Preprints of Proceedings of GWAI-92

This is a preprint of the proceedings of the German Workshop on Artificial Intelligence (GWAI) 1992. The final version will appear in the Lecture Notes in Artificial Intelligence

Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors

Boundary algebra [BA] is a simpler notation for Spencer-Brown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms ‘()’ and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are "()()=()" (A1), and "(()) [=?] may be written or erased at will” (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),?} called its simplification. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. The BA basis abc= bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies normal forms and Quine’s (1982) truth value analysis. () ? true [false] yields boundary logic.G. Spencer Brown; boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; C.S. Peirce; existential graphs.

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

Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives

Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs

Agoric computation: trust and cyber-physical systems

In the past two decades advances in miniaturisation and economies of scale have led to the emergence of billions of connected components that have provided both a spur and a blueprint for the development of smart products acting in specialised environments which are uniquely identifiable, localisable, and capable of autonomy. Adopting the computational perspective of multi-agent systems (MAS) as a technological abstraction married with the engineering perspective of cyber-physical systems (CPS) has provided fertile ground for designing, developing and deploying software applications in smart automated context such as manufacturing, power grids, avionics, healthcare and logistics, capable of being decentralised, intelligent, reconfigurable, modular, flexible, robust, adaptive and responsive. Current agent technologies are, however, ill suited for information-based environments, making it difficult to formalise and implement multiagent systems based on inherently dynamical functional concepts such as trust and reliability, which present special challenges when scaling from small to large systems of agents. To overcome such challenges, it is useful to adopt a unified approach which we term agoric computation, integrating logical, mathematical and programming concepts towards the development of agent-based solutions based on recursive, compositional principles, where smaller systems feed via directed information flows into larger hierarchical systems that define their global environment. Considering information as an integral part of the environment naturally defines a web of operations where components of a systems are wired in some way and each set of inputs and outputs are allowed to carry some value. These operations are stateless abstractions and procedures that act on some stateful cells that cumulate partial information, and it is possible to compose such abstractions into higher-level ones, using a publish-and-subscribe interaction model that keeps track of update messages between abstractions and values in the data. In this thesis we review the logical and mathematical basis of such abstractions and take steps towards the software implementation of agoric modelling as a framework for simulation and verification of the reliability of increasingly complex systems, and report on experimental results related to a few select applications, such as stigmergic interaction in mobile robotics, integrating raw data into agent perceptions, trust and trustworthiness in orchestrated open systems, computing the epistemic cost of trust when reasoning in networks of agents seeded with contradictory information, and trust models for distributed ledgers in the Internet of Things (IoT); and provide a roadmap for future developments of our research

Advanced Proof Viewing in ProofTool

Sequent calculus is widely used for formalizing proofs. However, due to the proliferation of data, understanding the proofs of even simple mathematical arguments soon becomes impossible. Graphical user interfaces help in this matter, but since they normally utilize Gentzen's original notation, some of the problems persist. In this paper, we introduce a number of criteria for proof visualization which we have found out to be crucial for analyzing proofs. We then evaluate recent developments in tree visualization with regard to these criteria and propose the Sunburst Tree layout as a complement to the traditional tree structure. This layout constructs inferences as concentric circle arcs around the root inference, allowing the user to focus on the proof's structural content. Finally, we describe its integration into ProofTool and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785

Equity price predictions of selected African emerging markets

Abstract : Predicting equity share prices could be useful to various stakeholders. The common methods used to forecast equity share price besides the naïve model are the Autoregressive Conditional Heteroskedasticity (ARCH) and General Autoregressive Conditional Heteroskedasticity (GARCH) models, however, no conclusion has been reached as to which model produces the most accurate predictions. In this research, ARCH and GARCH forecasting models (and their extended variants), as well as the Monte Carlo Simulation, were used to forecast price-weighted equity indices that were constructed from the South African, Nigerian, and Kenyan share markets. These three countries were selected based on their significance in the African continent due to the relative size of their economies and the liquidity of their share markets. The daily closing share prices for companies listed on the FTSE/JSE Top 40 Index, NSE Top 30 Index, and the NrSE Top 20 Index were collected between the 4th of January 2010 and the 30th of June 2015. The companies that were selected from each of these indices to construct the price-weighted indices for each country, were based on criteria to eliminate bias. Different autoregressive models were fitted for the mean equation. The EViews statistical programme was used to analyse the data. The ARCH effects were tested using the ARCH LM test. The ARCH/GARCH family models selected were GARCH (2,1), EGARCH (2,2), and EGARCH (2,1) for Nigeria, Kenya, and South Africa respectively. A Monte Carlo Simulation with 1 200 iterations was also performed to forecast the equity share prices. Post estimation and performance evaluation metrics were performed using the RMSE, MSE, MAD, and MAPE. The results based on the evaluation metrics indicated that the ARCH/GARCH models in-sample forecasts were more accurate than out-of-sample forecasts. The accuracy of the ARCH/GARCH models’ predictions was sounder than that of the Monte Carlo Simulation based on the evaluation metrics. Comparing the forecasting models to the actual graphs, in most cases the ARCH/GARCH models were closer to the actuals than the Monte Carlo II Simulation. The accuracy of the model predictions were also influenced by the sample size, the nature of the data, the leverage effect, and the macro economic conditions. In conclusion, the African equity markets cannot be predicted accurately using the ARCH/GARCH models and the Monte Carlo Simulation. The predictions from the forecasting models are not sufficiently accurate for investors, traders, and company management to use to make informed decisions. However, these predictions are better than the naïve model. The researcher also concluded that the markets are efficient, as the publicly available information cannot be used to gain abnormal returns. This study’s findings are similar to those of previous studies carried out in South Africa and globally.M.Com. (Finance

Spatial Aggregation: Theory and Applications

Visual thinking plays an important role in scientific reasoning. Based on the research in automating diverse reasoning tasks about dynamical systems, nonlinear controllers, kinematic mechanisms, and fluid motion, we have identified a style of visual thinking, imagistic reasoning. Imagistic reasoning organizes computations around image-like, analogue representations so that perceptual and symbolic operations can be brought to bear to infer structure and behavior. Programs incorporating imagistic reasoning have been shown to perform at an expert level in domains that defy current analytic or numerical methods. We have developed a computational paradigm, spatial aggregation, to unify the description of a class of imagistic problem solvers. A program written in this paradigm has the following properties. It takes a continuous field and optional objective functions as input, and produces high-level descriptions of structure, behavior, or control actions. It computes a multi-layer of intermediate representations, called spatial aggregates, by forming equivalence classes and adjacency relations. It employs a small set of generic operators such as aggregation, classification, and localization to perform bidirectional mapping between the information-rich field and successively more abstract spatial aggregates. It uses a data structure, the neighborhood graph, as a common interface to modularize computations. To illustrate our theory, we describe the computational structure of three implemented problem solvers -- KAM, MAPS, and HIPAIR --- in terms of the spatial aggregation generic operators by mixing and matching a library of commonly used routines.Comment: See http://www.jair.org/ for any accompanying file