Kleene under a Demonic Star

In relational semantics, the input-output semantics of a program is a relation on its set of states. We generalize this in considering elements of Kleene algebras as semantical values. In a nondeterministic context, the demonic semantics is calculated by considering the worst behavior of the program. In this paper, we concentrate on while loops. Calculating the semantics of a loop is diffcult, but showing the correctness of any candidate abstraction is much easier. For deterministic programs, Mills has described a checking method known as the while statement verification rule. A corresponding programming theorem for nondeterministic iterative constructs is proposed, proved and applied to an example. This theorem can be considered as a generalization of the while statement verification rule to nondeterministic loops.

[1]  Roland Carl Backhouse,et al.  Demonic operators and monotype factors , 1993, Mathematical Structures in Computer Science.

[2]  Roland Carl Backhouse,et al.  Fixed-Point Calculus , 1995, Inf. Process. Lett..

[3]  Gunther Schmidt,et al.  Relations and Graphs: Discrete Mathematics for Computer Scientists , 1993 .

[4]  Theodore S. Norvell.  Predicative semantics of loops , 1997, Algorithmic Languages and Calculi.

[5]  J. Conway.  Regular algebra and finite machines , 1971 .

[6]  Roland Carl Backhouse.  The calculational method , 1995 .

[7]  Jerzy Tiuryn,et al.  Dynamic logic , 2001, SIGA.

[8]  Gunther Schmidt,et al.  Relations and Graphs , 1993, EATCS Monographs on Theoretical Computer Science.

[9]  Jules Desharnais,et al.  Embedding a demonic semilattice in a relation algebra , 1995 .

[10]  Rudolf Berghammer,et al.  Relational Algebraic Semantics of Deterministic and Nondeterministic Programs , 1986, Theor. Comput. Sci..

[11]  Ralph-Johan Back,et al.  Refinement Calculus: A Systematic Introduction , 1998 .

[12]  C. j. Aarts,et al.  Galois Connections Presented Calculationally , 1992 .

[13]  Harlan D. Mills,et al.  Princples of Computer Programming , 1987 .

[14]  Emil Sekerinski.  A Calculus for Predicative Programming , 1992, MPC.

[15]  Harlan D. Mills,et al.  Principles of computer programming : a mathematical approach , 1987 .

[16]  Dexter Kozen,et al.  Kleene algebra with tests , 1997, TOPL.

[17]  Harlan D. Mills,et al.  Trace: Tennessee Research and CreativeExchange , 2022 .

[18]  Wolfram Kahl,et al.  Relational Methods in Computer Science , 1997, Advances in Computing Sciences.

[19]  Kern W. Dickman.  Principles of computer programming , 1964 .

[20]  Harlan D. Mills,et al.  Teaching principles of computer programming , 1987, CSC '87.

[21]  Eric C. R. Hehner,et al.  Predicative programming Part I , 1984, CACM.

[22]  Jules Desharnais,et al.  Characterizing Functions in Kleene Algebras , 2000, RelMiCS.

[23]  J. Desharnais,et al.  Applying a generalization of a theorem of Mills to generalized looping structures , 1999, Proceedings. Science and Engineering for Software Development: A Recognition of Harlin D. Mills Legacy (Cat. No. PR00010).