Abstract
We investigate models for programming and specifying in which higher-order functions and nondeterminacy (both demonic and angelic) coexist. The models are built using predicate transformers, binary multirelations, state transformers, and free lattices over a poset. We show there exist suitable models in each approach, and that they are isomorphic. The models support an algebra of nondeterminacy which we use to prove that the classical list fusion law holds even in the presence of nondeterminacy.
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Abramsky S., Jung A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Clarendon Press, Oxford (1994)
Apt K.R., Plotkin G.D.: Countable nondeterminism and random assignment. J. ACM 33(4), 724–767 (1986)
Back R.-J., von Wright J.: Refinement Calculus: A Systematic Introduction. Springer, Heidelberg (1998)
Bird R., de Moor O.: Algebra of Programming. Prentice Hall, London (1997) ISBN 0-13-507245-X
Birkhoff, G.: Lattice Theory, 3rd edn. Colloquium Publications, vol. 25. American Mathematical Society, New York (1967)
Bonsangue M.: Topological Duality in Semantics. Electronic Notes in Theoretical Computer Science, vol. 8. Elsevier, Amsterdam (1998) ISBN 0-13-507245-X
Cavalcanti A.L.C., Woodcock J.C.P., Dunne S.: Angelic nondeterminism in the unifying theories of programming. Formal Aspects Comput. 18(3), 288–307 (2006) ISBN 0-13-507245-X
Davey B.A., Priestley H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002) ISBN 0-13-507245-X
de Moor, O., Gibbons, J.: Invited talk: Pointwise relational programming. In: Proceedings of the 8th International Conference on Algebraic Methodology and Software Technology, vol. 1816, pp. 371–390. Springer, Heidelberg (2000)
Dijkstra, E.W.: A Discipline of Programming. Prentice Hall (1976)
Dijkstra E.W., Scholten C.S.: Predicate Calculus and Program Semantics. Springer, New York (1990) ISBN 0-13-507245-X
Dunne S.: Chorus angelorum. In: Julliand, J., Kouchnarenko, O.B. (eds) Lecture Notes in Computer Science, vol. 4355, pp. 19–33. Springer, Heidelberg (2007) ISBN 0-13-507245-X
Heckmann R.: Power domain constructions. Sci. Comput. Program. 17(1–3), 77–117 (1991) ISBN 0-13-507245-X
Hehner, E.C.R.: A Practical Theory of Programming, 2nd edn. Springer, New York. ISBN 0387941061 1993. 2004 at http://www.cs.toronto.edu/~hehner/aPToP/
Hennessy M., Plotkin G.D.: Full abstraction for a simple parallel programming language. In: Becvár, J. (eds) MFCS. Lecture Notes in Computer Science, vol. 74, pp. 108–120. Springer, Heidelberg (1979) ISBN 0-13-507245-X
Hesselink, W.: Multirelations are predicate transformers. Technical report, Dept. of Computing Science, University of Groningen, The Netherlands (2004)
Hesselink W.H.: Modalities of nondeterminacy. In: Feijen, W.H.J., van Gasteren, A.J.M., Gries, D., Misra, J. (eds) Beauty is our Business: A Birthday Salute to E.W. Dijkstra, pp. 182–192. Springer, Heidelberg (1990) ISBN 0-13-507245-X
Hoare C.A.R., Jifeng H.: Unifying Theories of Programming. Prentice Hall, New Jersey (1998) ISBN 0-13-507245-X
Hoofman, R.: Powerdomains. Technical Report RUU-CS-87-23, Institute of Information and Computing Sciences, Utrecht University (1987)
Hoogerwoord, R.R.: The design of functional programs: a calculational approach. PhD Thesis, Technische Universiteit Eindhoven (1989)
Hughes, J., Moran, A.: Making choices lazily. In: Proceedings of the Seventh International Conference on Functional Programming Languages and Computer Architecture, pp. 108–119. ACM Press, New York (1995)
Hughes, J., O’Donnell, J.: Nondeterministic functional programming with sets. In: Proceedings of the 1990 Banf Conference on Higher Order Reasoning (1990)
Kozen, D., Shankland, C. (eds.) Mathematics of program construction, 7th International Conference, MPC 2004, Stirling, Scotland, UK, July 12–14, 2004. In: Proceedings of Lecture Notes in Computer Science, vol. 3125. Springer, Heidelberg (2004)
Larsen P.G., Hansen B.S.: Semantics of under-determined expressions. Formal Aspects Comput. 8(1), 47–66 (1996) ISBN 0-13-507245-X
