This paper describes Turing’s Halting Problem (HP), and reviews the classic proof that no function exists that can solve HP. The concept of a “Context-Dependent Function” (CDF), whose behavior varies based on seemingly irrelevant changes to a program calling that function, is introduced, and the proof of HP’s undecidability is re-examined in light of CDFs. The existence of CDFs is established v...

We consider the termination/non-termination property of a class of loops. Such loops are commonly used abstractions of real program pieces. Second-order logic is a convenient language to express non-termination. Of course, such property is generally undecidable. However, by restricting the language to known decidable cases, we exhibit new classes of loops, the non-termination of which is decida...

Since many real-world problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by si...

In the s Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object s structure In particular Bennett identi ed the classes of weakly deep and strongly deep sequences and showed that the halting problem is strongly deep Juedes Lath rop and Lutz subsequently extended this result by de ning the class of weakly useful sequences an...

The aim of this paper is to study the power of parallel multiset-rewriting systems with permitting or forbidding context (or P systems with non-cooperative rules with promoters or inhibitors). The main results obtained say that if we use promoters or inhibitors of weight two, then the systems are computational universal. Moreover, both constructions satisfy a special property we define: they ar...

The introduction. The introduction usually starts with a description of and a motivation for the general problem area. After introducing the general problem area, you zoom in to the specific problem studied in the paper. I like to already discuss previous work here. This way you can explain where the specific problem fits into the state-of-the-art and why it is interesting. Ideally, the discuss...

Subexponential logic is a variant of linear logic with a family of exponential connectives—called subexponentials—that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that classical propositional multiplicative linear logic extended with one unrestricted and two incomparable linear subexponentials c...

Let A be any nonrecursive set. We deene a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible to A based on bounding the number of queries to A that an oracle machine can make. When A is the halting problem K our hierarchy of sets interleaves with the diierence hierarchy 1 on the r.e. sets in a logarithmic way; this follows from a tradeoo between the number of parall...

In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units { formulas labelled according to a labelling algebra. In the system described here, labels are either ground terms or variabl...

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in a planar exterior domain in a half space with appropriate boundary conditions on the wall, the body, and at infinity. We focus on the case where the size of the body is small. We ...