Proving total correctness of nondeterministic programs in infinitary logic

Summary It is shown how the weakest precondition approach to proving total correctness of nondeterministic programs can be formalized in infinitary logic. The weakest precondition technique is extended to hierarchically structured programs by adding a new primitive statement for operational abstraction, the nondeterministic assignment statement, to the guarded commands of Dijkstra. The infinitary logic L ₁ is shown to be strong enough to express the weakest preconditions for Dijkstra's guarded commands, but too weak for the extended guarded commands. Two possible solutions are considered: going to the essentially stronger infinitary logic L ₁₁ and restricting the power of the nondeterministic assignment statement in a way which allows the weakest preconditions to be expressed in L ₁.     

On a random Volterra integral equation

Summary Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ) = h(t; ) + _o ^t k(t, ; )f(, x(; )) d, where , the supporting set of a probability measure space (,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (, )-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

Extremal Relations between Additive Loss Functions and the Kolmogorov Complexity

Conditions are presented under which the maximum of the Kolmogorov complexity (algorithmic entropy) K(₁... _N ) is attained, given the cost f( _i ) of a message ₁... _N . Various extremal relations between the message cost and the Kolmogorov complexity are also considered; in particular, the minimization problem for the function f( _i ) – K(₁... _N ) is studied. Here, is a parameter, called the temperature by analogy with thermodynamics. We also study domains of small variation of this function.     

The growth of configurations

The transition ruleF of a cellular automaton may sometimes be regarded as a rule of growth of a crystal from a seed. A study is made of the iterates,F,F ² .For certain one-dimensional growth rules, the limiting shapes of the crystals are computed, and an asymptotic formula for the size of the crystal as a function of time is obtained.     

Exponential Queuing Network with Dependent Servicing, Negative Customers, and Modification of the Customer Type

The open exponential queuing network with negative customers (G-network) was considered.For each arriving customer, given was a set of its random parameters such as the route defining the sequence of network nodes passed by the customer, route length, size, and servicing duration at each stage of the route. It was assumed that the negative customer arriving to the sth node with the probabilities _s and _s ⁺ either kills the positive customer in a randomly chosen server or does not affect it at all and with the probability _s ^– =1 – _s – _s ⁺ transforms it into a negative customer which after an exponentially distributed time arrives to the sth node with the given probability. The multidimensional stationary probability distribution of the network states was proved to be representable in the multiplicative form.     

On problem transformability in VLSI

The two basic performance parameters that capture the complexity of any VLSI chip are the area of the chip,A, and the computation time,T. A systematic approach for establishing lower bounds onA is presented. This approach relatesA to the bisection flow, . A theory of problem transformation based on , which captures bothAT ² andA complexity, is developed. A fundamental problem, namely, element uniqueness, is chosen as a computational prototype. It is shown under general input/output protocol assumptions that any chip that decides ifn elements (each with (1+)lognbits) are unique must have =(nlogn), and thus, AT²=(n ²log² n), andA= (nlogn). A theory of VLSI transformability reveals the inherentAT ² andA complexity of a large class of related problems.This work was supported in part by the Semiconductor Research Corporation under contract RSCH 84-06-049-6.     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

Definability of Polyadic Lifts of Generalized Quantifiers

We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type < 1 >.We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms of simpler quantifiers.     

Constructing strongly convex approximate hulls with inaccurate primitives

The first half of this paper introducesEpsilon Geometry, a framework for the development of robust geometric algorithms using inaccurate primitives. Epsilon Geometry is based on a very general model of imprecise computations, which includes floating-point and rounded-integer arithmetic as special cases. The second half of the paper introduces the notion of a (–)-convex polygon, a polygon that remains convex even if its vertices are all arbitrarily displaced by a distance of of less, and proves some interesting properties of such polygons. In particular, we prove that for every point set there exists a (–)-convex polygonH such that every point is at most 4 away fromH. Using the tools of Epsilon Geometry, we develop robust algorithms for testing whether a polygon is (–)-convex, for testing whether a point is inside a (–)-convex polygon, and for computing a (–)-convex approximate hull for a set of points.     

ω-Perfect graphs

-Perfect graph is defined and some classes of -perfect graphs are described, although the characterization of the complete class of -perfect graphs remains an open question. A bound on the chromatic number for graphs without even holes is derived.Translated from Kibernetika, No. 2, pp. 8–11, March–April, 1990.     

A Parallel ODE Solver Adapted to Oscillatory Problems

We propose a multistep method for solving special second-order ordinary differential equations with damped oscillatory solutions. The proposed methods integrate exactly (with only round-off error) ordinary polynomials and the product of trigonometric functions at a frequency by exponentials of a parameter g. When =g=0 they reduce to the classical Nyströn and Cowell methods. Although there exist several methods with these properties, the proposed method allows independent computation of predictor and corrector which motivates parallel implementation.     

