Kolmogorov Complexity in perspective. Part II: Classification, Information Processing and Duality

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts published in a same volume. Part II is dedicated to the relation between logic and information system, within the scope of Kolmogorov algorithmic information theory. We present a recent application of Kolmogorov complexity: classification using compression, an idea with provocative implementation by authors such as Bennett, Vitanyi and Cilibrasi. This stresses how Kolmogorov complexity, besides being a foundation to randomness, is also related to classification. Another approach to classification is also considered: the so-called "Google classification". It uses another original and attractive idea which is connected to the classification using compression and to Kolmogorov complexity from a conceptual point of view. We present and unify these different approaches to classification in terms of Bottom-Up versus Top-Down operational modes, of which we point the fundamental principles and the underlying duality. We look at the way these two dual modes are used in different approaches to information system, particularly the relational model for database introduced by Codd in the 70's. This allows to point out diverse forms of a fundamental duality. These operational modes are also reinterpreted in the context of the comprehension schema of axiomatic set theory ZF. This leads us to develop how Kolmogorov's complexity is linked to intensionality, abstraction, classification and information system.Comment: 43 page

Variables, Generality and Existence: considerations on the notion of a concept-script

A defense of the Frege / Russell idea of logic as a 'concept=script' or 'ideal language', and a discussion of the relationship of this project to the formalisation of mass nouns or non-count noun

Kantian Philosophy and ‘Linguistic Kantianism’

The expression “linguistic Kantianism” is widely used to refer to ideas about thought and cognition being determined by language — a conception characteristic of 20th century analytic philosophy. In this article, I conduct a comparative analysis of Kant’s philosophy and views falling under the umbrella expression “linguistic Kantianism.” First, I show that “linguistic Kantianism” usually presupposes a relativistic conception that is alien to Kant’s philosophy. Second, I analyse Kant’s treatment of linguistic determinism and the place of his ideas in the 18th century intellectual milieu and provide an overview of relevant contemporary literature. Third, I show that authentic Kantianism and “linguistic Kantianism” belong to two different types of transcendentalism, to which I respectively refer as the “transcendentalism of the subject” and the “transcendentalism of the medium.” The transcendentalism of the subject assigns a central role to the faculties of the cognising subject. The transcendentalism of the medium assigns the role of an “active” element neither to the external world nor to the faculties of the cognising subject, but to something in between — language, in the case of “linguistic Kantianism.” I conclude that the expression “linguistic Kantianism” can be misleading when it comes to the origins of this theory. It would be more appropriate to refer to this theory by the expression “linguistic transcendentalism,” thus avoiding an incorrect reference to Kant

Fexprs as the basis of Lisp function application; or, $vau: the ultimate abstraction

Abstraction creates custom programming languages that facilitate programming for specific problem domains. It is traditionally partitioned according to a two-phase model of program evaluation, into syntactic abstraction enacted at translation time, and semantic abstraction enacted at run time. Abstractions pigeon-holed into one phase cannot interact freely with those in the other, since they are required to occur at logically distinct times. Fexprs are a Lisp device that subsumes the capabilities of syntactic abstraction, but is enacted at run-time, thus eliminating the phase barrier between abstractions. Lisps of recent decades have avoided fexprs because of semantic ill-behavedness that accompanied fexprs in the dynamically scoped Lisps of the 1960s and 70s. This dissertation contends that the severe difficulties attendant on fexprs in the past are not essential, and can be overcome by judicious coordination with other elements of language design. In particular, fexprs can form the basis for a simple, well-behaved Scheme-like language, subsuming traditional abstractions without a multi-phase model of evaluation. The thesis is supported by a new Scheme-like language called Kernel, created for this work, in which each Scheme-style procedure consists of a wrapper that induces evaluation of operands, around a fexpr that acts on the resulting arguments. This arrangement enables Kernel to use a simple direct style of selectively evaluating subexpressions, in place of most Lisps\u27 indirect quasiquotation style of selectively suppressing subexpression evaluation. The semantics of Kernel are treated through a new family of formal calculi, introduced here, called vau calculi. Vau calculi use direct subexpression-evaluation style to extend lambda calculus, eliminating a long-standing incompatibility between lambda calculus and fexprs that would otherwise trivialize their equational theories. The impure vau calculi introduce non-functional binding constructs and unconventional forms of substitution. This strategy avoids a difficulty of Felleisen\u27s lambda-v-CS calculus, which modeled impure control and state using a partially non-compatible reduction relation, and therefore only approximated the Church-Rosser and Plotkin\u27s Correspondence Theorems. The strategy here is supported by an abstract class of Regular Substitutive Reduction Systems, generalizing Klop\u27s Regular Combinatory Reduction Systems