Light Ray\u27s

An occasional collection from the master of whimsy

lim+, delta+, and Non-Permutability of beta-Steps

Using a human-oriented formal example proof of the (lim+) theorem, i.e. that the sum of limits is the limit of the sum, which is of value for reference on its own, we exhibit a non-permutability of beta-steps and delta+-steps (according to Smullyan's classification), which is not visible with non-liberalized delta-rules and not serious with further liberalized delta-rules, such as the delta++-rule. Besides a careful presentation of the search for a proof of (lim+) with several pedagogical intentions, the main subject is to explain why the order of beta-steps plays such a practically important role in some calculi.Comment: ii + 36 page

Emergent bipartiteness in a society of knights and knaves

We propose a simple model of a social network based on so-called knights-and-knaves puzzles. The model describes the formation of networks between two classes of agents where links are formed by agents introducing their neighbours to others of their own class. We show that if the proportion of knights and knaves is within a certain range, the network self-organizes to a perfectly bipartite state. However, if the excess of one of the two classes is greater than a threshold value, bipartiteness is not observed. We offer a detailed theoretical analysis for the behaviour of the model, investigate its behaviou r in the thermodynamic limit, and argue that it provides a simple example of a topology-driven model whose behaviour is strongly reminiscent of a first-order phase transitions far from equilibrium.Comment: 12 pages, 5 figure

Towards an efficient prover for the C1 paraconsistent logic

The KE inference system is a tableau method developed by Marco Mondadori which was presented as an improvement, in the computational efficiency sense, over Analytic Tableaux. In the literature, there is no description of a theorem prover based on the KE method for the C1 paraconsistent logic. Paraconsistent logics have several applications, such as in robot control and medicine. These applications could benefit from the existence of such a prover. We present a sound and complete KE system for C1, an informal specification of a strategy for the C1 prover as well as problem families that can be used to evaluate provers for C1. The C1 KE system and the strategy described in this paper will be used to implement a KE based prover for C1, which will be useful for those who study and apply paraconsistent logics.Comment: 16 page

Advanced Modalizing De Dicto and De Re

Lewis’ (1968, 1986) analysis of modality faces a problem in that it appears to confer unintended truth values to certain modal claims about the pluriverse: e.g. ‘It is possible that there are many worlds’ is false when we expect truth. This is the problem of advanced modalizing. Divers (1999, 2002) presents a principled solution to this problem by treating modal modifiers as semantically redundant in some such cases. However, this semantic move does not deal adequately with advanced de re modal claims. Here, we motivate and detail a comprehensive semantics (a la Lewis 1968) for advanced modalizing de dicto and de re. The generalized semantic feature of the initial solution is not redundancy but absence from counterpart-theoretic translations of world-constrictions

From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions

Permissive-Nominal Logic (PNL) extends first-order predicate logic with term-formers that can bind names in their arguments. It takes a semantics in (permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are just term-formers satisfying axioms, and their denotation is functions on nominal atoms-abstraction. Then we have higher-order logic (HOL) and its models in ordinary (i.e. Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full or partial function spaces. This raises the following question: how are these two models of binding connected? What translation is possible between PNL and HOL, and between nominal sets and functions? We exhibit a translation of PNL into HOL, and from models of PNL to certain models of HOL. It is natural, but also partial: we translate a restricted subsystem of full PNL to HOL. The extra part which does not translate is the symmetry properties of nominal sets with respect to permutations. To use a little nominal jargon: we can translate names and binding, but not their nominal equivariance properties. This seems reasonable since HOL---and ordinary sets---are not equivariant. Thus viewed through this translation, PNL and HOL and their models do different things, but they enjoy non-trivial and rich subsystems which are isomorphic

Efficient Implementation and the Product State Representation of Numbers

The relation between the requirement of efficient implementability and the product state representation of numbers is examined. Numbers are defined to be any model of the axioms of number theory or arithmetic. Efficient implementability (EI) means that the basic arithmetic operations are physically implementable and the space-time and thermodynamic resources needed to carry out the implementations are polynomial in the range of numbers considered. Different models of numbers are described to show the independence of both EI and the product state representation from the axioms. The relation between EI and the product state representation is examined. It is seen that the condition of a product state representation does not imply EI. Arguments used to refute the converse implication, EI implies a product state representation, seem reasonable; but they are not conclusive. Thus this implication remains an open question.Comment: Paragraph in page proof for Phys. Rev. A revise