# American Institute of Mathematical Sciences

December  2015, 8(6): 1357-1371. doi: 10.3934/dcdss.2015.8.1357

## Nondeterministic semantics of compound diagrams

 1 Mathematics department, King Saud University, P.O.Box 22452, Riyadh 11495, Saudi Arabia

Received  May 2015 Revised  July 2015 Published  December 2015

We presented a unified description of flow control and single steps of a program is given to obtain flexible definitions of algebraic manipulations. This is achieved by using the notion of relational diagram. We show how the notion of relational diagram, introduced by Schmidt, can be used to give a demonic definition for a wide range of programming constructs. It is shown that the input-output relation of a compound diagram is equal to that of the diagram in which each sub-diagram has been replaced by its input-output relation. This process is repeated until elementary diagrams is obtained.
Citation: Fairouz Tchier. Nondeterministic semantics of compound diagrams. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1357-1371. doi: 10.3934/dcdss.2015.8.1357
##### References:
 [1] C. Aarts, R. Backhouse, P. Hoogendijk, E. Voermans and J. van der Woude, A Relational Theory of Datatypes, Department of Computing Science, Eindhoven University of Technology, 1992. [2] R. J. R. Back, On the correctness of refinement in program development, Thesis, Department of Computer Science, University of Helsinki, 1978. [3] R. J. R. Back, Combining angels, demons and miracles in program specifications, Theoretical Computer Science, 100 (1992), 365-383. doi: 10.1016/0304-3975(92)90309-4. [4] R. J. R. Back, A continuous semantics for unbounded nondeterminism, Theoretical Computer Science, 23 (1983), 187-210. doi: 10.1016/0304-3975(83)90055-5. [5] R. C. Backhouse and H. Doornbos, Mathematical Induction Made Calculational, Computing Science Note 94/16, Dept. of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands, 1994. [6] R. C. Backhouse and J. van der Woude, Demonic operators and monotype factors, Mathematical Structures in Computer Science, 3 (1993), 417-433. Also: Computing Science Note 92/11, Dept. of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands, 1992. doi: 10.1017/S096012950000030X. [7] R. Berghammer, Relational Specification of Data Types and Programs, Technical report 9109, Fakultät für Informatik, Universität der Bundeswehr München, Germany, September 1991. [8] R. Berghammer and G. Schmidt, Relational specifications, in Algebraic Logic, Banach Center Publications (ed. C. Rauszer), 28, Polish Academy of Sciences, 1993, 167-190. [9] R. Berghammer and H. Zierer, Relational algebraic semantics of deterministic and nondeterministic programs, Theoretical Computer Science, 43 (1986), 123-147. doi: 10.1016/0304-3975(86)90172-6. [10] J. Bharat and T. Pallavi, Compositional semantics for diagrams using constrained objects, in International Conference No 2, Callaway Gardens GA, ETATS-UNIS (18/04/2002), Lecture Notes in Computer Science, Vol. 2317, 2002, 94-96. [11] C. Böhm, On a family of Turing machines and the related programming languages, ICC Bull., 3 (1964), 185-194. [12] N. Boudriga, F. Elloumi and A. Mili, On the lattice of specifications: Applications to a specification methodology, Formal Aspects of Computing, 4 (1992), 544-571. [13] C. Brink, W. Kahl and G. Schmidt, eds., Relational Methods in Computer Science, Springer, 1997. doi: 10.1007/978-3-7091-6510-2. [14] L. H. Chin and A. Tarski, Distributive and modular laws in the arithmetic of relation algebras, University of California Publications, 1 (1951), 341-384. [15] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1990. [16] W. P. De Roever, Recursive Program Schemes: Semantics and Proof Theory, Math. Centrum Tracts, Amsterdam, Theses, 1974. [17] J. Desharnais, B. Möller and F. Tchier, Kleene under a modal demonic star, Journal of Logic and Algebraic Programming, 66 (2006), 127-160. doi: 10.1016/j.jlap.2005.04.006. [18] J. Desharnais, B. Möller and F. Tchier, Kleene under a demonic star, in 8th International Conference