Experiment with an automatic theorem-prover having partial ordering inference rules

Abstract

Step 2. Place a dot at coordinate X, Y.Step 3. If Y equals the y coordinate of the bottom end of the line, stop. Otherwise, go to Step 4. Step 4. lfD is negative, set D = D + 2 lAX [, and set Y = Y + 1.Go to Step 2. If Dis positive, set D = D + 2 [ AX ] --2 I AY I, setY = Y+ 1, andsetX = X+ l. GotoStep2.By careful initialization of constants it is possible to combine the procedures given here for slopes less than 45 degrees and slopes greater than 45 degrees into a single procedure for all cases. However, the procedures are so simple and compact to start with that the additional code required to combine them is difficult to justify. Experience has shown that a faster and more compact program is obtained if the two cases are treated separately.The precision with which the calculations in the procedures must be carried out depends on the resolution of the particular plotter to be used and the characteristics of the dot-generating algorithm. If there are k dots along the maximum dimension of the rectangular raster, the maximum value of AX or AY for any line in the drawing will he k units. Since the values of the variables in the Bresenham algorithm may range between -t-2 AX or -t-2 AY, the arithmetic of the procedure will encounter signed integers with a maximum range of -I-2k units. The number of bits required to represent these signed integers in 2's complement arithmetic is V-logs 4k-1 bits. The largest plotting area in general use is 1024 × 1024 units, which would require 12 bit calculations. Arithmetic of this magnitude is within the single precision integer arithmetic capability of most popular minicomputers.

A program verification system

“…The subgoal solver is a resolution theorem prover which was developed at NIH by Lewis M. Norton and is similar to the program described in [13]. RES was designed to make efficient deductions in a theory of total ordering.…”

“…Rather than add these axioms, a modification was made to the theorem prover described in [13]. This modification simplifies clauses as they are generated by normalizing arithmetic expressions and relations.…”

“…

A program verification system has been developed consisting of three major components.The verification condition generator (which is patterned after the work of Igarashi, London, and Luckham [lO],is used to generate verification conditions from asserted PASCAL programs.The sub6oal generator (which is similar to the program described in [2])processes the verification conditions to produce simpler subgoals, a significant number of which it proves using restricted techniques.The sub~oal solver is a resolution theorem prover which was developed at NIH by L. Norton [13]. The more powerful techniques of the subgoal solver are used to prove subgoals which the subgoal generator was unable to prove,

The verification system has been used to verify a number of programs.

These include all but one of the examples from King [ii], a Bubble Sort program, and Hoare's FIND program.

…”

“…The sub~oal solver is a resolution theorem prover which was developed at NIH by L. Norton [13]. The more powerful techniques of the subgoal solver are used to prove subgoals which the subgoal generator was unable to prove,…”

Resolution in Solving Graph Problems

“…A first refinement, hyper-resolution, has been introduced by Robinson himself the same year as Resolution [10]. More recently ordered resolution [9,12], (polarized ) resolution modulo (PRM) [5,6], and finally ordered polarized resolution modulo (OPRM) [1] introduced more restrictions. However, as we shall see, these kind of refinements are still redundant.…”