Ordinary interactive small-step algorithms, I

Abstract

This is the first in a series of papers extending the Abstract State Machine Thesis -that arbitrary algorithms are behaviorally equivalent to abstract state machines -to algorithms that can interact with their environments during a step rather than only between steps. In the present paper, we describe, by means of suitable postulates, those interactive algorithms that (1) proceed in discrete, global steps, (2) perform only a bounded amount of work in each step, (3) use only such information from the environment as can be regarded as answers to queries, and (4) never complete a step until all queries from that step have been answered.We indicate how a great many sorts of interaction meet these requirements. We also discuss in detail the structure of queries and replies and the appropriate definition of equivalence of algorithms.Finally, motivated by our considerations concerning queries, we discuss a generalization of first-order logic in which the arguments of function and relation symbols are not merely tuples of elements but orbits of such tuples under groups of permutations of the argument places.

Model–Based Testing of Cryptographic Protocols

“…. The full theory of (ordinary) interactive algorithms is developed in [BG04a,BG04b]; overviews are given in [Gur05] and [RR05].…”

“…All algorithms in this paper are assumed to be small-step ordinary interactive algorithms in the sense of [BG04a,BG04b,Gur05].…”

“…Here we shall repeatedly use the following results from [RR05](where Y −X is the set of differences of two states over the same carrier, see [Gur00,BG04a,RR05] for definitions):…”

“…Gurevich,A.Blass [Gur00,BG03,BG04a,BG04b,Gur05], and also B.Rossman and the authors [RR05]. The gist is that algorithms can be mathematically captured at their own native level of abstraction -ex.…”

“…Unlike the static abstract models, which necessarily invoke additional proof-theoretic devices to capture dynamic aspects, the behavioral theory explicitly targets the dynamic behavior of algorithms semantically. By recent work on behavioral theory [BG04a,BG04b,RR05,Gur05], this also includes interactive algorithms talking to an environment between steps, and within a step, allowing us to represent the abstract content of oracle algorithms and adversary games typical of computational cryptography directly. In the framework of intra-step interactive algorithms exact abstract representations of computational security notions, defined in terms of adversary games, emerge clearly.…”