Abstract
It can be very advantageous to borrow key components of a logic for use in another logic. The advantages may be not only conceptual; due to the existence of software systems supporting mechanized reasoning in a given logic, it may be possible to reuse a system developed for one logic—for example, a theorem-prover—to obtain a new system for another. Translations between logics by appropriate mappings provide a first way of reusing tools of one logic in another. This paper generalizes this idea to the case where entire components—for example, the proof theory—of one of the logics involved may be completely missing, so that the appropriate mapping could not even be defined. The idea then is to borrow the missing components (as well as their associated tools if they exist) from a logic that has them in order to create the full-fledged logic and tools that we desire. The relevant structure is transported using maps that only involve a limited aspect of the two logics in question—for example, their model theory. The constructions accomplishing this kind of borrowing of logical structure are very general and simple. They only depend upon a few abstract properties that hold under very general conditions given a pair of categories linked by adjoint functors.
Partially supported by Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo of C.N.R. (Italy), MURST-40% Modelli e Specifiche di Sistemi Concorrenti, by the US Office of Naval Research under contracts N00014-90-C-0086 and N00014-92-C-0518, and by the Information Technology Promotion Agency, Japan, as a part of the R & D of Basic Technology for Future Industries “New Models for Software Architecture” sponsored by NEDO (New Energy and Industrial Technology Development Organization).
Preview
Unable to display preview. Download preview PDF.
References
E. Astesiano and M. Cerioli. Partial higher-order specifications. Fundamenta Informaticae, 16(2):101–126, 1992.
E. Astesiano and M. Cerioli. Relationships between logical frames. In M. Bidoit and C. Choppy, editors, Recent Trends in Data Type Specification, number 655 in LNCS, pages 126–143, Berlin, 1993. Springer Verlag.
R.M. Burstall and J.A. Goguen. The semantics of Clear, a specification language. In D. Bjørner, editor, Proceedings of the 1979 Copenhagen Winter School on Abstract Software Specification, number 86 in LNCS, pages 292–332, Berlin, 1980. Springer Verlag.
M. Cerioli and J. Meseguer. May I borrow your logic? Technical report, SRI International, Computer Science Laboratory, 1993. To appear.
H. Ehrig, M. Baldamus, and F. Cornelius. Theory of algebraic module specification including behavioural semantics, constraints and aspects of generalized morphisms. In Proceedings of 2nd International Conference on Algebraic Methodology and Software Technology, pages 101–125, Iowa City, Iowa, USA, 1991.
H. Ehrig and B. Mahr. Fundamentals of Algebraic Specifications 1: Equations and Initial semantics, volume 6 of EATCS Monographs on Theoretical Computer Science. Springer Verlag, Berlin, 1985.
J. Fiadeiro and A. Sernadas. Structuring theories on consequence. In D. Sannella and A. Tarlecki, editors, Recent Trends in Data Type Specification, number 332 in LNCS, pages 44–72, Berlin, 1987. Springer Verlag.
J.A. Goguen and R.M. Burstall. Introducing institutions. In E. Clarke and D. Kozen, editors, Logics of Programming Workshop, number 164 in LNCS, pages 221–255, Berlin, 1984. Springer Verlag.
J.A. Goguen and R.M. Burstall. A study in the foundations of programming methodology: Specifications, institutions, charters and parchments. In D. Pitt, S. Abramsky, A. Poigné, and D. Rydehard, editors, Proceedings of Summer Workshop on Category Theory and Computer Programming, number 240 in LNCS, pages 313–333, Berlin, 1986. Springer Verlag.
J.A. Goguen and R.M. Burstall. Institutions: Abstract model theory for specification and programming. Journal of the ACM, 39(1):95–146, 1992.
J.A. Goguen and J. Meseguer. Eqlog: Equality, types, and generic modules for logic programming. In D. DeGroot and G. Lindstrom, editors, Logic Programming: Functions, Relations and Equations, pages 295–363. Prentice-Hall, 1986.
J.A. Goguen, T. Winkler, J. Meseguer, K. Futatsugi, and J.-P. Jouannaud. Introducing OBJ. Technical Report SRI-CSL-92-03, SRI International, Computer Science Laboratory, 1992. To appear in J.A. Goguen, editor, Applications of Algebraic Specification Using OBJ, Cambridge University Press.
R. Harper, D. Sannella, and A. Tarlecki. Logic representation in LF. In D.H. Pitt et al., editor, Category Theory and Computer Science, number 389 in LNCS, pages 250–272. Springer Verlag, 1989.
S. MacLane. Categories for the Working Mathematician. Springer Verlag, 1971.
B. Mayoh. Galleries and institutions. Technical Report DAIMI PB — 191, Aarhus University, 1985.
J. Meseguer. General logics. In Logic Colloquium '87, pages 275–329, Amsterdam, 1989. North Holland.
J. Meseguer. A logical theory of concurrent objects and its realization in the Maude language. SRI Technical Report SRI-CSL-92-08, July 1992. To appear in G. Agha, P. Wegner, and A. Yonezawa, editors, Research Directions in Object-Based Concurrency, MIT Press, 1993.
P. Padawitz and M. Wirsing. Completeness of many-sorted equational logic revisited. Bulletin EATCS, (24), 1984.
A. Poigné. Foundations are rich institutions, but institutions are poor foundations. In H. Ehrig, H. Herrlich, Kreowski H.-J., and G. Preuß, editors, Categorical Methods in Computer Science, number 393 in LNCS, pages 82–101, Berlin, 1989. Springer Verlag.
A. Salibra and G. Scollo. A soft stairway to institutions. In Recent Trends in Data Type Specification, number 655 in LNCS, pages 310–329, Berlin, 1992. Springer Verlag.
D. Sannella and A. Tarlecki. On observational equivalence and specifications. Journal of Comp. and Sys. Sciences, 34:150–178, 1987.
A. Tarlecki. Free constructions in algebraic institutions. In M.P. Chytil and V. Koubek, editors, Proceedings of Mathematical Foundation of Computer Science '84, number 176 in LNCS, pages 526–534, Berlin, 1984. Springer Verlag.
A. Tarlecki. On the existence of free models in abstract algebraic institutions. Theoretical Computer Science, 37(3):269–304, 1985.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cerioli, M., Meseguer, J. (1993). May I borrow your logic?. In: Borzyszkowski, A.M., Sokołowski, S. (eds) Mathematical Foundations of Computer Science 1993. MFCS 1993. Lecture Notes in Computer Science, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57182-5_26
Download citation
DOI: https://doi.org/10.1007/3-540-57182-5_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57182-7
Online ISBN: 978-3-540-47927-7
eBook Packages: Springer Book Archive