Abstract
First-order logic (FOL) is evidently insufficient for the many applications of logic in computer science, mainly due to its inability to provide inductive definitions. Therefore, only an extension of FOL which allows finitary inductive definitions can be used as a framework for automated reasoning. The minimal logic that is suitable for this goal is Ancestral Logic (AL), which is an extension of FOL by a transitive closure operator. In order for AL to be able to serve as a reasonable (and better) substitute to the use of FOL in computer science, it is crucial to develop adequate, user-friendly proof systems for it. While the expressiveness of AL renders any effective proof system for it incomplete with respect to the standard semantics, there are useful approximations. In this paper we show that such a Gentzen-style approximation is both sound and complete with respect to a natural, computationally-meaningful Henkin-style semantics for AL.
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
Similar content being viewed by others
Notes
- 1.
Actually, at this point we are only referring to the formal languages used in the applications, ignoring (for the time being) other essential components of the notion of a ‘logic’, like the corresponding consequence relation.
- 2.
A great deal of attention has been given to AL in the area of finite model theory, and in related areas of computer science, like complexity classes (see, e.g., [4]). However, not much has been done so far about it in the context of arbitrary structures, or from a proof theoretical point-of-view.
- 3.
To demonstrate one such application of AL in computer science, in [3] a constructive version of AL was shown to subsume Kleene algebra with tests [11] (as the reflexive transitive closure operator is essentially Kleene’s star operator), while offering much more expressive power. This demonstrates that AL can serve as a natural programming logic for specifying, developing and reasoning about programs.
- 4.
In fact, [2] presented several proof systems for different variations of AL, and the connection between them was investigated.
- 5.
To be precise, we take here an equivalent variant of a system presented in [2].
- 6.
In fact, it was shown in [2] that in the case of arithmetics the ordinal number of \(AL_{G}\) is \(\varepsilon _{0}\), like in the case of PA.
- 7.
An assignment v in M is defined as in the standard semantics.
References
Avron, A.: Transitive closure and the mechanization of mathematics. In: Kamareddine, F.D. (ed.) Thirty Five Years of Automating Mathematics. Applied Logic Series, vol. 28, pp. 149–171. Springer, Netherlands (2003). doi:10.1007/978-94-017-0253-9_7
Cohen, L., Avron, A.: The middle ground-ancestral logic. Synthese, 1–23 (2015)
Cohen, L., Constable, R.L.: Intuitionistic ancestral logic. J. Logic Comput. (2015)
Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Springer Science & Business Media, New York (2005)
Feferman, S.: Finitary inductively presented logics. Stud. Logic Found. Math. 127, 191–220 (1989)
Gentzen, G.: Investigations into Logical Deduction (1934). (in German). An English translation appears in ‘The Collected Works of Gerhard Gentzen’, edited by M.E. Szabo. North-Holland (1969)
Girard, J.Y.: Proof Theory and Logical Complexity, vol. 1. Humanities Press, London (1987)
Halpern, J.Y., Harper, R., Immerman, N., Kolaitis, P.G., Vardi, M.Y., Vianu, V.: On the unusual effectiveness of logic in computer science. Bull. Symb. Logic 7(02), 213–236 (2001)
Henkin, L.: Completeness in the theory of types. J. Symb. Logic 15(2), 81–91 (1950)
Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16(4), 760–778 (1987)
Kozen, D.: Kleene algebra with tests. ACM Trans. Program. Lang. Syst. (TOPLAS) 19(3), 427–443 (1997)
Martin, R.M.: A homogeneous system for formal logic. J. Symb. Logic 8(1), 1–23 (1943)
Martin, R.M.: A note on nominalism and recursive functions. J. Symb. Logic 14(1), 27–31 (1949)
Myhill, J.: A derivation of number theory from ancestral theory. J. Symb. Logic 17(3), 192–197 (1952)
Shapiro, S.: Foundations Without Foundationalism: A Case for Second-Order Logic. Oxford University Press, Oxford (1991)
Tait, W.W.: A nonconstructive proof of gentzen’s hauptsatz for second order predicate logic. Bull. Am. Math. Soc. 72(6), 980–983 (1966)
Takeuti, G.: Proof Theory. Courier Dover Publications, Mineola (1987)
Wohrle, S., Thomas, W.: Model checking synchronized products of infinite transition systems. In: Logic in Computer Science, pp. 2–11 (2004)
Acknowledgments
This research was supported by: Ministry of Science, Technology and Space, Israel; Fulbright Post-doctoral Scholar program; Weizmann Institute of Science – National Postdoctoral Award Program for Advancing Women in Science; Eric and Wendy Schmidt Postdoctoral Award program for Women in Mathematical and Computing Sciences; and Cornell University PRL Group.
The author is indebt to A. Avron for his invaluable comments and expertise that greatly assisted this research.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Cohen, L. (2017). Completeness for Ancestral Logic via a Computationally-Meaningful Semantics. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-66902-1_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66901-4
Online ISBN: 978-3-319-66902-1
eBook Packages: Computer ScienceComputer Science (R0)