Abstract
The relationship between Boolean proof nets of multiplicative linear logic ( APN ) and Boolean circuits has been studied [Ter04] in a non-uniform setting. We refine this results by taking care of uniformity: the relationship can be expressed in term of the (Turing) polynomial hierarchy. We give a proofs-as-programs correspondence between proof nets and deterministic as well as non-deterministic Boolean circuits with a uniform depth-preserving simulation of each other. The Boolean proof nets class m &BN (poly ) is built on multiplicative and additive linear logic with a polynomial amount of additive connectives as the non-deterministic circuit class NNC (poly) is with non-deterministic variables. We obtain uniform-APN = NC and m & BN (poly ) = NNC (poly)= NP.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Allender, E.W.: P-uniform circuit complexity. Journal of the Association for Computing Machinery 36(4), 912–928 (1989)
Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC1. J. of Comput. and System Science 41(3), 274–306 (1990)
Boppana, R.B., Sipser, M.: The complexity of finite functions. MIT Press, Cambridge (1990)
Cook, S., Krajicek, J.: Consequences of the provability of NP⊆P/poly (2006)
Danos, V.: La logique linéaire appliquée à l’étude de divers processus de normalisation (et principalement du λ-calcul). PhD thesis, Univ. Paris VII (1990)
Danos, V., Regnier, L.: The structure of multiplicatives. Archive for Mathematical Logic 28(3), 181–203 (1989)
Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50(1), 1–102 (1987)
Hughes, D.J.D., van Glabbeek, R.J.: Proof nets for unit-free multiplicative-additive linear logic. In: Proc. IEEE Logic in Comput. Sci. (2003)
Hughes, D.J.D., van Glabbeek, R.J.: Proof nets for unit-free multiplicative-additive linear logic. ACM Trans. on Comput. Logic (2005)
Karp, R., Lipton, R.: Some connections between nonuniform and uniform complexity classes. In: Proc. 12th ACM Symp. on Theory of Computing, pp. 302–309 (1980)
Laurent, O., Tortora de Falco, L.: Slicing polarized additive normalization. In: Linear Logic in Computer Science (2004)
Mairson, H., Terui, K.: On the computational complexity of cut-elimination in linear logic. Theoretical Computer Science 2841, 23–36 (2003)
Ruzzo, W.: On uniform circuit complexity. J. of Computer and System Science 21, 365–383 (1981)
Terui, K.: Proof nets and boolean circuits. In: Proc. IEEE Logic in Comput. Sci., pp. 182–191 (2004)
Tortora De Falco, L.: Additives of linear logic and normalization - part i: a (restricted) church-rosser property. T.C.S. 294(3), 489–524 (2003)
Venkateswaran, H.: Circuit definitions of nondeterministic complexity classes. Siam J. Comput. 21(4), 655–670 (1992)
Vollmer, H.: Introduction to Circuit Complexity – A Uniform Approach. Texts in Theoretical Computer Science. Springer, Heidelberg (1999)
Wolf, M.J.: Nondeterministic circuits, space complexity and quasigroups. Theoretical Computer Science 125(2), 295–313 (1994)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Mogbil, V., Rahli, V. (2007). Uniform Circuits, & Boolean Proof Nets. In: Artemov, S.N., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2007. Lecture Notes in Computer Science, vol 4514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72734-7_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-72734-7_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72732-3
Online ISBN: 978-3-540-72734-7
eBook Packages: Computer ScienceComputer Science (R0)