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.

y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the Church-Turing thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.

Abstract State Machines (ASMs) allow modeling system behaviors at any desired level of abstraction, including a level with rich data types, such as sets, sequences, maps, and user-defined data types. The availability of high-level data types allow state elements to be represented both abstractly and faithfully at the same time. In this paper we look at symbolic analysis of ASMs. We consider ASMs over a fixed state background T that includes linear arithmetic, sets, tuples, and maps. For symbolic analysis, ASMs are translated into guarded update systems called model programs. We formulate the problem of bounded path exploration of model programs, or the problem of Bounded Model Program Checking (BMPC) as a satisfiability problem modulo T. Then we investigate the boundaries of decidable and undecidable cases for BMPC. In a general setting, BMPC is shown to be highly undecidable (Σ 1 1-complete); and even when restricting to finite sets the problem remains re-hard (Σ 0 1-hard). On the other hand, BMPC is shown to be decidable for a class of basic model programs that are common in practice. We use Satisfiability Modulo Theories (SMT) for solving BMPC; an instance of the BMPC problem is mapped to a formula, the formula is satisfiable modulo T if and only if