Abdul Rahim Nizamani

Bounded kolmogorov complexity based on cognitive models

Computable versions of Kolmogorov complexity have been used in the context of pattern discovery [1]. However, these complexity measures do not take the psychological dimension of pattern discovery into account. We propose a method for pattern discovery based on a version of Kolmogorov complexity where computations are restricted to a cognitive model with limited computational resources. The potential of this method is illustrated by implementing it in a system used to solve number sequence problems. The system was tested on the number sequence problems of the IST IQ test [2], and it scored 28 out of 38 problems, above average human performance, whereas the mathematical software packages Maple, Mathematica, and WolframAlpha scored 9, 9, and 12, respectively. The results obtained and the generalizability of the method suggest that this version of Kolmogorov complexity is a useful tool for pattern discovery in the context of AGI.

Reasoning About Truth in First-Order Logic

First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof systems when capacity limits are put on the cognitive resources. Finally, we investigate the correlation between a number of mathematical complexity measures defined on graphs and sentences and some psychological complexity measures that were recorded in the experiment.

Bounded Cognitive Resources and Arbitrary Domains

When Alice in Wonderland fell down the rabbit hole, she entered a world that was completely new to her. She gradually explored that world by observing, learning, and reasoning. This paper presents a simple system Alice in Wonderland that operates analogously. We model Alices Wonderland via a general notion of domain and Alice herself with a computational model including an evolving belief set along with mechanisms for observing, learning, and reasoning. The system operates autonomously, learning from arbitrary streams of facts from symbolic domains such as English grammar, propositional logic, and simple arithmetic. The main conclusion of the paper is that bounded cognitive resources can be exploited systematically in artificial general intelligence for constructing general systems that tackle the combinatorial explosion problem and operate in arbitrary symbolic domains.

A General System for Learning and Reasoning in Symbolic Domains

We present the system O ⋆ that operates in arbitrary symbolic domains, including arithmetic, logic, and grammar. O ⋆ can start from scratch and learn the general laws of a domain from examples. The main learning mechanism is a formalization of Occam’s razor. Learning is facilitated by working within a cognitive model of bounded rationality. Computational complexity is thereby dramatically reduced, while preserving human-level performance. As illustration, we describe the learning process by which O ⋆ learns elementary arithmetic. In the beginning, O ⋆ knows nothing about the syntax or laws of arithmetic; by the end, it has constructed a theory enabling it to solve previously unseen problems such as “what is 67*8?” and “which number comes next in the sequence 8,11,14?”.

Integrating Symbolic and Sub-symbolic Reasoning

This paper proposes a way of bridging the gap between symbolic and sub-symbolic reasoning. More precisely, it describes a developing system with bounded rationality that bases its decisions on sub-symbolic as well as symbolic reasoning. The system has a fixed set of needs and its sole goal is to stay alive as long as possible by satisfying those needs. It operates without pre-programmed knowledge of any kind. The learning mechanism consists of several meta-rules that govern the development of its network-based memory structure. The decision making mechanism operates under time constraints and combines symbolic reasoning, aimed at compressing information, with sub-symbolic reasoning, aimed at planning.

Learning and Reasoning in Unknown Domains

Abstract In the story Alice in Wonderland, Alice fell down a rabbit hole and suddenly found herself in a strange world called Wonderland. Alice gradually developed knowledge about Wonderland by observing, learning, and reasoning. In this paper we present the system Alice In Wonderland that operates analogously. As a theoretical basis of the system, we define several basic concepts of logic in a generalized setting, including the notions of domain, proof, consistency, soundness, completeness, decidability, and compositionality. We also prove some basic theorems about those generalized notions. Then we model Wonderland as an arbitrary symbolic domain and Alice as a cognitive architecture that learns autonomously by observing random streams of facts from Wonderland. Alice is able to reason by means of computations that use bounded cognitive resources. Moreover, Alice develops her belief set by continuously forming, testing, and revising hypotheses. The system can learn a wide class of symbolic domains and challenge average human problem solvers in such domains as propositional logic and elementary arithmetic.

Integrating Axiomatic and Analogical Reasoning

We present a computational model of a developing system with bounded rationality that is surrounded by an arbitrary number of symbolic domains. The system is fully automatic and makes continuous observations of facts emanating from those domains. The system starts from scratch and gradually evolves a knowledge base consisting of three parts: (1) a set of beliefs for each domain, (2) a set of rules for each domain, and (3) an analogy for each pair of domains. The learning mechanism for updating the knowledge base uses rote learning, inductive learning, analogy discovery, and belief revision. The reasoning mechanism combines axiomatic reasoning for drawing conclusions inside the domains, with analogical reasoning for transferring knowledge from one domain to another. Thus the reasoning processes may use analogies to jump back and forth between domains.