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Hybrid linear logic, revisited

Published online by Cambridge University Press:  22 April 2019

KAUSTUV CHAUDHURI Open the ORCID record for KAUSTUV CHAUDHURI [Opens in a new window] ,
JOËLLE DESPEYROUX ,
CARLOS OLARTE Open the ORCID record for CARLOS OLARTE [Opens in a new window]  and
ELAINE PIMENTEL Open the ORCID record for ELAINE PIMENTEL [Opens in a new window]
KAUSTUV CHAUDHURI
Affiliation:
Inria & LIX/École Polytechnique, Palaiseau, France Email: kaustuv.chaudhuri@inria.fr
JOËLLE DESPEYROUX
Affiliation:
INRIA and CNRS, I3S, Sophia-Antipolis, France Email: joelle.despeyroux@i3s.unice.fr
CARLOS OLARTE
Affiliation:
ECT – Universidade Federal do Rio Grande do Norte, Natal, Brazil Email: carlos.olarte@gmail.com
ELAINE PIMENTEL
Affiliation:
Departamento de Matemática – Universidade Federal do Rio Grande do Norte, Natal, Brazil Email: elaine.pimentel@gmail.com
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Abstract

HyLL (Hybrid Linear Logic) is an extension of intuitionistic linear logic (ILL) that has been used as a framework for specifying systems that exhibit certain modalities. In HyLL, truth judgements are labelled by worlds (having a monoidal structure) and hybrid connectives (at and ↓) relate worlds with formulas. We start this work by showing that HyLL's axioms and rules can be adequately encoded in linear logic (LL), so that one focused step in LL will correspond to a step of derivation in HyLL. This shows that any proof in HyLL can be exactly mimicked by a LL focused derivation. Another extension of LL that has extensively been used for specifying systems with modalities is Subexponential Linear Logic (SELL). In SELL, the LL exponentials (!, ?) are decorated with labels representing locations, and a pre-order on such labels defines the provability relation. We propose an encoding of HyLL into SELL⋒ (SELL plus quantification over locations) that gives better insights about the meaning of worlds in HyLL. More precisely, we identify worlds as locations, and show that a flat subexponential structure is sufficient for representing any world structure in HyLL. This shows that HyLL's monoidal structure is not reflected in LL derivations, hence not increasing the expressiveness of LL, from a proof theoretical point of view. We conclude by proposing the notion of fixed points in multiplicative additive HyLL (μHyMALL), which can be encoded into multiplicative additive linear logic with fixed points (μMALL). As an application, we propose encodings of Computational Tree Logic (CTL) into both μMALL and μHyMALL. In the former, states are represented as atoms in the linear context, hence reflecting a more operational view of CTL connectives. In the latter, worlds represent states of the transition system, thus exhibiting a pleasant similarity with the semantics of CTL.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Special Issue 8: A special issue on structural proof theory, automated reasoning and computation in celebration of Dale Miller’s 60th birthday , September 2019 , pp. 1151 - 1176
Copyright
© Cambridge University Press 2019 

