Computing with Infinite Objects: the Gray Code Case

Infinite Gray code has been introduced by Tsuiki as a redundancy-free representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also H. Tsuiki, K. Sugihara, "Streams with a bottom in functional languages"). Berger and Tsuiki reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates \bS and \bG are defined and the inclusion \bS \subseteq \bG is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version $\S_{2}$ of $S$ and \bG \subseteq \bS_{2} is shown. What one is looking for, however, is an equivalence proof of the involved concepts. One of the main aims of the present paper is to close the gap. A concurrent version \bG^{*} of \bG and a modification \bS^{*} of \bS_{2} are presented such that \bS^{*} = \bG^{*}. A crucial tool in U. Berger, H. Tsuiki, "Intuitionistic fixed point logic" is a formulation of the Archimedean property of the real numbers as an induction principle. We introduce a concurrent version of this principle which allows us to prove that \bS^{*} and \bG^{*} coincide. A further central contribution is the extension of the above results to the hyperspace of non-empty compact subsets of the reals

Computing with Infinite Objects: the Gray Code Case

Infinite Gray code has been introduced by Tsuiki as a redundancy-free representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also H. Tsuiki, K. Sugihara, "Streams with a bottom in functional languages"). Berger and Tsuiki reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates \bS and \bG are defined and the inclusion \bS⊆\bG is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version §2 of S and \bG⊆\bS2 is shown. What one is looking for, however, is an equivalence proof of the involved concepts. One of the main aims of the present paper is to close the gap. A concurrent version \bG∗ of \bG and a modification \bS∗of \bS2 are presented such that \bS∗=\bG∗. A crucial tool in U. Berger, H. Tsuiki, "Intuitionistic fixed point logic" is a formulation of the Archimedean property of the real numbers as an induction principle. We introduce a concurrent version of this principle which allows us to prove that \bS∗ and \bG∗ coincide. A further central contribution is the extension of the above results to the hyperspace of non-empty compact subsets of the reals

Computing with Infinite Objects: the Gray Code Case

Infinite Gray code has been introduced by Tsuiki as a redundancy-free representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also H. Tsuiki, K. Sugihara, "Streams with a bottom in functional languages"). Berger and Tsuiki reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates \bS and \bG are defined and the inclusion \bS \subseteq \bG is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version $\S_{2}$ of $S$ and \bG \subseteq \bS_{2} is shown. What one is looking for, however, is an equivalence proof of the involved concepts. One of the main aims of the present paper is to close the gap. A concurrent version \bG^{*} of \bG and a modification \bS^{*} of \bS_{2} are presented such that \bS^{*} = \bG^{*}. A crucial tool in U. Berger, H. Tsuiki, "Intuitionistic fixed point logic" is a formulation of the Archimedean property of the real numbers as an induction principle. We introduce a concurrent version of this principle which allows us to prove that \bS^{*} and \bG^{*} coincide. A further central contribution is the extension of the above results to the hyperspace of non-empty compact subsets of the reals

Infinitely Complex Machines

Infinite machines (IMs) can do supertasks. A supertask is an infinite series of operations done in some finite time. Whether or not our universe contains any IMs, they are worthy of study as upper bounds on finite machines. We introduce IMs and describe some of their physical and psychological aspects. An accelerating Turing machine (an ATM) is a Turing machine that performs every next operation twice as fast. It can carry out infinitely many operations in finite time. Many ATMs can be connected together to form networks of infinitely powerful agents. A network of ATMs can also be thought of as the control system for an infinitely complex robot. We describe a robot with a dense network of ATMs for its retinas, its brain, and its motor controllers. Such a robot can perform psychological supertasks - it can perceive infinitely detailed objects in all their detail; it can formulate infinite plans; it can make infinitely precise movements. An endless hierarchy of IMs might realize a deep notion of intelligent computing everywhere

The Role of the Environment in Tissue P Systems with Cell Division

Classical tissue P systems with cell division have a special alphabet whose elements appear at the initial configuration of the system in an arbitrary large number of copies. These objects are shared in a distinguished place of the system, called the environment. Besides, the ability of these computing devices to have infinite copies of some objects has been widely exploited in the design of efficient solutions to computationally hard problems. This paper deals with computational aspects of tissue P systems with cell division where there is not an environment having the property mentioned above. Specifically, we establish the relationships between the polynomial complexity class associated with tissue P systems with cell division and with or without environment. As a consequence, we prove that it is not necessary to have infinite copies of some objects at the initial configuration in order to solve NP–complete problems in an efficient way.Ministerio de Ciencia e Innovación TIN2009-13192Junta de Andalucía P08 – TIC 0420

Constructive Preference Elicitation over Hybrid Combinatorial Spaces

Preference elicitation is the task of suggesting a highly preferred configuration to a decision maker. The preferences are typically learned by querying the user for choice feedback over pairs or sets of objects. In its constructive variant, new objects are synthesized "from scratch" by maximizing an estimate of the user utility over a combinatorial (possibly infinite) space of candidates. In the constructive setting, most existing elicitation techniques fail because they rely on exhaustive enumeration of the candidates. A previous solution explicitly designed for constructive tasks comes with no formal performance guarantees, and can be very expensive in (or unapplicable to) problems with non-Boolean attributes. We propose the Choice Perceptron, a Perceptron-like algorithm for learning user preferences from set-wise choice feedback over constructive domains and hybrid Boolean-numeric feature spaces. We provide a theoretical analysis on the attained regret that holds for a large class of query selection strategies, and devise a heuristic strategy that aims at optimizing the regret in practice. Finally, we demonstrate its effectiveness by empirical evaluation against existing competitors on constructive scenarios of increasing complexity.Comment: AAAI 2018, computing methodologies, machine learning, learning paradigms, supervised learning, structured output

Integration of the kenzo system within sagemath for new algebraic topology computations

This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before

Limits on Efficient Computation in P Systems with Symport/Antiport Rules

Classical membrane systems with symport/antiport rules observe the con- servation law, in the sense that they compute by changing the places of objects with respect to the membranes, and not by changing the objects themselves. In these systems the environment plays an active role because the systems not only send objects to the environment, but also bring objects from the environment. In the initial configuration of a system, there is a special alphabet whose elements appear in an arbitrary large number of copies. The ability of these computing devices with infinite copies of some objects has been widely exploited in the design of efficient solutions to computationally hard problems. This paper deals with computational aspects of P systems with symport/antiport rules and membrane division rules or membrane separation rules. Specifically, we study the limitations of such P systems when the only communication rules allowed have length 1.Ministerio de Ciencia e Innovación TIN2012-3743