on Algebraic Methodology And Software Technology (AMAST 2000), May 2000, Iowa City, Iowa, USA, Lecture Notes in Computer Science, 1816, Springer-Verlag, 2000, 355-370. doi: 10.1007/3-540-45499-3_26. [19] J. Desharnais, N. Belkhiter, S. B. M. Sghaier, F. Tchier, A. Jaoua, A. Mili and N. Zaguia, Embedding a demonic semilattice in a relation algebra, Theoretical Computer Science, 149 (1995), 333-360. doi: 10.1016/0304-3975(94)00271-J. [20] J. Desharnais, F. Tchier and R. Khédri, Demonic Relational Semantics of Sequential Programs, Rapport de recherche DIUL-RR-9406, Département d'informatique, Université Laval, Québec, QC, G1K 7P4, Canada, 1994. [21] J. Desharnais, A. Jaoua, F. Mili, N. Boudriga and A. Mili, A Relational division operator: The conjugate kernel, Theoretical Comput. Sci., 114 (1993), 247-272. doi: 10.1016/0304-3975(93)90074-4. [22] J. Desharnais, Abstract Relational Semantics, Ph D. thesis, School of Computer Science, Univ. McGill, Montréal, 1989. [23] E. W. Dijkstra, A Discipline of Programming, Prentice-Hall, 1976. [24] H. Doornbos, Reductivity, Science of Computer Programming, 26 (1996), 217-236. doi: 10.1016/0167-6423(95)00027-5. [25] H. Doornbos, A relational model of programs without the restriction to Egli-Milner monotone constructs, in IFIP Transactions, A-56: North-Holland, 1994, 363-382. [26] H. Doornbos, R. Backhouse and J. van der Woude, A calculational approach to mathematical induction, Theoretical Computer Science, 179 (1997), 103-135. doi: 10.1016/S0304-3975(96)00154-5. [27] R. W. Floyd, Assigning meanings to programs, Proceedings AMS Symposium in Applied Mathematics, 19 (1967), 19-32. [28] M. Frappier, A Relational Basis for Program Construction by Parts, Dept. of Computer Science, University of Ottawa, 1994. [29] C. Gunter, Semantics of Programming Languages, MIT Press, 1992. [30] H. Riis Nielson and F. Nielson, Semantics with Applications: An Appetizer, Undergraduate Topics in Computer Science, Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2007. [31] C. A. R. Hoare and J. He, The weakest prespecification, Part I, Fundamenta Informaticae IX, 9 (1986), 51-84. [32] C. A. R. Hoare and J. He, The weakest prespecification, Part II, Fundamenta Informaticae IX, 9 (1986), 217-251. [33] C. A. R. Hoare, et al., Laws of programming, Communications of the ACM, 30 (1987), 672-686. doi: 10.1145/27651.27653. [34] Y. I. Ianov, On the equivalence and transformation of program schemes, Dokl. Akad. Nauk, 1 (1958), 8-12. doi: 10.1145/368924.368930. [35] J. S. Reich and J. L. Jacob, Relational Denotational Semantics of the While Language, Department of Computer Science, University of York, UK. [36] R. D. Maddux, Relation-algebraic semantics, Theoretical Computer Science, 160 (1996), 1-85. doi: 10.1016/0304-3975(95)00082-8. [37] A. Mili, A relational approach to the design of deterministic programs, Acta Informatica, 20 (1983), 315-328. doi: 10.1007/BF00264277. [38] A. Mili, J. Desharnais and F. Mili, Relational heuristics for the design of deterministic programs, Acta Informatica, 24 (1987), 239-276. doi: 10.1007/BF00265990. [39] D. L. Parnas, A generalized control structure and its formal definition, Communications of the ACM, 26 (1983), 572-581. doi: 10.1145/358161.358168. [40] J. Riguet, Programmation et théorie des catégories, in Proc. ICC Symp. Symbolic Languages in Data Processing, Gordon and Breach, New York, 1962, 83-98. [41] G. Schmidt, Programs as partial graphs I: Flow equivalence and correctness, Theoretical Computer Science, 15 (1981), 1-25. doi: 10.1016/0304-3975(81)90060-8. [42] G. Schmidt and T. Ströhlein, Relations and Graphs, EATCS Monographs in Computer Science, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-77968-8. [43] E. Sekerinski, A calculus for predicative programming, in Second International Conf. on the Mathematics of Program Construction, Oxford, June 1992 (eds. R. S. Bird, C. C. Morgan and J. C. P. Woodcock), Lecture Notes in Computer Science, 669, Springer-Verlag, 1993, 302-322. doi: 10.1007/3-540-56625-2_20. [44] A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics, 