Abstract
Graph algorithms that involve complex conditions on subgraphs can be specified much easier, if the specification allows expressions in higher-order logic to be used. In this paper an extension of Abstract State Machines by such expressions is introduced and its usefulness is demonstrated by examples of computations on graphs, such as graph factoring and checking self-similarity. In a naïve way these high-level specifications can be refined using submachines for the evaluation of the higher-order expressions. We show that refinements can be obtained in an automatic way for well-defined fragments of higher-order logic that collapse to second-order, by means of which the naïve refinement is only necessary for second-order logic expressions.
The research reported in this paper was partially supported by the Austrian Science Fund (FWF) [I2420-N31] for the project: Higher-Order Logics and Structures.
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Notes
- 1.
In the case of non-determinsm using the choice rule we actually obtain a set of update sets.
- 2.
Note that even when the factoring of a graph is unique up to isomorphism, the third-order relation \(\mathcal {F}_I^{\mathcal {A}}\) provides one of the possible set of graphs which are factors of \((V_I^{\mathcal {A}}, E_I^{\mathcal {A}})\).
- 3.
To make this subformula more understandable we chose to use a standard algebraic notation mixed with the syntax of third-order.
References
Abrial, J.R.: The B-Book - Assigning Programs to Meanings. Cambridge University Press, Cambridge (2005)
Abrial, J.R.: Modeling in Event-B - System and Software Engineering. Cambridge University Press, Cambridge (2010)
Abu-Khzam, F.N., Langston, M.A.: Graph coloring and the immersion order. In: Warnow, T., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 394–403. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-45071-8_40
Beaudry, M., McKenzie, P.: Circuits, matrices, and nonassociative computation. In: Proceedings of the Seventh Annual Structure in Complexity Theory Conference, pp. 94–106 (1992)
Blass, A., Gurevich, Y.: Background of computation. Bull. EATCS 92, 82–114 (2007)
Bollobás, B.: Modern Graph Theory. Graduate Texts in Mathematics, vol. 184. Springer, Heidelberg (2002). https://doi.org/10.1007/978-1-4612-0619-4
Börger, E., Stärk, R.F.: Abstract State Machines. A Method for High-Level System Design and Analysis. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-642-18216-7
Dill, S., Kumar, R., Mccurley, K.S., Rajagopalan, S., Sivakumar, D., Tomkins, A.: Self-similarity in the web. ACM Trans. Internet Technol. 2(3), 205–223 (2002)
Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. In: Karp, R. (ed.) Complexity of Computations. SIAM-AMS Proceedings, vol. 7, pp. 27–41. American Mathematical Society (1974)
Ferrarotti, F., Schewe, K.D., Tec, L., Wang, Q.: A new thesis concerning synchronised parallel computing - simplified parallel ASM thesis. Theor. Comput. Sci. 649, 25–53 (2016)
Ferrarotti, F.: Expressibility of higher-order logics on relational databases: proper hierarchies. Ph.D. thesis, Massey University, Wellington, New Zealand (2008). http://hdl.handle.net/10179/799
Ferrarotti, F., González, S., Turull-Torres, J.M.: On fragments of higher order logics that on finite structures collapse to second order. In: Kennedy, J., de Queiroz, R.J.G.B. (eds.) WoLLIC 2017. LNCS, vol. 10388, pp. 125–139. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-55386-2_9
Ferrarotti, F., Ren, W., Turull Torres, J.M.: Expressing properties in second- and third-order logic: hypercube graphs and SATQBF. Logic J. IGPL 22(2), 355–386 (2014)
Grohe, M., Kawarabayashi, K., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC 2011), pp. 479–488. ACM (2011)
Guimerà, R., Danon, L., Díaz-Guilera, A., Giralt, F., Arenas, A.: Self-similar community structure in a network of human interactions. Phys. Rev. E 68, 065103 (2003)
Hella, L., Turull Torres, J.M.: Computing queries with higher-order logics. Theor. Comput. Sci. 355(2), 197–214 (2006)
Immerman, N.: Languages which capture complexity classes (preliminary report). In: Johnson, D.S., et al. (eds.) Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC 1983), pp. 347–354. ACM (1983)
Immerman, N.: Descriptive Complexity. Graduate texts in computer science. Springer, Heidelberg (1999). https://doi.org/10.1007/978-1-4612-0539-5
Lamport, L.: Specifying Systems, The TLA\(^+\) Language and Tools for Hardware and Software Engineers. Addison-Wesley, Boston (2002)
Leivant, D.: Higher order logic. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A., Siekmann, J.H. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming. Deduction Methodologies, vol. 2, pp. 229–322. Oxford University Press (1994)
Réka, A.: Scale-free networks in cell biology. J. Cell Sci. 118(21), 4947–4957 (2005)
Schewe, K.D., Torres, J.M.T.: Fixed-point quantifiers in higher order logics. In: Kiyoki, Y., et al. (eds.) Information Modelling and Knowledge Bases XVII. Frontiers in Artificial Intelligence and Applications, vol. 136, pp. 237–244. IOS Press (2006)
Schewe, K.D., Wang, Q.: A customised ASM thesis for database transformations. Acta Cybernetica 19(4), 765–805 (2010)
Schewe, K.D., Wang, Q.: XML database transformations. J. Univers. Comput. Sci. 16(20), 3043–3072 (2010)
Song, C., Havlin, S., Makse, H.A.: Self-similarity of complex networks. Nature 433, 392–395 (2005)
Vardi, M.Y.: The complexity of relational query languages (extended abstract). In: Lewis, H.R., et al. (eds.) Proceedings of the 14th Annual ACM Symposium on Theory of Computing (STOC 2014), pp. 137–146. ACM (1982)
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Ferrarotti, F., González, S., Schewe, KD., Turull-Torres, J.M. (2018). Systematic Refinement of Abstract State Machines with Higher-Order Logic. In: Butler, M., Raschke, A., Hoang, T., Reichl, K. (eds) Abstract State Machines, Alloy, B, TLA, VDM, and Z. ABZ 2018. Lecture Notes in Computer Science(), vol 10817. Springer, Cham. https://doi.org/10.1007/978-3-319-91271-4_14
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