Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.

Efficient Algebraic Operations on Programs

Abstract: A symbolic version of an operation on values is a corresponding operation on program texts. For example, symbolic composition of two programs p, q yields a program whose meaning is the (mathematical) composition of the meanings of p and q. Another example is symbolic specialization of a function to a known first argument value. This operation, given the first argument, transforms a two-input program into an equivalent one-input program. Computability of both of these symbolic operations has long been established in recursive function theory [12,16]; the latter is known as Kleene’s “s-m-n” theorem, also known as partial evaluation. In addition to computability we are concerned with efficient symbolic operations, in particular applications of the two just mentioned to compiling and compiler generation. Several examples of symbolic composition are given, culminating in nontrivial applications to compiler generation [14], [18]. Partial evaluation has recently become the subject of considerable interest [1]. Reasons include simplicity, efficiency and the surprising fact that self-application can be used in practice to generate compilers, and a compiler generator as well. This paper makes three contributions: First, it introduces a new notation to describe the types of symbolic operations, one that makes an explicit distinction between the types of program texts and the values they denote. This leads to natural definitions of what it means for an interpreter or compiler to be type correct—a tricky problem in a multilanguage context. Second, it uses the notation to give a clear overview of several earlier applications of symbolic computation. For example, it is seen that the new type notation can satisfactorily explain the types involved when generating a compiler by self-applying a partial evaluator. Finally, a number of problems for further research are stated along the way. The paper ends by suggesting Cartesian categorical combinators as a unifying framework in which to study symbolic operations.

Towards a Theory of Peer-to-Peer Computability.

Abstract: A peer-to-peer system is a distributed system with no physical or logical central authority. We give a formal model of a peer-to-peer system where agents communicate through read-modify-write registers that can be accessed by exactly two agents. For this model, we study so-called ordering decision tasks for wait-free agents. We show how agents can determine their position in a total linearizable order in the peer-to-peer model. We also show that electing a leader among the agents and finding a predecessor agent in the total ordering cannot be implemented without a central authority. Our peer-to-peer model is related to other models of distributed computing, specifically to concurrent objects in asynchronous shared memory and to switching (counting) networks.

Approximate decidability in euclidean spaces

Abstract: We study concepts of decidability (recursivity) for subsets of Euclidean spaces ℝk within the framework of approximate computability (type two theory of effectivity). A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets, and it modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of sets can equivalently be expressed by computability of the characteristic functions by means of appropriately working oracle Turing machines. The notion fulfills some natural requirements and is hereditary under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces.

Computing Cliques and Cavities in Networks

Abstract: Complex networks have complete subgraphs such as nodes, edges, triangles, etc., referred to as cliques of different orders. Notably, cavities consisting of higher-order cliques have been found playing an important role in brain functions. Since searching for the maximum clique in a large network is an NP-complete problem, we propose using k-core decomposition to determine the computability of a given network subject to limited computing resources. For a computable network, we design a search algorithm for finding cliques of different orders, which also provides the Euler characteristic number. Then, we compute the Betti number by using the ranks of the boundary matrices of adjacent cliques. Furthermore, we design an optimized algorithm for finding cavities of different orders. Finally, we apply the algorithm to the neuronal network of C. elegans in one dataset, and find all of its cliques and some cavities of different orders therein, providing a basis for further mathematical analysis and computation of the structure and function of the C. elegans neuronal network.

Freezing, Bounded-Change and Convergent Cellular Automata

Abstract: This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.