M., Schild, U.: Analysis of the Talmudic argumentum a fortiori inference rule (Kal Vachomer) using matrix abduction. Stud. Log. 92(3), 281–364 (2009)
Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9(6), 717–772 (2009)
Caseau, Y., Habib, M., Nourine, L., Raynaud, O.: Encoding of multiple inheritance hierarchies and partial orders. Comput. Intell. 15(1), 50–62 (1999)
Filman, R.E.: Polychotomic encoding: a better quasi-optimal bit-vector encoding of tree hierarchies. In ECOOP 2002 – Object-oriented Programming, pp. 545–561. Springer, Berlin, Heidelberg (2002)
Fulkerson, D.R.: Note on Dilworth’s decomposition theorem for partially ordered sets. Proc. Am. Math. Soc. 7, 701–702 (1956)
Habib, M., Nourine, L., Raynaud, O., Thierry, E.: Computational aspects of the two dimension of partially ordered sets. Theor. Comput. Sci. 312(2), 401–431 (2004)
Hegde, R., Jain, K.: The hardness of approximating poset dimension. Electron. Notes Discret. Math. 29, 435–443 (2007)
Hill, A.: Reasoning by analogy in inductive logic. Dissertation, University of Manchester (2013)
Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)
Johnson, C.R.: Matrix completion problems: a survey. In: Proceedings of Symposia in Applied Mathematics, vol. 40, pp. 171–198. American Mathematical Society (1990)
Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Springer, New York (1972)
Krall, A., Vitek, J., Horspool, R.N.: Near optimal hierarchical encoding of types. In: ECOOP’97 – Object-oriented Programming, pp. 128–145. Springer, Berlin, Heidelberg (1997)
Laufer, A., Gabbay, D.M.: Topological aspects of matrix abduction 1. In: Koslow, A., Buchsbaum, A. (eds.) The Road to Universal Logic – Festschrift for the 50th Birthday of Jean-Yves Beziau, Volume II. Birkhäuser, Basel (2015)
Prade, H., Richard, G.: Homogeneous logical proportions: their uniqueness and their role in similarity-based prediction. Thirteenth International Conference on the Principles of Knowledge Representation and Reasoning, 2012
Raynaud, O., Thierry, E.: The complexity of embedding orders into small products of chains. Order 27(3), 365–381 (2010)
Ribeiro, H.J. (ed.): Systematic Approaches to Argument by Analogy. Springer, Berlin (2014)
Schlechta, K.: Karl’s remarks on Kal-algorithm. Personal communication (2011)
Trotter, W.T.: Combinatorics and Partially Ordered Sets. The John Hopkins University Press (1992)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Ben Adar Bessos, M., Gabbay, D.M. (2015). Topological Aspects of Matrix Abduction 2. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-15368-1_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-15368-1_17
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-15367-4
Online ISBN: 978-3-319-15368-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)