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

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

Rejection in Łukasiewicz's and Słupecki's Sense

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic

Roots and (re)sources of value (in)definiteness versus contextuality. A contribution to the Pitowsky Volume in memory of Itamar Pitowsky (1950--2010)

In Itamar Pitowsky's reading of the Gleason and the Kochen-Specker theorems, in particular, his Logical Indeterminacy Principle, the emphasis is on the value indefiniteness of observables which are not within the preparation context. This is in stark contrast to the prevalent term {\em contextuality} used by many researchers in informal, heuristic yet omni-realistic and potentially misleading ways. This paper discusses both concepts and argues in favor of value indefiniteness in all but a continuum of contexts intertwining in the vector representing a single pure (prepared) state. Even more restrictively, and inspired by operationalism but not justified by Pitowsky's Logical Indeterminacy Principle or similar, one could identify with a "quantum state" a single quantum context -- aka the respective maximal observable, or, in terms of its spectral decomposition, the associated orthonormal basis - from the continuum of intertwining context, as per the associated maximal observable actually or implicitly measured.Comment: 11 pages, revised and polished, discussion on joint probabilities of observables in different contexts adde