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.
Yiannis Moschovakis @cite_23 proposed that the informal notion of algorithm be identified with the formal notion of . A recursor is a monotone operator over partial functions whose least fixed point includes (as one component) the function that the algorithm computes. The approach does not seem to scale to algorithms interacting with an unknown environment. See [Section 4.3] abs for a critique of Moschovakis's computation model.
@cite_23
2345322282 2128992690 1762545898 2128659236
Maximal monotone operators are set-valued mappings which extend (but are not limited to) the notion of subdifferential of a convex function. The proximal point algorithm is a method for finding a zero of a maximal monotone operator. The algorithm consists in fixed point iterations of a mapping called the resolvent which depends on the maximal monotone operator of interest. The paper investigates a stochastic version of the algorithm where the resolvent used at iteration k is associated to one realization of a random maximal monotone operator. We establish the almost sure ergodic convergence of the iterates to a zero of the expectation (in the Aumann sense) of the latter random operator. Application to constrained stochastic optimization is considered. A fertile field of research in theoretical computer science investigates the rep- resentation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations, implementation of operational semantics, formalization of domain theory, and inductive definition of domain predicates. Here, a different solution is proposed: exploiting coinductive types to model infinite com- putations. To every type A we associate a type of partial elements A � , coinductively generated by two constructors: the first, p aq just returns an element a: A; the second, ⊲ x, adds a computation step to a recursive element x: A � . We show how this simple device is sufficient to formalize all recursive functions between two given types. It allows the definition of fixed points of finitary, that is, continuous, operators. We will compare this approach to different ones from the literature. Finally, we mention that the formalization, with appropriate structural maps, defines a strong monad. We consider the problem of partially recovering hidden binary variables from the observation of (few) censored edge weights, a problem with applications in community detection, correlation clustering and synchronization. We describe two spectral algorithms for this task based on the non-backtracking and the Bethe Hessian operators. These algorithms are shown to be asymptotically optimal for the partial recovery problem, in that they detect the hidden assignment as soon as it is information theoretically possible to do so. We describe a recursive algorithm to compute representations of functions with respect to nonorthogonal and possibly overcomplete dictionaries of elementary building blocks e.g. affine (wavelet) frames. We propose a modification to the matching pursuit algorithm of Mallat and Zhang (1992) that maintains full backward orthogonality of the residual (error) at every step and thereby leads to improved convergence. We refer to this modified algorithm as orthogonal matching pursuit (OMP). It is shown that all additional computation required for the OMP algorithm may be performed recursively. >