P systems with energy accounting ∗

We consider P systems where each evolution rule “Produces” or “Consumes” some quantity of energy, in amounts which are expressed as integer numbers. In each moment and in each membrane the total energy involved in an evolution step should be positive, but if “Soo much” energy is present in a membrane, then the membrane will be destroyed (dissolved). We show that this feature is rather powerful. In the case of multisets of symbol-objects we find that systems with two membranes and arbitrary energy associated with rules, or with arbitrarily many membranes and a bounded energy associated with rules characterize the recursively enumerable sets of vectors of natural numbers (catalysts and priorities are used). In the case of string-objects we have only proved that the recursively enumerable languages can be generated by systems with arbitrarily many membranes and bounded energy; when bounding the number of membranes and leaving free the quantity of energy associated with each rule we have only generated all matrix languages. Several research topics are also pointed out.     

A Class of Functions Computable by Index Grammars

The classes of functions IGF and IGF ₀ are investigated that can be computed by means of special index grammars. The class IGF ₀ is demonstrated to coincide with the class of strictly increasing functions (of a natural argument), which can be represented by systems of linear recurrence relations with natural coefficients, and to be closed with respect to the operations of addition, multiplication, series summation, and addition of a natural number to their arguments or results.     

Transcending Turing Computability

It has been argued that neural networks and other forms of analog computation may transcend the limits of Turing-machine computation; proofs have been offered on both sides, subject to differing assumptions. In this article I argue that the important comparisons between the two models of computation are not so much mathematical as epistemological. The Turing-machine model makes assumptions about information representation and processing that are badly matched to the realities of natural computation (information representation and processing in or inspired by natural systems). This points to the need for new models of computation addressing issues orthogonal to those that have occupied the traditional theory of computation.     

Comments on some theories of fuzzy computation

In classical computability theory, there are several (equivalent) definitions of computable function, decidable subset and semi-decidable subset. This paper is devoted to the discussion of some proposals for extending these definitions to the framework of fuzzy set theory. The paper mainly focuses on the notions of fuzzy Turing machine and fuzzy computability by limit processes. The basic idea of this paper is that the presence of real numbers in the interval [0,1] forces us to refer to endless approximation processes (as in recursive analysis) and not to processes terminating after a finite number of steps and giving the exact output (as in recursive arithmetic). In accordance with such a point of view, an extension of the famous Church thesis is proposed.     

Enhancing dependency pair method using strong computability in simply-typed term rewriting

We enhance the dependency pair method in order to prove termination using recursive structure analysis in simply-typed term rewriting systems, which is one of the computational models of functional programs. The primary advantage of our method is that one can exclude higher-order variables which are difficult to analyze theoretically, from recursive structure analysis. The key idea of our method is to analyze recursive structure from the viewpoint of strong computability. This property was introduced for proving termination in typed λ-calculus, and is a stronger condition than the property of termination. The difficulty in incorporating this concept into recursive structure analysis is that because it is defined inductively over type structure, it is not closed under the subterm relation. This breaks the correspondence between strong computability and recursive structure. In order to guarantee the correspondence, we propose plain function-passing as a restriction, which is satisfied by many non-artificial functional programs.     

Hypercomputation

A survey of the field of hypercomputation, including discussion of a variety of objections.     

Unexpected universality results for three classes of P systems with symport/antiport

Symport and antiport are biological ways of transporting molecules through membranesin ``collaborating' pairs; in the case of symport the two molecules pass in the same direction, in the case of antiport the two molecules pass in opposite directions. Here we first survey the results about the computing power of membrane systems (P systems) using only symport/antiport rules (hence these systems compute by communication only), then we consider a recently introduced, way of defining the result of a computation in a membrane system: looking for the trace of certain objects in their movement through membranes. Rather unexpected, in this way we get characterizations of recursively enumerable languages by means of membrane systems with symport/antiport which work with multisets of objects (note the qualitative difference between the data structure used by computations – multisets: no ordering– and the data structure of the output – strings: linear ordering). A similar remark holds true for the case of analysing P systems, which work in an automata-like manner: the sequence of certain distinguished objects taken from the environment during acomputation is the string recognized by the computation. We also survey universality results from this area, with sketched proofs. Some open problems are also formulated. This revised version was published online in June 2006 with corrections to the Cover Date.     

蒙古文整词编码研究

作者基于蒙古文黏着记录其词汇方式和按书面音节拼读书写整词规则,提出了蒙古文整词编码方法。本文依据可计算性理论,提出了拼音文字非键盘映射编码方法,将整词编码分为输写码与计算码。整词输写码设计模仿传统蒙古文整词固有拼读书写规则,达到了最佳人机键盘交互目的。整词计算码既可载荷整词复杂特征知识信息、又可保证信息的可计算性,从而为蒙古文整词复杂特征合一计算和并行处理奠定了可行性科学基础。     

A study of homogeneity in relational databases

We define four different properties of relational databases which are related tothe notion of homogeneity in classical model theory. The main question for their definition is, for any given database to determine the minimum integer k, such that whenever two k-tuples satisfy the same properties which are expressible in first order logic with up to k variables (FO ^k ), then there is an automorphism which maps each of these k-tuples onto each other. We study these four properties as a means to increase the computational power of subclasses of the reflective relational machines (RRMs) of bounded variable complexity. These were introduced by S. Abiteboul, C. Papadimitriou and V. Vianu and are known to be incomplete. For this sake we first give a semantic characterization of the subclasses of total RRM with variable complexity k (RRM ^k ) for every natural number k. This leads to the definition of classes of queries denoted as Q C Q ^k . We believe these classes to be of interest in their own right. For each k>0, we define the subclass Q C Q ^k as the total queries in the class C Q of computable queries which preserve realization of properties expressible in FO ^k . The nature of these classes is implicit in the work of S. Abiteboul, M. Vardi and V. Vianu. We prove Q C Q ^k =total(RRM ^k ) for every k>0. We also prove that these classes form a strict hierarchy within a strict subclass of total(C Q). This hierarchy is orthogonal to the usual classification of computable queries in time-space-complexity classes. We prove that the computability power of RRM ^k machines is much greater when working with classes of databases which are homogeneous, for three of the properties which we define. As to the fourth one, we prove that the computability power of RRM with sublinear variable complexity also increases when working on databases which satisfy that property. The strongest notion, pairwise k-homogeneity, allows RRM ^k machines to achieve completeness.     

Divergence bounded computable real numbers

A real x$x$ is called h$h$-bounded computable  , for some function h:N→N$h:\mathbb{N}\to \mathbb{N}$, if there is a computable sequence (_xs)$(x_s)$ of rational numbers which converges to x$x$ such that, for any n∈N$n\in \mathbb{N}$, at most h(n)$h(n)$ non-overlapping pairs of its members are separated by a distance larger than 2^-n$2^{-n}$. In this paper we discuss properties of h$h$-bounded computable reals for various functions h$h$. We will show a simple sufficient condition for a class of functions h$h$ such that the corresponding h$h$-bounded computable reals form an algebraic field. A hierarchy theorem for h$h$-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h$h$-bounded computability for special functions h$h$.