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An Axiomatic Basis for Computer Programming. Robert Stewart. The most important property of a program is whether it accomplishes the intentions of its user. Current method of verification A better method is using an axiomatic basis for programs. Computer programming as an exact science
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An Axiomatic Basis for Computer Programming Robert Stewart
The most important property of a program is whether it accomplishes the intentions of its user. • Current method of verification • A better method is using an axiomatic basis for programs.
Computer programming as an exact science • Deductive reasoning
Outline • Computer Arithmetic • Program Execution • General Reservations • Proofs of Program Correctness • Formal Language Definition
Computer Arithmetic • The foundation upon which valid reasoning about a program can be developed. • Uses a set of axioms • Examples: x + y = y + x; x + 0 = x; x * y = y * x • These axioms are true for both the infinite set of integers in mathematics and the finite set of integers in computer programming • They work independent of the size of the set
Program Execution • One of the most important properties of a program is if it carries out its intended function. • The intended function can be specified by making assertions about the values of relevant variables after the program is finished. • P{Q}R
Program Execution – Axioms + Rules • Axiom of Assignment • x := f • Rule of Consequence • If P{Q}R and R implies S, then P{Q}S • Rule of Composition • If P{Q1}R1 and R1 {Q2}R, then P{Q1 ; Q2}R • Rule of Iteration
General Reservations • Drawbacks to the axiomatic based approach • The axioms assume the absence of side effects • They do not prove that a program terminates • Areas that present difficulty are labels, jumps, and pointers
Proofs of Program Correctness • The most important property of a program is whether it accomplishes the intentions of its user. • When the correctness of a program, its compiler, and the hardware have been established, one can place great reliance on the results of the program.
Proofs of Program Correctness • Time spent on the current method of verifying correctness of a program is often more than half the time spent on the entire project • The cost of removing errors during this phase is close to two thirds the cost of the project • Removing errors discovered after a program is distributed often costs much more
Proofs of Program Correctness • The practice of proving programs can alleviate other problems in the computing world • Reliability • Documentation • Compatibility • At present, however, program proving is a difficult endeavor, even for skilled programmers.
Formal Language Definition • High level programming languages are usually intended to be implemented on a variety of computers • The problem is defining those languages well enough to ensure compatibility across all computers.
Formal Language Definition • One way to achieve compatibility of a language would be to insist that all implementations of said language satisfy the axioms and rules of inference underlying the properties of programs created with the language.
Formal Language Definition • An advantage to an axiomatic approach is that axioms offer a simple and flexible technique for leaving certain aspects of a language undefined • Range of integers • Accuracy of floating point • Choice of overflow technique • This flexibility for uncertainty is necessary for the language to be able to be implemented on differing hardware designs
Formal Language Definition • An objective of formal language definition is to assist in the design of better programming languages. • The use of axioms can lead to an advantage in that area • A language that can be described by a few “self-evident” axioms is preferable to a language with many obscure axioms
Formal Language Definition • Axioms allow the designer to express his or her intentions, without the mass of detail that comes with algorithmic descriptions.