Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-19T20:22:51.869Z Has data issue: false hasContentIssue false
  • English
  • Français

Rewriting on cyclic structures: Equivalence between the operational and the categorical description

Published online by Cambridge University Press:  15 August 2002

Andrea Corradini  and
Fabio Gadducci
Andrea Corradini
Affiliation:
Dipartimento di Informatica, University of Pisa, Corso Italia 40, 56125 Pisa, Italy; andrea@di.unipi.it.
Fabio Gadducci
Affiliation:
Division of Informatics, University of Edinburgh, Mayfield Road, EH9 3JZ Edinburgh, U.K.; fabio@dcs.ed.ac.uk.
Get access

Abstract

We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted operational definition proposed by Barendregt et al. – but for the case of circular redexes , for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms.

Keywords

Term graphsdirected acyclic graphsterm graph rewritingcategorical modelstraced monoidal categories2-categoriesalgebraic theoriesgs-monoidal theories.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 33 , Issue 4-5 , July 1999 , pp. 467 - 493
Copyright
© EDP Sciences, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ariola, Z.M. and Klop, J.W., Equational term graph rewriting. Fund. Inform. 26 (1996) 207-240.
Z.M. Ariola, J.W. Klop and D. Plump, Confluent rewriting of bisimilar term graphs, C. Palamidessi and J. Parrow, Eds., Expressiveness in Concurrency. Elsevier Sciences, Electron. Notes Theoret. Comput. Sci. 7 (1997). Available at http://www.elsevier.nl/locate/entcs/volume7.html/.
Barendregt, H.P., van Eekelen, M.C.J.D., Glauert, J.R.W., Kennaway, J.R., Plasmeijer, M.J. and Sleep, M.R., Term graph reduction, J.W. de Bakker, A.J. Nijman and P.C. Treleaven, Eds., Parallel Architectures and Languages Europe. Springer Verlag, Lecture Notes in Comput. Sci. 259 (1987) 141-158. CrossRef
Bernátsky, L. and Ésik, Z., Semantics of flowchart programs and the free Conway theories. Theoret. Informatics Appl. 32 (1998) 35-78. CrossRef
Berry, G. and Lévy, J.-J., Minimal and optimal computations of recursive programs. J. ACM 26 (1979) 148-175. CrossRef
Bloom, S. and Ésik, Z., Axiomatizing schemes and their behaviors. J. Comput. System Sci. 31 (1985) 375-393. CrossRef
S. Bloom and Z. Ésik, Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer Verlag (1993).
Bloom, S. and Ésik, Z., The equational logic of fixed points. Theoret. Comput. Sci. 179 (1997) 1-60. CrossRef
Bloom, S.L., Ésik, Z., Labella, A. and Manes, E.G., Iteration 2-theories, M. Johnson, Ed., Algebraic Methodology and Software Technology. Springer Verlag, Lecture Notes in Comput. Sci. 1349 (1997) 30-44. CrossRef
G. Boudol, Computational semantics of term rewriting systems, M. Nivat and J. Reynolds, Eds., Algebraic Methods in Semantics. Cambridge University Press (1985) 170-235.
Corradini, A., Term rewriting in CT Σ, M.-C. Gaudel and J.-P. Jouannaud, Eds., Trees in Algebra and Programming. Springer Verlag, Lecture Notes in Comput. Sci. 668 (1993) 468-484. CrossRef
A. Corradini and F. Drewes, (Cyclic) term graph rewriting is adequate for rational parallel term rewriting, Technical Report TR-97-14. Dipartimento di Informatica, Pisa (1997).
Corradini, A. and Gadducci, F., A 2-categorical presentation of term graph rewriting, E. Moggi and G. Rosolini, Eds., Category Theory and Computer Science. Springer Verlag, Lecture Notes in Comput. Sci. 1290 (1997) 87-105. CrossRef
