The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status of the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s).

Residual stress states in engineering structures are usually determined by measuring components of stress tensors with depth below the material surface. There are destructive and nondestructive methods to measure strain tensor components and convert them into stress tensor components by a variety of techniques derived from constitutive (material) equations. In this study, four methods for determining the strain tensor components are presented: X-ray diffraction method (XRDM), magnetic Barkhausen noise method (MBNM), hole drilling method (HDM), and cut-and-section method (CSM); the first two are nondestructive, and the third and fourth are semidestructive and destructive, respectively. A complementarity of the experimental and two numerical methods such as boundary element method and finite element method is explained. An application of the experimental and numerical methods to measure residual stress states in an industrial component, an L-shaped part of a supporting column in a high voltage structure, is presented.

This paper revisits the problem of determinacy inference addressing the problem of how to uniformly handle cut. To this end a new semantics is introduced for cut, which is abstracted to systematically derive a backward analysis that derives conditions sufficient for a goal to succeed at most once. The method is conceptionally simpler and easier to implement than the existing techniques, while improving the latter's handling of cut. Formal arguments substantiate correctness and experimental work, and a tool called ‘RedAlert’ demonstrates the method's generality and applicability.

Prolog's very useful expressive power is not captured by traditional logic programming semantics, due mainly to the cut and goal and clause order. Several alternative semantics have been put forward, exposing operational details of the computation state. We propose instead to redesign Prolog around structured alternatives to the cut and clauses, keeping the expressive power and computation model but with a compositional denotational semantics over much simpler states—just variable bindings. This considerably eases reasoning about programs, by programmers and tools such as a partial evaluator, with safe unfolding of calls through predicate definitions. An if-then-else across clauses replaces most uses of the cut, but the cut's full power is achieved by an until construct. Disjunction, conjunction and until, along with unification, are the primitive goal types with a compositional semantics yielding sequences of variable-binding solutions. This extends to programs via the usual technique of a least fixpoint construction. A simple interpreter for Prolog in the alternative language, and a definition of until in Prolog, establish the identical expressive power of the two languages. Many useful control constructs are derivable from the primitives, and the semantic framework illuminates the discussion of alternative ones. The formalisation rests on a term language with variable abstraction as in the λ-calculus. A clause is an abstraction on the call arguments, a continuation, and the local variables. It can be inclusive or exclusive, expressing a local case bound to a continuation by either a disjunction or an if-then-else. Clauses are open definitions, composed (and closed) with simple functional application β-reduction). This paves the way for a simple account of flexible module composition mechanisms. Cube, a concrete language with the exposed principles, has been implemented on top of a Prolog engine and successfully used to build large real-world applications.

In this paper we study the number of random records in an arbitrary split tree (or, equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for the convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree in Holmgren (2010), which is one specific case of split trees. Other important examples of split trees include m-ary search trees, quad trees, medians of (2k + 1)-trees, simplex trees, tries, and digital search trees.

Termination is an important and well-studied property for logic programs. However, almost all approaches for automated termination analysis focus on definite logic programs, whereas real-world Prolog programs typically use the cut operator. We introduce a novel pre-processing method which automatically transforms Prolog programs into logic programs without cuts, where termination of the cut-free program implies termination of the original program. Hence after this pre-processing, any technique for proving termination of definite logic programs can be applied. We implemented this pre-processing in our termination prover AProVE and evaluated it successfully with extensive experiments.

The semantics of the Prolog ‘cut’ construct is explored in the context of some desirable properties of logic programming systems, referred to as the witness properties. The witness properties concern the operational consistency of responses to queries. A generalization of Prolog with negation as failure and cut is described, and shown not to have the witness properties. A restriction of the system is then described, which preserves the choice and first-solution behaviour of cut but allows the system to have the witness properties. The notion of cut in the restricted system is more restricted than the Prolog hard cut, but retains the useful first-solution behaviour of hard cut, not retained by other proposed cuts such as the ‘soft cut’. It is argued that the restricted system achieves a good compromise between the power and utility of the Prolog cut and the need for internal consistency in logic programming systems. The restricted system is given an abstract semantics, which depends on the witness properties; this semantics suggests that the restricted system has a deeper connection to logic than simply permitting some computations which are logical. Parts of this paper appeared previously in a different form in the Proceedings of the 1995 International Logic Programming Symposium (Andrews, 1995).

Answers are given to two questions concerning the existence of some sparse subsets of ℋ = {0, 1 … ., H – 1} ⊆ *ℕ. where H is a hyperfinite integer. In §1. we answer a question of Kanovei by showing that for a given cut U in ℋ, there exists a countably determined set X ⊆ ℋ which contains exactly one element in each U-monad, if and only if U = a · ℕ for some a Є ℋ ∖ {0}. In §2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut V ⊆ ℋ such that for any cut U, (i) there exists a U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided , (ii) there does not exist any U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided . We obtain some partial results for the case U = V.