Finite Presentations of Infinite Structures: Automata and Interpretations

We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by model-theoretic interpretations. These two ways of presenting a structure are related. Indeed, a structure is automatic if, and only if, it is first-order interpretable in an appropriate expansion of Presburger arithmetic or, equivalently, in the infinite binary tree with prefix order and equal length predicate. Similar results hold for -automatic structures and appropriate expansions of the real ordered group. We also discuss the relationship to automatic groups. The model checking problem for FO(), first-order logic extended by the quantifier there are infinitely many, is proved to be decidable for automatic and -automatic structures. Further, the complexity for various fragments of first-order logic is determined. On the other hand, several important properties not expressible in FO, such as isomorphism or connectedness, turn out to be undecidable for automatic structures. Finally, we investigate methods for proving that a structure does not admit an automatic presentation, and we establish that the class of automatic structures is closed under Feferman–Vaught-like products.     

An evaluation ofalgol 68 for interrogation process algorithms

Given a random processT taking a finite number of statesT _j on a population , we consider a family,D, of random processes defined on . Each elementq ofD has a finite number of possible statesq _i, and we suppose that the conditional probabilitiesp(q _i|T _j) are known. We consider arborescent decision processes using the elements ofD as questions and aiming at determining the state of the systemT during a given experiment of . The questionnaires of Picard treat the cases where all the conditional probabilities are equal to 0 or 1; we introduce the pseudoquestionnaires to handle the general case. Certain recognition problems of structure can also lead to the construction of a pseudoquestionnaire. We will show here in an example that algol 68 appears to be well suited to the writing of such algorithms.     

A Fat Boundary Method for the Poisson Problem in a Domain with Holes

We consider the Poisson equation with Dirichlet boundary conditions, in a domain , where ⁿ , and B is a collection of smooth open subsets (typically balls). The objective is to split the initial problem into two parts: a problem set in the whole domain , for which fast solvers can be used, and local subproblems set in narrow domains around the connected components of B, which can be solved in a fully parallel way. We shall present here a method based on a multi-domain formulation of the initial problem, which leads to a fixed point algorithm. The convergence of the algorithm is established, under some conditions on a relaxation parameter . The dependence of the convergence interval for upon the geometry is investigated. Some 2D computations based on a finite element discretization of both global and local problems are presented.     

Is Solar System Stable? A Remark

It is not yet known (1997) whether the Solar system is stable or not. Common belief is that the Solar system is stable if and only if it is not a resonant system, i.e., whenever its orbital frequencies i satisfy an inequality | nii| for i|ni| N; a similar inequality is true for randomly chosen frequencies. In this paper, we show that the Solar system does not have such resonances, and therefore (if the above-mentioned belief is correct), it is stable.     

On characterizations of recursively enumerable languages

Summary Geffert has shown that earch recursively enumerable languageL over can be expressed in the formL{h(x) ^–1 g(x)x in ⁺} ^* where is an alphabet andg, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations ofL. For instance,L =(L ₀) ^* whereL ₀ is a minimal linear language and is the Dyck reductiona, A.     

Modelling Asynchrony with a Synchronous Model

The I/O automaton paradigm of Lynch and Tuttle models asynchrony through an interleaving parallel composition. The recognition that such interleaving models in fact can be viewed as special cases of synchronous parallel composition has been very limited. Let be any set of finite-state I/O automata drawing actions from a fixed finite set containing a subset . In this article we establish a translation T : to a class of -automata closed under a synchronous parallel composition, for which T is monotonic with respect to implementation relative to , and linear with respect to composition. Thus, for A¹, ..., A, B¹, ..., B and A = A¹ ... A, B = B¹ ... B, if is the set of actions common to both A and B, then A implements B (in the sense of I/O automata) if and only if the -automaton language containment (T(A¹) ... T(A)) (T(B¹) ... T(B)) obtains, where denotes the interleaving parallel composition on and denotes the synchronous parallel composition on . For the class , we use the L-process model of -automata. This result enables one to verify systems specified by I/O automata through model-checkers such as COSPAN or SMV, that operate on models with synchronous parallel composition. The translation technique generalizes to other interleaving models, although in each case, the translation map must match the specific model.     

Constructing strongly convex hulls using exact or rounded arithmetic

One useful generalization of the convex hull of a setS ofn points is the -strongly convex -hull. It is defined to be a convex polygon with vertices taken fromS such that no point inS lies farther than outside and such that even if the vertices of are perturbed by as much as , remains convex. It was an open question as to whether an -strongly convexO()-hull existed for all positive . We give here anO(n logn) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an -strongly convexO( + )-hull inO(n logn) time using rounded arithmetic with rounding unit . This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent ofn.