The Omega testi s ani nteger programmi ng algori thm that can determi ne whether a dependence exi sts between two array references, and i so, under what condi7: ns. Conventi nalwi[A m holds thati nteger programmiB techni:36 are far too expensi e to be used for dependence analysi6 except as a method

In studying the loe-.,..ical foundations of ' Canputer Science., it '15 inporLnnt to lmOd how powerful a fannal system 'We need:to derlve 'the important facts about corrputing. In the past) all interesting concrete theolY~rns about ccmputing could in principle be cast as theorems cibout 'PurJng machines) encoded by GMel's techniques as statements "about-the _inteL~t!r3) and proved using standard Peano Arithmetic (finite argtmJerits plus induction). By Omel's famous Incorrpleteness result, there "are true Gtatel1Ents about numbers which cannot be proved 'in Peano Ari:thm.::!tic. Hm'tnunls and Hopcroft [HI-! 76] pointed out that 50JTE such statements have tile form of tennination and complexity properties of ' prograxns., but the prop;rarro involved were cantriV-=d to make a point, and-not iJit:r1.nsi-cal~y interesting. 'Jhis paper establishes that the tenninatlon of "a natural 'and.inten~stinp; class of programs is independent of ' (Le., eannot 'be Proved 1n) Peano Arithm:!tie. This termination property and the variation on B;)Jn:~p..y·s theorem recently discovered by Paris [Pa 77J are the only-lmown ca~-;C'~; or inherentl..v interesting results, outside of lagle, whiCh are cxpn~:,::;lble but not provable in Peano Arithmetic. I'ror:;rams in Algol-like languages usually fail to halt because or jnfJldtc while loops or infinite recursions. Languages which allow functions or procedures _to be passed as parameters may provide stibtler Conm] of nontennim·.tion, such as the following example of ' a program with a function f which tal{es as argument a f'tmction. i

I wish to ~hank ~t. ~, ~. Glass, who supervised this research, for his encouragement and ~nterest in this work. I also wish to thank Dr. G. N. Patterson, Director of the Institute, for the opportunity to work ~nd study at urIAS, I wauld especially lik ~ to thank Mr. P. Crouse for his continuous and dedicated technical as~istance tQroughout ~he investigation. Special thanks are due to Mr. B. Whitten for his help durlng the course of the research. The advice reç~ived from Dr. M. P, E. B~istow and the a~sistance of my colleagues, Messrs. D. Appel, J. J. Go~tlieb and R. L. Evans are very much appreciated. My thanks go also tp Mrs. G. Gottlieb for many long hours of typing, and to Mr~. M. Stewaft for ~he f~st printing which enabled the work to be completed on time.

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