5 (1955), 285-309. doi: 10.2140/pjm.1955.5.285. [45] A. Tarski, On the calculus of relations, J. Symbolic Logic, 6 (1941), 73-89. doi: 10.2307/2268577. [46] F. Tchier and J. Desharnais, A generalisation of a theorem of Mills, in Proceedings of the Tenth International Symposium on Computer and Information Sciences, ISCIS X, Turkey, October 1995, 27-34. [47] F. Tchier, Sémantiques Relationnelles Démoniaques et Vérification de Boucles Non Déterministes, Ph. D. thesis, Département de mathématiques et de statistique, Université Laval, Canada, 1996. Available from: http://www2.ift.ulaval.ca/~desharnais/Recherche/Theses/these.Fairouz.pdf. [48] F. Tchier and J. Desharnais, Applying a generalization of a theorem of Mills to generalized looping structures, in Science and Engineering in Software Development. A recognition of Harlan D. Mills' Legacy, Los Angeles, CA, May 1999, IEEE Computer Society Press, 1999, 31-38. doi: 10.1109/SESD.1999.781109. [49] F. Tchier, La sémantique démoniaque relationnelle des diagrammes composés, in Proc. 5th Seminar on Relational Methods in computer Science (RelMICS'5), Université Laval, Canada, January 2000. [50] F. Tchier, While loop demonic relational semantics monotype/residual style, 2003 International Conference on Software Engineering Research and Practice (SERP'03), Las Vegas, Nevada, USA, June 2003. [51] F. Tchier, Demonic Semantics: Using monotypes and residuals, International Journal of Mathematics and Mathematical Sciences, IJMMS, 3 (2004), 135-160. doi: 10.1155/S016117120420415X. [52] F. Tchier, Nondeterministic programming theorem, WSEAS Trans, On Mathematics, 5 (2006), 1035-1044. [53] F. Tchier, From Operational to Denotational Demonic Semantics of Nondeterministic While Loops, 10th WSEAS International Conference on Communications and Computers, Athens, Greece, 2006. [54] F. Tchier, Demonic fixed points, Acta Cybernitica Journal, 17 (2006), 533-555. [55] F. Tchier, Demonic semantics are equal, in The 2008 International Conference on Foundations of Computer Science (FCS'08), Monte Carlo Resort, Las Vegas, Nevada, USA, July, 2008. [56] M. Walicki and S. Medal, Algebraic approches to nondeterminism: An overview, ACM Computing Surveys, 29 (1997), 30-81. [57] N. T. E. Ward, A refinement Calculus for Nondeterministic Expressions, Ph.D thesis, University of Queensland, Australia, 1994. [58] L. Xu, M. Takeichi and H. Iwasaki, Relational semantics for locally nondeterministic programs, New Generation Computing, 15 (1997), 339-362.

show all references

##### References:
 [1] C. Aarts, R. Backhouse, P. Hoogendijk, E. Voermans and J. van der Woude, A Relational Theory of Datatypes, Department of Computing Science, Eindhoven University of Technology, 1992. [2] R. J. R. Back, On the correctness of refinement in program development, Thesis, Department of Computer Science, University of Helsinki, 1978. [3] R. J. R. Back, Combining angels, demons and miracles in program specifications, Theoretical Computer Science, 100 (1992), 365-383. doi: 10.1016/0304-3975(92)90309-4. [4] R. J. R. Back, A continuous semantics for unbounded nondeterminism, Theoretical Computer Science, 23 (1983), 187-210. doi: 10.1016/0304-3975(83)90055-5. [5] R. C. Backhouse and H. Doornbos, Mathematical Induction Made Calculational, Computing Science Note 94/16, Dept. of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands, 1994. [6] R. C. Backhouse and J. van der Woude, Demonic operators and monotype factors, Mathematical Structures in Computer Science, 3 (1993), 417-433. Also: Computing Science Note 92/11, Dept. of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands, 1992. doi: 10.1017/S096012950000030X. [7] R. Berghammer, Relational Specification of Data Types and Programs, Technical report 9109, Fakultät für Informatik, Universität der Bundeswehr München, Germany, September 1991. [8] R. Berghammer and G. Schmidt, Relational specifications, in Algebraic Logic, Banach Center Publications (ed. C. Rauszer), 28, Polish Academy of Sciences, 1993, 