Lassen, S.B.: Relational Reasoning about Functions and Nondeterminism. PhD Thesis, Dept of Computer Science, University of Aarhus (1998)
Main M.G.: Free constructions of powerdomains. In: Melton, A. (eds) Mathematical Foundations of Programming Semantics. Lecture Notes in Computer Science, vol. 239, pp. 162–183. Springer, Heidelberg (1985) ISBN 0-13-507245-X
Martin C.E., Curtis S.A.: Nondeterministic folds. In: Uustalu, T. (eds) MPC. Lecture Notes in Computer Science, vol. 4014, pp. 274–298. Springer, Heidelberg (2006) ISBN 0-13-507245-X
Martin, C.E., Curtis, S.A., Rewitzky, I.: Modelling nondeterminism. In: Kozen and Shankland [23], pp. 228–251
Morris J.M., Bunkenburg A.: Specificational functions. ACM Trans. Programm. Lang. Syst. 21, 677–701 (1999) ISBN 0-13-507245-X
Morris J.M., Bunkenburg A.: A source of inconsistency in theories of nondeterministic functions. Sci. Comput. Program. 43(1), 77–89 (2002) ISBN 0-13-507245-X
Morris J.M., Tyrrell M.: Dual unbounded nondeterminacy, recursion, and fixpoints. Acta Inform. 44(5), 323–344 (2007) ISBN 0-13-507245-X
Morris J.M., Tyrrell M.: Terms with unbounded demonic and angelic nondeterminacy. Sci. Comp. Program. 65(2), 159–172 (2007) ISBN 0-13-507245-X
Morris, J.M., Tyrrell, M.: Dually nondeterministic functions. ACM Trans. Program. Lang. Syst. (2008) (in press)
Morris, J.M.: Augmenting types with unbounded demonic and angelic nondeterminacy. In: Kozen and Shankland [23], pp. 274–288
Naumann, D.A.: Ideal models for pointwise relational and state-free imperative programming. In: Proceedings of the 3rd ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming, pp. 4–15. ACM Press, New York (2001)
Norvell, T.S., Hehner, E.C.R.: Logical specifications for functional programs. In: Proceedings of the Second International Conference on Mathematics of Program Construction. Lecture Notes in Computer Science, vol. 669, pp. 269–290. Springer, Heidelberg (1993)
Partsch H.A.: Specification and Transformation of Programs. Springer, New York (1990) ISBN 0-13-507245-X
Plotkin G.: A Powerdomain Construction. SIAM J. Comput. 5(3), 452–487 (1976) ISBN 0-13-507245-X
Rewitzky I.: Binary multirelations. In: de Swart, H.C.M., Orlowska, E., Schmidt, G., Roubens, M. (eds) Theory and Applications of Relational Structures as Knowledge Instruments. Lecture Notes in Computer Science, vol. 2929, pp. 256–271. Springer, Heidelberg (2003) ISBN 0-13-507245-X
Rewitzky, I., Brink, C.: Monotone predicate transformers as up-closed multirelations. In: Schmidt [41], pp. 311–327
Schmidt, R.A., (ed.) Relations and Kleene algebra in computer science. In: 9th International Conference on Relational Methods in Computer Science and 4th International Workshop on Applications of Kleene Algebra, RelMiCS/AKA 2006, Manchester, UK, August 29–September 2, 2006, Proceedings. Lecture Notes in Computer Science, vol. 4136. Springer, Heidelberg (2006)
Smyth M.B.: Power domains. J. Comp. and Syst. Sci. 16(1), 23–26 (1978) ISBN 0-13-507245-X
Smyth M.B.: Power domains and predicate transformers: A topological view. In: Diaz, J. (eds) Proceedings of the 10th Colloquium on Automata, Languages and Programming, London. Lecture Notes in Computer Science, vol. 153, pp. 662–675. Springer, Heidelberg (1983) ISBN 0-13-507245-X
Solin, K.: On two dually nondeterministic refinement algebras. In: Schmidt [41], pp. 373–387
Tunnicliffe W.R.: The free completely distributive lattice over a poset. Algebra Universalis 21, 133–135 (1985) ISBN 0-13-507245-X
von Wright J.: Towards a refinement algebra. Sci. Comput. Program. 51(1–2), 23–45 (2004) ISBN 0-13-507245-X
Ward, N.: Adding specification constructs to the refinement calculus. In: FME’93: Industrial-Strength Formal Methods. Lecture Notes in Computer Science, vol. 670 (1993)
Ward, N.: A Refinement Calculus for Nondeterministic Expressions. PhD thesis, University of Queensland (1994)
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Morris, J.M., Tyrrell, M. Modelling higher-order dual nondeterminacy. Acta Informatica 45, 441–465 (2008). https://doi.org/10.1007/s00236-008-0076-1
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DOI: https://doi.org/10.1007/s00236-008-0076-1