Corradini, A. and Gadducci, F., Rational term rewriting, M. Nivat, Ed., Foundations of Software Science and Computation Structures. Springer Verlag, Lecture Notes in Comput. Sci. 1378 (1998) 156-171. CrossRef
A. Corradini and F. Gadducci, An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, to appear.
A. Corradini, U. Montanari, F. Rossi, H. Ehrig, R. Heckel and M. Löwe, Algebraic approaches to graph transformation I: Basic concepts and double pushout approach, G. Rozenberg, Ed., Handbook of Graph Grammars and Computing by Graph Transformation 1. World Scientific (1997).
Corradini, A. and Rossi, F., Hyperedge replacement jungle rewriting for term rewriting systems and logic programming. Theoret. Comput. Sci. 109 (1993) 7-48. CrossRef
Gh. Stefanescu, On flowchart theories: Part II. The nondeterministic case. Theoret. Comput. Sci. 52 (1987) 307-340. CrossRef
Gh. Stefanescu, Algebra of flownomials. Technical Report SFB-Bericht 342/16/94 A. Technical University of München, Institut für Informatik (1994).
Cazanescu, V.-E. and Stefanescu, Gh., Towards a new algebraic foundation of flowchart scheme theory. Fund. Inform. 13 (1990) 171-210.
Cazanescu, V.-E. and Stefanescu, Gh., A general result on abstract flowchart schemes with applications to the study of accessibility, reduction and minimization. Theoret. Comput. Sci. 99 (1992) 1-63. CrossRef
V.-E. Cazanescu and Gh. Stefanescu, Feedback, iteration and repetition, Gh. Paun, Ed., Mathematical Aspects of Natural and Formal Languages. World Scientific (1995) 43-62.
Elgot, C.C., Monadic computations and iterative algebraic theories, H.E. Rose and J.C. Shepherdson, Eds., Logic Colloquium 1973. North Holland, Stud. Logic Found. Math. 80 (1975) 175-230. CrossRef
Elgot, C.C., Structured programming with and without GO TO statements. IEEE Trans. Software Engrg. 2 (1976) 41-54. CrossRef
Elgot, C.C., Bloom, C.C. and Tindell, R., The algebraic structure of rooted trees. J. Comput. System Sci. 16 (1978) 362-339. CrossRef
Ésik, Z., Identities in iterative theories and rational algebraic theories. Computational Linguistics and Computer Languages 14 (1980) 183-207.
Ésik, Z., Group axioms for iteration. Inform. and Comput. 148 (1999) 131-180. CrossRef
Ésik, Z. and Labella, A., Equational properties of iteration in algebraically complete categories. Theoret. Comput. Sci. 195 (1998) 61-89. CrossRef
Farmer, W.M., Ramsdell, J.D. and Watro, R.J., A correctness proof for combinator reduction with cycles. ACM Trans. Program. Lang. Syst. 12 (1990) 123-134. CrossRef
Farmer, W.M. and Watro, R.J., Redex capturing in term graph rewriting, R.V. Book, Ed., Rewriting Techniques and Applications. Springer Verlag, Lecture Notes in Comput. Sci. 488 (1991) 13-24. CrossRef
Ginali, S., Regular trees and the free iterative theory. J. Comput. System Sci. 18 (1979) 222-242. CrossRef
Goguen, J.A., Tatcher, J.W., Wagner, E.G. and Wright, J.R, Initial algebra semantics and continuous algebras. J. ACM 24 (1977) 68-95. CrossRef
I. Guessarian, Algebraic Semantics. Springer Verlag, Lecture Notes in Comput. Sci. 99 (1981).
M. Hasegawa, Models of Sharing Graphs. Ph.D. Thesis, University of Edinburgh, Department of Computer Science (1997).
Hasegawa, M., Recursion from cyclic sharing: Traced monoidal categories and models of cyclic lambda-calculus, Ph. de Groote and R. Hindly, Eds., Typed Lambda Calculi and Applications. Springer Verlag, Lecture Notes in Comput. Sci. 1210 (1997) 196-213. CrossRef