167-190. [9] R. Berghammer and H. Zierer, Relational algebraic semantics of deterministic and nondeterministic programs, Theoretical Computer Science, 43 (1986), 123-147. doi: 10.1016/0304-3975(86)90172-6. [10] J. Bharat and T. Pallavi, Compositional semantics for diagrams using constrained objects, in International Conference No 2, Callaway Gardens GA, ETATS-UNIS (18/04/2002), Lecture Notes in Computer Science, Vol. 2317, 2002, 94-96. [11] C. Böhm, On a family of Turing machines and the related programming languages, ICC Bull., 3 (1964), 185-194. [12] N. Boudriga, F. Elloumi and A. Mili, On the lattice of specifications: Applications to a specification methodology, Formal Aspects of Computing, 4 (1992), 544-571. [13] C. Brink, W. Kahl and G. Schmidt, eds., Relational Methods in Computer Science, Springer, 1997. doi: 10.1007/978-3-7091-6510-2. [14] L. H. Chin and A. Tarski, Distributive and modular laws in the arithmetic of relation algebras, University of California Publications, 1 (1951), 341-384. [15] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1990. [16] W. P. De Roever, Recursive Program Schemes: Semantics and Proof Theory, Math. Centrum Tracts, Amsterdam, Theses, 1974. [17] J. Desharnais, B. Möller and F. Tchier, Kleene under a modal demonic star, Journal of Logic and Algebraic Programming, 66 (2006), 127-160. doi: 10.1016/j.jlap.2005.04.006. [18] J. Desharnais, B. Möller and F. Tchier, Kleene under a demonic star, in 8th International Conference on Algebraic Methodology And Software Technology (AMAST 2000), May 2000, Iowa City, Iowa, USA, Lecture Notes in Computer Science, 1816, Springer-Verlag, 2000, 355-370. doi: 10.1007/3-540-45499-3_26. [19] J. Desharnais, N. Belkhiter, S. B. M. Sghaier, F. Tchier, A. Jaoua, A. Mili and N. Zaguia, Embedding a demonic semilattice in a relation algebra, Theoretical Computer Science, 149 (1995), 333-360. doi: 10.1016/0304-3975(94)00271-J. [20] J. Desharnais, F. Tchier and R. Khédri, Demonic Relational Semantics of Sequential Programs, Rapport de recherche DIUL-RR-9406, Département d'informatique, Université Laval, Québec, QC, G1K 7P4, Canada, 1994. [21] J. Desharnais, A. Jaoua, F. Mili, N. Boudriga and A. Mili, A Relational division operator: The conjugate kernel, Theoretical Comput. Sci., 114 (1993), 247-272. doi: 10.1016/0304-3975(93)90074-4. [22] J. Desharnais, Abstract Relational Semantics, Ph D. thesis, School of Computer Science, Univ. McGill, Montréal, 1989. [23] E. W. Dijkstra, A Discipline of Programming, Prentice-Hall, 1976. [24] H. Doornbos, Reductivity, Science of Computer Programming, 26 (1996), 217-236. doi: 10.1016/0167-6423(95)00027-5. [25] H. Doornbos, A relational model of programs without the restriction to Egli-Milner monotone constructs, in IFIP Transactions, A-56: North-Holland, 1994, 363-382. [26] H. Doornbos, R. Backhouse and J. van der Woude, A calculational approach to mathematical induction, Theoretical Computer Science, 179 (1997), 103-135. doi: 10.1016/S0304-3975(96)00154-5. [27] R. W. Floyd, Assigning meanings to programs, Proceedings AMS Symposium in Applied Mathematics, 19 (1967), 19-32. [28] M. Frappier, A Relational Basis for Program Construction by Parts, Dept. of Computer Science, University of Ottawa, 1994. [29] C. Gunter, Semantics of Programming Languages, MIT Press, 1992. [30] H. Riis Nielson and F. Nielson, Semantics with Applications: An Appetizer, Undergraduate Topics in Computer Science, Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2007. [31] C. A. R. Hoare and J. He, The weakest prespecification, Part I, Fundamenta Informaticae IX, 9 (1986), 51-84. [32] C. A. R. Hoare and J. He, The weakest prespecification, Part II, Fundamenta Informaticae IX, 9 (1986), 217-251. [33] C. A. R. Hoare, et al., Laws of programming, Communications of the ACM, 30 (1987), 672-686. doi: 10.1145/27651.27653. [34] Y. I. Ianov, On the equivalence and transformation of program schemes, Dokl. Akad. Nauk, 1 (1958), 8-12. doi: 10.1145/368924.368930. [35] J. S. Reich and J. L. Jacob, Relational Denotational Semantics of the While