Some Things Algorithms Cannot Do

A new, ‘behavioral’ theory of algorithms, intending to capture algorithms at their intended abstraction level, has been developed in this century in a series of papers by Y.Gurevich, A.Blass and others, motivated initially by the goal of establishing the ASM thesis. A viable theory of algorithms must have its limitative results, algorithms, however abstract, cannot do just anything. We establish some nonclassical limitative results for the behavioral theory: – algorithms cannot distinguish some distinct structures; – algorithms cannot reach some existing states; – algorithms cannot access some existing objects. The algorithms studied are interactive, querying an environment, small–step, operating over different background classes. Since our primary motivation is abstract analysis of cryptographic algorithms, our examples come from this field – we believe however that the potential application field is much broader.