Weng Kin Ho

A Category Theoretical Argument against the Possibility of Artificial Life: Robert Rosen's Central Proof Revisited

One of Robert Rosens main contributions to the scientific community is summarized in his book Life itself. There Rosen presents a theoretical framework to define living systems; given this definition, he goes on to show that living systems are not realizable in computational universes. Despite being well known and often cited, Rosens central proof has so far not been evaluated by the scientific community. In this article we review the essence of Rosens ideas leading up to his rejection of the possibility of real artificial life in silico. We also evaluate his arguments and point out that some of Rosens central notions are ill defined. The conclusion of this article is that Rosens central proof is wrong.

The Localization Hypothesis and Machines

In a recent article in Artificial Life, Chu and Ho suggested that Rosens central result about the simulability of living systems might be flawed. This argument was later declared null and void by Louie. In this article the validity of Louies objections are examined.

Computational Realizations of Living Systems

Robert Rosens central theorem states that organisms are fundamentally different from machines, mainly because they are closed with respect to effcient causation. The proof for this theorem rests on two crucial assumptions. The first is that for a certain class of systems (mechanisms) analytic modeling is the inverse of synthetic modeling. The second is that aspects of machines can be modeled using relational models and that these relational models are themselves refined by at least one analytic model. We show that both assumptions are unjustified. We conclude that these results cast serious doubts on the validity of Rosens proof.

Operational domain theory and topology of sequential programming languages

A number of authors have exported domain-theoretic techniques from denotational semantics to the operational study of contextual equivalence and preorder. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact set and show that total programs with values on certain types are uniformly continuous on compact sets of total elements. We apply this and other conclusions to prove the correctness of non-trivial programs that manipulate infinite data. What is interesting is that the development applies to sequential programming languages.

On topologies defined by irreducible sets

Abstract In this paper, we define and study a new topology constructed from any given topology on a set, using irreducible sets. The manner in which this derived topology is obtained is inspired by how the Scott topology on a poset is constructed from its Alexandroff topology. This derived topology leads us to a weak notion of sobriety called k-bounded sobriety. We investigate the properties of this derived topology and k-bounded sober spaces. A by-product of our theory is a novel type of compactness, which involves crucially the Scott irreducible families of open sets. Some related applications on posets are also given.

Exponential function and its derivative revisited

Most of the available proofs for rely on results involving either power series, uniform convergence or a round-about definition of the natural logarithm function ln(x) by the definite integral , and are thus not readily accessible by high school teachers and students. Even instructors of calculus courses avoid showing the complete proof to their undergraduate students because a direct and elementary approach is missing. This short article fills in this gap by supplying a simple proof of the aforementioned basic calculus fact.

An elementary proof of the identity

This article gives an elementary proof of the famous identity Using nothing more than freshman calculus, the present proof is far simpler than many existing ones. This result also leads directly to Eulers and Nevilles identities, as well as the identity .

On the largest outscribed equilateral triangle

An outscribed triangle of a triangle 4ABC is a triangle 4DEF such that each side of 4DEF contains a vertex of 4ABC. In this article we study the equilateral outscribed triangles of an arbitrary triangle and determine the area of the largest such triangles. We prove that the largest outscribed equilateral triangle of 4ABC can be constructed by ruler and compass and its area equals a +b+c 2 √ 3 +2S4ABC where S4ABC denotes the area of 4ABC. Given two triangles 4ABC and 4DEF , if each side of 4DEF contains a vertex of 4ABC, then we call 4DEF an outscribed triangle of 4ABC. Given 4ABC, let Φ4ABC be the set of all outscibed equilateral triangles of 4ABC. Clearly Φ4ABC is non-empty. In the following we will determine the area of the largest member of Φ4ABC and show that this largest member can be constructed by ruler and compass from 4ABC. The corresponding problem on quadrilaterals has been considered in [1]. 1. Area of the largest outscribed equilateral triangle Given a triangle 4ABC, let a and b denote the lengths of the sides BC and AC, respectively, and θ denote the angle ∠ACB. Let 4DEF be any member in Φ4ABC as shown in Figure 1 and put t = ∠DCB.

An operational domain-theoretic treatment of recursive types

We develop an operational domain theory for treating recursive types with respect to contextual equivalence. The principal approach we take deviates from classical domain theory in that we do not produce the recursive types using the usual inverse limits constructions – we get them for free by working directly with the operational semantics. By extending type expressions to functors between some ‘syntactic’ categories, we establish algebraic compactness. To do this, we rely on an operational version of the minimal invariance property, for which we give a purely operational proof.

Characterising E-projectives via Comonads

This paper demonstrates the usefulness of a comonadic approach to give previously unknown characterisation of projective objects in certain categories over particular subclasses of epimorphisms. This approach is a simple adaptation of a powerful technique due to Escardo which has been used extensively to characterise injective spaces and locales over various kinds of embeddings, but never previously for projective structures. Using some examples, we advertise the versatility of this approach – in particular, highlighting its advantage over existing methods on characterisation of projectives, which is that the comonadic machinery forces upon us the structural properties of projectives without relying on extraneous characterisations of the underlying object of the coalgebra arising from the comonad.