Hoffmann, B. and Plump, D., Implementing term rewriting by jungle evaluation. Theoret. Informatics Appl. 25 (1991) 445-472. CrossRef
Joyal, A., Street, R.H. and Verity, D., Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119 (1996) 425-446. CrossRef
G. Kelly, Basic Concepts of Enriched Category Theory. Cambridge University Press (1982).
Kelly, G.M. and Street, R.H., Review of the elements of 2-categories, G.M. Kelly, Ed., Sydney Category Seminar. Springer Verlag, Lecture Notes in Math. 420 (1974) 75-103. CrossRef
Kennaway, J.R., On ``On Graph Rewritings". Theoret. Comput. Sci. 52 (1980) 37-58. CrossRef
Kennaway, J.R., Klop, J.W., Sleep, M.R. and de Vries, F.J., On the adequacy of graph rewriting for simulating term rewriting. ACM Trans. Program. Lang. Syst. 16 (1994) 493-523. CrossRef
Klop, J.W., Term rewriting systems, S. Abramsky, D. Gabbay and T. Maibaum, Eds. Oxford University Press, Handb. Log. Comput. Sci. 1 (1992) 1-116.
Martí-Oliet, N. and Meseguer, J., From Petri nets to linear logic through categories: A survey. Intrenat. J. Foundations Comput. Sci. 4 (1991) 297-399. CrossRef
Meseguer, J., Conditional rewriting logic as a unified model of concurrency. Theoret. Comput. Sci. 96 (1992) 73-155. CrossRef
Meseguer, J. and Montanari, U., Petri nets are monoids. Inform. and Comput. 88 (1990) 105-155. CrossRef
H. Miyoshi, Rewriting logic for cyclic sharing structures, T. Sato and A. Middeldorp, Eds., Fuji International Symposium on Functional and Logic Programming. World Scientific (1998) 167-186.
Montanari, U. and Rossi, F., Contextual nets. Acta Inform. 32 (1995) 545-596. CrossRef
M.J. Plasmeijer and M.C.J.D. van Eekelen, Functional Programming and Parallel Graph Rewriting. Addison Wesley (1993).
Power, A.J., An abstract formulation for rewrite systems, D.H. Pitt, D.E. Rydehard, P. Dybjer, A.M. Pitts and A. Poigné, Eds., Category Theory and Computer Science. Springer Verlag, Lecture Notes in Comput. Sci. 389 (1989) 300-312. CrossRef
W. Reisig, Petri Nets: An Introduction. EACTS Monographs on Theoretical Computer Science. Springer Verlag (1985).
Rydehard, D.E. and Stell, E.G., Foundations of equational deductions: A categorical treatment of equational proofs and unification algorithms, D.H. Pitt, A. Poigné and D.E. Rydehard, Eds., Category Theory and Computer Science. Springer Verlag, Lecture Notes in Comput. Sci. 283 (1987) 114-139. CrossRef
Scott, D., The lattice of flow diagrams, E. Engeler, Ed., Semantics of Algorithmic Languages, Springer Verlag, Lecture Notes in Math. 182 (1971) 311-366. CrossRef
M.R. Sleep, M.J. Plasmeijer and M.C. van Eekelen, Term Graph Rewriting: Theory and Practice. Wiley (1993).
Staples, J., Computation of graph-like expressions. Theoret. Comput. Sci. 10 (1980) 171-195. CrossRef
Staples, J., Optimal evaluation of graph-like expressions. Theoret. Comput. Sci. 10 (1980) 297-316. CrossRef
Staples, J., Speeding up subtree replacement systems. Theoret. Comput. Sci. 11 (1980) 39-47. CrossRef
R.H. Street, Categorical structures, M. Hazewinkel, Ed., Handbook of Algebra. Elsevier (1996) 529-577.
Turner, D.A., A new implementation technique for applicative languages. Software: Practice and Experience 9 (1979) 31-49.
M. Wand, A concrete approach to abstract recursive definitions, M. Nivat, Ed., Automata, Languages and Programming. North Holland (1973) 331-341.