Modular Calculus Research Articles

From Bion to Delany

Literature furnishes a particular vertex to see reality through narrative fiction. In particular, science fiction literature, which creates a fantastic situation starting from realistic data (history, science, cultures), may be considered a kind of creative process. It uses heterogeneous things and “integrates” them into a homogeneous, new and comprehensible product. Science fiction writing allows the objects of the real to be reprocessed in terms which are thinkable at the current moment. Using the terminology established in psychoanalysis by Wilfred Bion this reprocessing work is a transformation. According to Bion we can hypothesise that the writer of the science fiction literary work serves as a “container” and the science fiction novel, considered a different way to represent reality and not just a simple editorial product, serves as a alpha-function to make concepts that were not previously thinkable or understandable. Between the 70s and the 80s the writer Samuel Delany theorized and put into practice the use of a literary model called “modular calculus”. This model allows the literary work of making something unthinkable into thinkable. The purpose of this paper is to highlight how modular calculus is a particular type of Bionian transformation, and how the science fiction novel can play the role of alpha-function, transforming unthinkable concepts into thinkable ones.

A Random Bag Preserving Product Operation

The author's current research programme is the development of a modular calculus for the average-cost of data structuring. This modular calculus provides a novel foundation for the analysis of algorithms. Its applicability to the analysis of algorithms has been demonstrated at the Center for Efficiency-Oriented Languages (CEOL) through the design of the novel programming language MOQA and the associated average-case analysis tool DISTRI-TRACK [M. Schellekens, D. Hickey and G. Bollella, ACETT, a Linearly-Compositional Programming Language for (semi-)automated Average-Case analysis, IEEE Real-Time Systems Symposium - Work In Progress Session, 2004; M. Boubekeur, D. Hickey, J. Mc Enery and M. Schellekens, A new Approach for Modular Average-Case Timing of Real-Time Java Programs, Accepted for publication in WSEAS Transactions on Computers, 2007; M. Boubekeur, D. Hickey, J. Mc Enery and M. Schellekens, Towards Modular Average-Case Timing in Real-Time Languages: An Application to Real-Time Java, Accepted for publication on the 6th WSEAS International Conference on APPLIED COMPUTER SCIENCE (ACS'06), Tenerife, December, 2006; M. Schellekens, A Modular Calculus for the Average Cost of Data Structuring, Springer book to appear, 250 pages, May 2008 (to appear), M. Schellekens, Compositional Average-Case Analysis, preprint, under review, 2006]. Modular computations of the average cost of data structuring are possible through the fundamental notion of random bag preservation. Random bag preserving operations enable the constructive tracking of the data and the distribution of the data states during a MOQA computation. This in turn enables the (semi-)automated derivation of the average cost of the operations. Two fundamental MOQA operations enable the creation and destruction of data structures: the MOQA product operation, which is the subject of this paper, and the MOQA delete operation, which forms the subject of [M. Schellekens, Compositional Average-Case Analysis, preprint, under review, 2006]. The introduction of the entire MOQA language is well beyond the scope of this paper and will be reported in a book [M. Schellekens, A Modular Calculus for the Average Cost of Data Structuring, Springer book to appear, 250 pages, May 2008 (to appear)]. The language has been implemented at CEOL and automated derivations of average-cost of data structuring are under way. Here we report on a (simplified) version of the fundamental notion of random bag preservation and demonstrate that the central MOQA product operation possesses this crucial property.

Functionally-Generalised MOQA Operations

The programming language MOQA was designed by Michel Schellekens in [Schellekens M.: Compositional Average-Case Analysis, Preprint, submitted to the Journal of ACM (2006), 78p.; M.P. Schellekens, A Modular Calculus for the Average Cost of Data Structuring, Springer book to appear (2008), 250 pp.] specifically to facilitate the average execution time analysis of its programs. MOQA is based on a special data structure and an associated suite of operations. This special data structure, said here as labeled partial orders (lpo), is a pair (P,ℓ) such that P=(X,⊑) is a poset and ℓ:X↦L a strictly increasing bijection between X and a totally ordered set L. MOQA basic operations are defined on lpo's under some restrictions. These restrictions guarantee the main property of MOQA's operations, said here as Automatic Random Preservation. Random Preservation (RP) is a fundamental property used to calculate the average time of programs, opening the way to a (semi-)automation of this calculation in MOQA.In the paper, we introduce functionally-generalised versions of some MOQA basic operations. Those versions are defined with more natural restrictions and in a more general way than the correspondent MOQA operations. As a consequence of their greater generality, they are not RP on all their domains in contrast to MOQA operations. Nevertheless, they have two good properties. The first one is that they have the intended semantics. For instance, the generalised split actually splits any lpo into two parts using a label as a pivot. The second one is that the subdomains on which they are RP are strictly bigger than the domains where MOQA basic operations are defined and RP. A tractable characterisation of those RP subdomains is an open problem and its solution would be an important step toward an implementation of the functionally-generalised operations as Automatic RP operations.