Language, Department of Computer Science, University of York, UK. [36] R. D. Maddux, Relation-algebraic semantics, Theoretical Computer Science, 160 (1996), 1-85. doi: 10.1016/0304-3975(95)00082-8. [37] A. Mili, A relational approach to the design of deterministic programs, Acta Informatica, 20 (1983), 315-328. doi: 10.1007/BF00264277. [38] A. Mili, J. Desharnais and F. Mili, Relational heuristics for the design of deterministic programs, Acta Informatica, 24 (1987), 239-276. doi: 10.1007/BF00265990. [39] D. L. Parnas, A generalized control structure and its formal definition, Communications of the ACM, 26 (1983), 572-581. doi: 10.1145/358161.358168. [40] J. Riguet, Programmation et théorie des catégories, in Proc. ICC Symp. Symbolic Languages in Data Processing, Gordon and Breach, New York, 1962, 83-98. [41] G. Schmidt, Programs as partial graphs I: Flow equivalence and correctness, Theoretical Computer Science, 15 (1981), 1-25. doi: 10.1016/0304-3975(81)90060-8. [42] G. Schmidt and T. Ströhlein, Relations and Graphs, EATCS Monographs in Computer Science, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-77968-8. [43] E. Sekerinski, A calculus for predicative programming, in Second International Conf. on the Mathematics of Program Construction, Oxford, June 1992 (eds. R. S. Bird, C. C. Morgan and J. C. P. Woodcock), Lecture Notes in Computer Science, 669, Springer-Verlag, 1993, 302-322. doi: 10.1007/3-540-56625-2_20. [44] A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics, 5 (1955), 285-309. doi: 10.2140/pjm.1955.5.285. [45] A. Tarski, On the calculus of relations, J. Symbolic Logic, 6 (1941), 73-89. doi: 10.2307/2268577. [46] F. Tchier and J. Desharnais, A generalisation of a theorem of Mills, in Proceedings of the Tenth International Symposium on Computer and Information Sciences, ISCIS X, Turkey, October 1995, 27-34. [47] F. Tchier, Sémantiques Relationnelles Démoniaques et Vérification de Boucles Non Déterministes, Ph. D. thesis, Département de mathématiques et de statistique, Université Laval, Canada, 1996. Available from: http://www2.ift.ulaval.ca/~desharnais/Recherche/Theses/these.Fairouz.pdf. [48] F. Tchier and J. Desharnais, Applying a generalization of a theorem of Mills to generalized looping structures, in Science and Engineering in Software Development. A recognition of Harlan D. Mills' Legacy, Los Angeles, CA, May 1999, IEEE Computer Society Press, 1999, 31-38. doi: 10.1109/SESD.1999.781109. [49] F. Tchier, La sémantique démoniaque relationnelle des diagrammes composés, in Proc. 5th Seminar on Relational Methods in computer Science (RelMICS'5), Université Laval, Canada, January 2000. [50] F. Tchier, While loop demonic relational semantics monotype/residual style, 2003 International Conference on Software Engineering Research and Practice (SERP'03), Las Vegas, Nevada, USA, June 2003. [51] F. Tchier, Demonic Semantics: Using monotypes and residuals, International Journal of Mathematics and Mathematical Sciences, IJMMS, 3 (2004), 135-160. doi: 10.1155/S016117120420415X. [52] F. Tchier, Nondeterministic programming theorem, WSEAS Trans, On Mathematics, 5 (2006), 1035-1044. [53] F. Tchier, From Operational to Denotational Demonic Semantics of Nondeterministic While Loops, 10th WSEAS International Conference on Communications and Computers, Athens, Greece, 2006. [54] F. Tchier, Demonic fixed points, Acta Cybernitica Journal, 17 (2006), 533-555. [55] F. Tchier, Demonic semantics are equal, in The 2008 International Conference on Foundations of Computer Science (FCS'08), Monte Carlo Resort, Las Vegas, Nevada, USA, July, 2008. [56] M. Walicki and S. Medal, Algebraic approches to nondeterminism: An overview, ACM Computing Surveys, 29 (1997), 30-81. [57] N. T. E. Ward, A refinement Calculus for Nondeterministic Expressions, Ph.D thesis, University of Queensland, Australia, 1994. [58] L. Xu, M. Takeichi and H. Iwasaki, Relational semantics for locally nondeterministic programs, New Generation Computing, 15 (1997), 339-362.
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