An Inflationary Fixed Point Operator in XQuery

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

We shall say that a logic is “simply consistent” if there is no formula A such that both A and ∼ A are provable. “ω-consistent” will be used in the sense of Gödel. “General recursive” and “primitive recursive” will be used in the sense of Kleene, so that what Gödel calls “rekursiv” will be called “primitive recursive.” By an “Entscheidungsverfahren” will be meant a general recursive function ϕ(n) such that, if n is the Gödel number of a provable formula, ϕ(n) = 0 and, if n is not the Gödel number of a provable formula, ϕ(n) = 1. In specifying that ϕ must be general recursive we are following Church in identifying “general recursiveness” and “effective calculability.” First, a modification is made in Gödel's proofs of his theorems, Satz VI (Gödel, p. 187—this is the theorem which states that ω-consistency implies the existence of undecidable propositions) and Satz XI (Gödel, p. 196—this is the theorem which states that simple consistency implies that the formula which states simple consistency is not provable). The modifications of the proofs make these theorems hold for a much more general class of logics. Then, by sacrificing some generality, it is proved that simple consistency implies the existence of undecidable propositions (a strengthening of Gödel's Satz VI and Kleene's Theorem XIII) and that simple consistency implies the non-existence of an Entscheidungsverfahren (a strengthening of the result in the last paragraph of Church).

We are presenting a coherent framework for XQuery processing that incorporates IR-style approximate matching and allows the ordering of results by their relevance score. Our relevance ranking algorithm is based on both stem matching and term proximity. Our XQuery processor is stream-based, consisting of iterators connected into pipelines. In our framework, all values produced by XQuery expressions are assigned scores,and these scores propagate and are combined when piped through the iterators. The most important feature of our evaluation engine is the use of structural and content-based inverse indexes that deliver data in document order and facilitate the use of efficient merge joins to evaluate path expressions and search predicates. We present the rules for the translation from a large part of XQuery to iterator pipeline. Our modular approach of building pipelines to evaluate XQuery scales up to any query complexity because the pipes can be connected in the same way complex queries are formed from simpler ones.

The Pathfinder XQuery compiler has been enhanced by a new code generator that can target any SQL:1999-compliant relational database system(RDBMS). This code generator marks an important next step towards truly relational XQuery processing, a branch of database technology that aims to turn RDBMSs into highly efficient XML and XQuery processors without the need to invade the relational database kernel. Pathfinder, a retargetable front-end compiler, translates input XQuery expressions into DAG-shaped relational algebra plans. The code generator then turns these plans into sequences of either SQL:1999 statements or view definitions which jointly implement the (sometimes intricate) XQuery semantics. In a sense, this demonstration thus lets relational algebra and SQL swap their traditional roles in database query processing. The result is a code generator that (1) supports an almost complete dialect of XQuery, (2) can target any RDBMS with a SQL:1999 language interface, and (3) exhibits quite promising performance characteristics when run against high-volume XML data as well as complex XQuery expressions.

A fertile field of research in theoretical computer science investigates the representation 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 computations. To every type A we associate a type of partial elements Partial(A), coinductively generated by two constructors: the first, return(a) just returns an element a:A; the second, step(x), adds a computation step to a recursive element x:Partial(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 present Transitive Closure based visual word formation technique for obtaining robust object representations from smoothly varying multiple views. Each one of our visual words is represented by a set of feature vectors which is obtained by performing transitive closure operation on SIFT features. We also present range-reducing tree structure to speed up the transitive closure operation. The robustness of our visual word representation is demonstrated for Structure from Motion (SfM) and location identification in video images.

Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic

Reviews - Hartley RogersJr., Theory of recursive functions and effective computability.McGraw-Hill Book Company, New York, St. Louis, San Francisco, Toronto, London, and Sydney, 1967, xix + 482 pp. - Volume 36 Issue 1

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Computation of recursive functionals using minimal initial segments

4 - Recursive Enumerability

In this paper we proved that the function class CFRF and its proper subclass CFPRF are respectively the partial recursive functions and primitive recursive functions of context free languages (CFLs). Also we discussed the relation between them and recursive functions defined on other domains. It is indicated that the functions of natural numbers and/or symbol strings (words) are functions of CFLs. Several frequently used primitive recursive functions on words were given, including logical connectives, conditional expressions. Also the powerful operators (bounded maximization and minimization operators) for constructing primitive recursive functions were defined. Two important nontrivial algorithms, the characteristic function of arbitrary CFL and the parse function of CFL sentences were constructed. Based on them, the method for extending or restricting function domain was described.

Minimization and NP multifunctions

We highlight the limitations of formal methods by exhibiting two results in recursive function theory: that there is no effective means of finding a program that satisfies a given formal specification; or checking that a program meets a specification. We exhibit a simple MAS which has all the power of a Turing machine. We argue that any pure design methodology will face insurmountable difficulties in today's open and complex MAS. We recommend instead a methodology based on experimental method - scientific foundations for MAS construction and control.

Recursive function theory studies functions that can be evaluated by an algorithm based on some computing model such as a Turing machine or computer program. These functions are called partial recursive (p.r.) and are defined on subsets of N, where N = {0, 1,... }, with values in N. (A good introduction to the theory may be found in [2].) This paper explains the Parameter Theorem, illustrates some unusual computer programs, and explains their connection to recursive function theory. Basing our computing model on a modern high level language, we can construct programs whose existence is known through more formal means. Our functions will operate on strings rather than numbers, however. There is no loss in generality since strings can be viewed as members of N, and vice-versa. From a practical aspect, this approach is somewhat more desirable since conventional programming languages often allow making use of long strings, but place more restrictive limits on the size of integers. We first pose a problem: Does there exist a program (written in some conventional programming language) which halts if and only if its input is identical to the program's text (i.e., source code)? For the remainder of this note X is the set of ASCII characters 'a,' 'b, +,' space, line feed, and so forth, needed to write programs in most programming languages. We denote by X * the set of all strings (finite sequences) over X. Let P1 be the set of all one variable partial recursive functions on X *. A function I belongs to P1 if and only if there is a subset of S of X* and some algorithm which computes 't by associating with each element of S a well-defined element of X *. Similarly, 't belongs to P2 if and only if there is a set S c X * X X *, and an algorithm for 't which associates with each element of S a well-defined element of X *. A partial recursive function I in P1 (or in P2) is recursive if the algorithm is defined on all of X * (or X * X X *). This is equivalent to saying that the algorithm which evaluates I always halts. Our computing model will be based on Adat; its high-level string-handling properties simplify later examples. For readability, however, we will express algorithms in a more familiar Pascal-like style. Readers familiar with either programming language should find forthcoming algorithms sufficiently clear. A formal presentation in Ada is given in the Appendix. A string of ASCII characters, such as in FIGURE 1, represents a function in our language which takes strings to strings. Of course, strings of ASCII characters such as The quick brown fox and %*!! represent nothing at all in our programming language. Let A1 be the set of all strings in X * representing functions in our language of one input STRING which outputs a STRING. Similarly let A2 be the set of all strings in