The Construction and Computation of Conditional Statements for SCMPDS

Jing-Chao Chen

Shanghai Jiaotong University

Summary.

We construct conditional statements like the usual high level program language by program blocks of SCMPDS. Roughly speaking, the article justifies such a fact that when the condition of a conditional statement is true (false), and the true (false) branch is shiftable, parahalting and does not contain any halting instruction, and the false branch is shiftable, then it is halting and its computation result equals that of the true (false) branch. The parahalting means some program halts for all states, this is strong condition. For this reason, we introduce the notions of "is\_closed\_on" and "is\_halting\_on". The predicate "A is\_closed\_on B" denotes program A is closed on state B, and "A is\_halting\_on B" denotes program A is halting on state B. We obtain a similar theorem to the above fact by replacing parahalting by "is\_closed\_on" and "is\_halting\_on".

This research is partially supported by the National Natural Science Foundation of China Grant No. 69873033.

MML Identifier: SCMPDS_6

Contents (PDF format)

1. Preliminaries
2. The Predicates of is\_closed\_on and is\_halting\_on
3. The Construction of Conditional Statements
4. The Computation of ``if var=0 then block1 else block2''
5. The Computation of ``if var=0 then block''
6. The Computation of ``if var<>0 then block''
7. The Computation of ``if var>0 then block1 else block2''
8. The Computation of ``if var>0 then block''
9. The Computation of ``if var<=0 then block''
10. The Computation of ``if var<0 then block1 else block2''
11. The Computation of ``if var<0 then block''
12. The Computation of ``if var>=0 then block''

Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.

[3] Grzegorz Bancerek and Piotr Rudnicki. Development of terminology for \bf scm. Journal of Formalized Mathematics, 5, 1993.

[4] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Journal of Formalized Mathematics, 8, 1996.

[5] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[6] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.

[7] Jing-Chao Chen. Computation and program shift in the SCMPDS computer. Journal of Formalized Mathematics, 11, 1999.

[8] Jing-Chao Chen. Computation of two consecutive program blocks for SCMPDS. Journal of Formalized Mathematics, 11, 1999.

[9] Jing-Chao Chen. The construction and shiftability of program blocks for SCMPDS. Journal of Formalized Mathematics, 11, 1999.

[10] Jing-Chao Chen. The SCMPDS computer and the basic semantics of its instructions. Journal of Formalized Mathematics, 11, 1999.

[11] Yatsuka Nakamura and Andrzej Trybulec. A mathematical model of CPU. Journal of Formalized Mathematics, 4, 1992.

[12] Yatsuka Nakamura and Andrzej Trybulec. On a mathematical model of programs. Journal of Formalized Mathematics, 4, 1992.

[13] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989.

[14] Yasushi Tanaka. On the decomposition of the states of SCM. Journal of Formalized Mathematics, 5, 1993.

[15] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[16] Andrzej Trybulec and Yatsuka Nakamura. Some remarks on the simple concrete model of computer. Journal of Formalized Mathematics, 5, 1993.

[17] Michal J. Trybulec. Integers. Journal of Formalized Mathematics, 2, 1990.

[18] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.