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References

Arias, J., Desainte-Catherine, M., Olarte, C. and Rueda, C. (2015). Foundations for reliable and flexible interactive multimedia scores. In: Collins, T., Meredith, D. and Volk, A. (eds.) MCM 2015, Lecture Notes in Computer Science, vol. 9110, Springer, 29–41.Google Scholar
Andreoli, J.-M. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2(3) 297–347.CrossRefGoogle Scholar
Baelde, D. (2012). Least and greatest fixed points in linear logic. ACM Transactions on Computational Logic 13 (1) 2.CrossRefGoogle Scholar
Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L. and Hwang, L.J. (1992). Symbolic model checking: 10∧20 states and beyond. Information and Computation 98(2) 142–170.CrossRefGoogle Scholar
Clarke, E.M. and Emerson, E.A. (1981). Design and synthesis of synchronization skeletons using branching-time temporal logic. In: Kozen, D. (ed.) Logics of Programs, Workshop, Yorktown Heights, May 1981, Lecture Notes in Computer Science, vol. 131, New York: Springer, 52–71.Google Scholar
Chaudhuri, K. (2010). Classical and intuitionistic subexponential logics are equally expressive. In: Dawar, A. and Veith, H. (eds.) CSL 2010, Lecture Notes in Computer Science, vol. 6247, Springer, 185–199.Google Scholar
Church, A. (1940). A formulation of the simple theory of types. The Jornal of Symbolic Logic 5 56–68.CrossRefGoogle Scholar
Cervesato, I. and Pfenning, F. (2002). A linear logical framework. Information and Computation 179(1) 19–75.CrossRefGoogle Scholar
Caires, L., Pfenning, F. and Toninho, B. (2016). Linear logic propositions as session types. Mathematical Structures in Computer Science 26(3) 367–423.CrossRefGoogle Scholar
Chaudhuri, K. and Reis, G. (2015). An adequate compositional encoding of bigraph structure in linear logic with subexponentials. In: Davis, M., Fehnker, A., McIver, A. and Voronkov, A. (eds.) LPAR-20 2015, Lecture Notes in Computer Science, vol. 9450, Springer, 146–161.Google Scholar
Despeyroux, J. and Chaudhuri, K. (2014). A hybrid linear logic for constrained transition systems. In: Post-Proc. of TYPES 2013, Leibniz Intl. Proceedings in Informatics, vol. 26, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 150–168.Google Scholar
Danos, V., Joinet, J.-B. and Schellinx, H. (1993). The structure of exponentials: Uncovering the dynamics of linear logic proofs. In: Gottlob, G., Leitsch, A. and Mundici, D. (eds.) Kurt Gödel Colloquium, Lecture Notes in Computer Science, vol. 713, Springer, 159–171.Google Scholar
de Maria, E., Despeyroux, J. and Felty, A. (2014). A logical framework for systems biology. In: Proceedings of the 1st Intl. Conference on Formal Methods in Macro-Biology (FMMB), Lecture Notes in Computer Science, vol. 8738, Springer, 136–155.CrossRefGoogle Scholar
Despeyroux, J., Olarte, C. and Pimentel, E. (2017). Hybrid and subexponential linear logics. Electronic Notes in Theoretical Computer Science 332 95–111.CrossRefGoogle Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 1–102.CrossRefGoogle Scholar
Miller, D. (1992). Unification under a mixed prefix. Journal of Symbolic Computation 14(4) 321–358.CrossRefGoogle Scholar
Miller, D. and Pimentel, E. (2013). A formal framework for specifying sequent calculus proof systems. Theoretical Computer Science 474 98–116.CrossRefGoogle Scholar
Nigam, V. (2014). A framework for linear authorization logics. Theoretical Computer Science 536 21–41.CrossRefGoogle Scholar
Nigam, V. and Miller, D. (2009). Algorithmic specifications in linear logic with subexponentials. In: Porto, A. and López-Fraguas, F.J. (eds.) PPDP, ACM, 129–140.CrossRefGoogle Scholar
Nigam, V. and Miller, D. (2010). A framework for proof systems. Journal of Automated Reasoning 45(2) 157–188.CrossRefGoogle Scholar
Nigam, V., Olarte, C. and Pimentel, E. (2013). A general proof system for modalities in concurrent constraint programing. In: CONCUR, Lecture Notes in Computer Science, vol. 8052, Springer Verlag, 410–424.Google Scholar
Nigam, V., Olarte, C. and Pimentel, E. (2017). On subexponentials, focusing and modalities in concurrent systems. Theoretical Computer Science 693 35–58.CrossRefGoogle Scholar
Nigam, V., Pimentel, E. and Reis, G. (2011). Specifying proof systems in linear logic with subexponentials. Electronic Notes in Theoretical Computer Science 269 109–123.CrossRefGoogle Scholar
Nigam, V., Pimentel, E. and Reis, G. (2016). An extended framework for specifying and reasoning about proof systems. Journal of Logic and Computation 26(2) 539–576.CrossRefGoogle Scholar
Olarte, C., Chiarugi, D., Falaschi, M. and Hermith, D. (2016). A proof theoretic view of spatial and temporal dependencies in biochemical systems. Theoretical Computer Science 641 25–42.CrossRefGoogle Scholar
Olarte, C., Pimentel, E. and Nigam, V. (2015). Subexponential concurrent constraint programming. Theoretical Computer Science 606 98–120.CrossRefGoogle Scholar
Olarte, C., Pimentel, E. and Rueda, C. (2018). A concurrent constraint programming interpretation of access permissions. TPLP 18(2) 252–295.Google Scholar
Pimentel, E., Olarte, C. and Nigam, V. (2014). A proof theoretic study of soft concurrent constraint programming. Theory and Practice of Logic Programming 14 475–308.CrossRefGoogle Scholar
Reed, J. (2006). Hybridizing a logical framework. In: International Workshop on Hybrid Logic (HyLo), Electronic Notes in Theoretical Computer Science, Seattle, USA, Elsevier, 135–148.Google Scholar