In earlier work, the Abstract State Machine Thesis -- that arbitrary algorithms are behaviorally equivalent to abstract state machines -- was established for several classes of algorithms, including ordinary, interactive, small-step algorithms. This was accomplished on the basis of axiomatizations of these classes of algorithms. Here we extend the axiomatization and, in a companion paper, the proof, to cover interactive small-step algorithms that are not necessarily ordinary. This means that the algorithms (1) can complete a step without necessarily waiting for replies to all queries from that step and (2) can use not only the environment's replies but also the order in which the replies were received.
The says that all states are first-order structures of a fixed vocabulary, the transition function does not change the base set of a state, and isomorphism of structures preserves everything, which here means states, initial states and the transition function. It reflects the vast experience of mathematics and mathematical logic according to which every static mathematical situation can be adequately represented as a first-order structure. The idea behind the second requirement is that, even when the base set seems to increase with the creation of new objects, those objects can be regarded as having been already present in a reserve'' part of the state. What looks like creation is then regarded as taking an element from out of the reserve and into the active part of the state. (The nondeterministic choice of the element is made by the environment.) See @cite_15 @cite_12 @cite_27 @cite_21 and the next section for discussion. The idea behind the third requirement is that all relevant state information is reflected in the vocabulary: if your algorithm can distinguish red integers from green integers, then it is not just about integers.
@cite_27 @cite_15 @cite_21 @cite_12
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Abstract Define a (∀1 unary)-sentence to be a prenex first-order sentence of unary type (i.e., a type which only contains unary relation and function symbols and constant symbols) with only one (universal) quantifier. A successor structure is a structure 〈 B , S 〉 such that S is a function which is a permutation of the basis B with only one cycle. We exhibit a (∀1, unary)-sentence φ of type S , U 1 , …, U p such that if B is finite then 〈 B , S 〉 is a successor structure if 〈 B , S 〉 satisfies ∃ U 1 , …, ∃ U p ϕ . It implies that ⋃ NRAM( cn )=SPECTRA(∀1, unary), c ⩾1 where NRAM(cn) denotes the class of sets of positive integers accepted by a nondeterministic random access machine in time cn (where n is the input integer) and SPECTRA(∀1, unary) is the class of finite spectra of (∀1, unary)-sentences. Another consequence is that some graph properties (hamiltonicity, connectedness) can be characterised by sentences with unary function symbols and constant symbols and only one variable. This contrasts with the result (by Fagin and De Rougemont) that these two graph properties are not definable by monadic generalized spectra (without function symbols) even in the presence of an underlying successor relation. In this paper, we present a new visual way of exploring state sequences in large observational time-series. A key advantage of our method is that it can directly visualize higher-order state transitions. A standard first order state transition is a sequence of two states that are linked by a transition. A higher-order state transition is a sequence of three or more states where the sequence of participating states are linked together by consecutive first order state transitions. Our method extends the current state-graph exploration methods by employing a two dimensional graph, in which higher-order state transitions are visualized as curved lines. All transitions are bundled into thick splines, so that the thickness of an edge represents the frequency of instances. The bundling between two states takes into account the state transitions before and after the transition. This is done in such a way that it forms a continuous representation in which any subsequence of the timeseries is represented by a continuous smooth line. The edge bundles in these graphs can be explored interactively through our incremental selection algorithm. We demonstrate our method with an application in exploring labeled time-series data from a biological survey, where a clustering has assigned a single label to the data at each time-point. In these sequences, a large number of cyclic patterns occur, which in turn are linked to specific activities. We demonstrate how our method helps to find these cycles, and how the interactive selection process helps to find and investigate activities. The spectrum of a first-order sentence is the set of cardinalities of its finite models. We refine the well-known equality between the class of spectra and the class of sets (of positive integers) accepted by nondeterministic Turing machines in polynomial time. Let @math denote the class of spectra of sentences with d universal quantifiers. For any integer @math and each set of positive integers, A, we obtain: [ A NTIME (n^d ) A Sp (d ) A NTIME (n^d ( n)^2 ). ] Further the first implication holds even if we use multidimensional nondeterministic Turing machines. These results hold similarly for generalized spectra. As a consequence, we obtain a simplified proof of a hierarchy result of P. Pudlak about (generalized) spectra. We also prove that the set of primes is the spectrum of a certain sentence with only one variable. We present a novel approach for generalizing the IC3 algorithm for invariant checking from finite-state to infinite-state transition systems, expressed over some background theories. The procedure is based on a tight integration of IC3 with Implicit (predicate) Abstraction, a technique that expresses abstract transitions without computing explicitly the abstract system and is incremental with respect to the addition of predicates. In this scenario, IC3 operates only at the Boolean level of the abstract state space, discovering inductive clauses over the abstraction predicates. Theory reasoning is confined within the underlying SMT solver, and applied transparently when performing satisfiability checks. When the current abstraction allows for a spurious counterexample, it is refined by discovering and adding a sufficient set of new predicates. Importantly, this can be done in a completely incremental manner, without discarding the clauses found in the previous search.