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Imaginary groups: lazy monoids and reversible computation

Published online by Cambridge University Press:  15 May 2013

MURDOCH J. GABBAY
Affiliation:
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom Website: http://www.gabbay.org.uk
PETER H. KROPHOLLER
Affiliation:
Mathematical Sciences, Southampton University, Highfield, Southampton SO17 1BJ, United Kingdom Email: p.h.kropholler@southampton.ac.uk

Abstract

We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups.

We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m ○ m′)f(m) ○ f(m′).

Every monoid can be endowed with the discrete partial ordering (m ≤ m′ if and only if m=m′), so our constructions provide a way of embedding monoids into groups. A simple counterexample (the two-element monoid with a non-trivial idempotent) and some calculations show that one can never hope for such an embedding to be a monoid homomorphism, so the price paid for injecting a monoid into a group is that we must weaken the notion of a homomorphism to this new notion of a lazy homomorphism.

The computational significance of this is that a monoid is an abstract model of computation – or at least of ‘operations’ – and, similarly, a group models reversible computations/operations. With this reading, the adjunction with its injective unit gives a systematic high-level way of faithfully translating an irreversible system into a ‘lazy’ reversible one.

Informally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add ‘undo’ (in the sense of ‘control-Z’).

Type
Paper
Copyright
Copyright © Cambridge University Press 2013 

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References

Abramsky, S. and Jung, A. (1994) Domain theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science: Semantic Structures, volume 3, Oxford University Press 1168.Google Scholar
Baader, F. and Nipkow, T. (1998) Term Rewriting and All That, Cambridge University Press.CrossRefGoogle Scholar
Bennett, C. H. (1973) Logical reversibility of computation. IBM journal of Research and Development 17 (6)525532.CrossRefGoogle Scholar
Cheng, E. and Lauda, A. (2004) Higher-dimensional categories: an illustrated guide book. Notes prepared for IMA workshop.Google Scholar
Danos, V. and Krivine, J. (2004) Reversible communicating systems. In: Gardner, P. and Yoshida, N. (eds.) Proceedings of the 15th International Conference on Concurrency Theory (CONCUR 2004). Springer-Verlag Lecture Notes in Computer Science 3170 292307.CrossRefGoogle Scholar
Dowek, G. (2001) Higher-order unification and matching. In: Robinson, A. and Voronkov, A. (eds.) Handbook of automated reasoning, volume 2, Elsevier 10091062.CrossRefGoogle Scholar
Gabbay, D. M., Rodrigues, O. and Woods, J. (2004) Belief contraction, anti-formulae and resource overdraft: Part II, deletion in resource unbounded logics. In: Rahman, S., Symons, J., Gabbay, D. M. and van Bendegem, J. P.Logic, Epistemology, and the Unity of Science, volume 1, Springer-Verlag 291326.CrossRefGoogle Scholar
Glass, A. M. W. (1999) Partially ordered groups, Series in Algebra volume 7, World Scientific.CrossRefGoogle Scholar
Herlihy, M. and Moss, J. E. B. (1993) Transactional memory: Architectural support for lock-free data structures. ACM SIGARCH Computer Architecture News 21 (2)289300.CrossRefGoogle Scholar
Hindley, J. R. and Seldin, J. P. (2008) Lambda-Calculus and Combinators, An Introduction, 2nd edition, Cambridge University Press.CrossRefGoogle Scholar
Huet, G. (1975) A unification algorithm for typed λ-calculus. Theoretical Computer Science 1 2757.CrossRefGoogle Scholar
Johnstone, P. T. (2008) On embedding categories in groupoids. Mathematical Proceedings of the Cambridge Philosophical Society 145 273294.CrossRefGoogle Scholar
Keyes, R. W. and Landauer, R. (1970) Minimal energy dissipation in logic. IBM Journal of Research and Development 14 153157.CrossRefGoogle Scholar
Kitaev, A. Y., Shen, A. H. and Vyalyi, M. N. (2002) Classical and quantum computation, Graduate Studies in Mathematics volume 47, American Mathematical Society. (Translated from the Russian by Lester J. Senechal.)CrossRefGoogle Scholar
Lambek, J. (1951) The immersibility of a semigroup into a group. Canadian Journal of Mathematics 3 3443.CrossRefGoogle Scholar
Landauer, R. (1961) Irreversibility and heat generation in the computational process. IBM Journal of Research and Development 5 183191.CrossRefGoogle Scholar
Lawson, M. V. (1998) Inverse semigroups: the theory of partial symmetries, World Scientific.CrossRefGoogle Scholar
Leinster, T. (2004) Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series volume 298, Cambridge University Press.CrossRefGoogle Scholar
Liang, C. and Miller, D. (2009) Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science 410 (46)47474768.CrossRefGoogle Scholar
Mal'cev, A. I. (1937) On the immersion of an algebraic ring into a field. Mathematische Annalen 113 686691.CrossRefGoogle Scholar
Mal'cev, A. I. (1939) On the embedding of associative systems in groups I. Matematicheskii Sbornik 6 (48) (2) 331336. (In German: Über die Einbettung von assoziativen Systemen in Gruppen.)Google Scholar
Manzonetto, G. and Salibra, A. (2010) Applying universal algebra to lambda calculus. Journal of Logic and computation 20 (4)877915.CrossRefGoogle Scholar
Milner, R. (1999) Communicating and mobile systems: the pi-calculus, Cambridge University Press.Google Scholar
Nummela, E. (1980) Cayley's theorem for topological groups. American Mathematical Monthly 87 (3)202203.CrossRefGoogle Scholar
Phillips, I. and Ulidowski, I. (2007) Reversing algebraic process calculi. Journal of Logic and Algebraic Programming 73 (1–2)7096.CrossRefGoogle Scholar
Restall, G. (1999) An Introduction to Substructural Logics, Routledge.Google Scholar
Spiller, T. P. (1998) Basic elements of quantum information technology. In: Lo, H. K., Spiller, T. and Popescu, S. (eds.) Introduction to Quantum Computation and Information, World Scientific 128.Google Scholar
Storme, L., Vos, A. D. and Jacobs, G. (1999) Group theoretical aspects of reversible logic gates. Journal of Universal Computer Science (J.UCS) 5 (5)307321.Google Scholar
Toffoli, T. (1980) Reversible computing. In: de Bakker, J. and van Leeuwen, J. (eds.) Proceedings of the 7th Colloquium on Automata, Languages and Programming. Springer-Verlag Lecture Notes in Computer Science 85 632644.CrossRefGoogle Scholar
Urban, C., Pitts, A. M. and Gabbay, M. J. (2004) Nominal Unification. Theoretical Computer Science 323 (1–3)473497.CrossRefGoogle Scholar