Reasoning is one of the important core capabilities mentioned in the Turing test for intelligence. Reasoning allows an agent to infer or deduce new information from information already known to be true. Reasoning occurs in different "logics": languages of varying power and other constructs to describe the known characteristics, which include Boolean or propositional logic and first-order logic (FOL). FOL is better capable of compactly describing known information about a system. This unit will help you understand these logics and how inference is carried out in propositional logic and FOL. Finally, we will look at uncertainty models such as Bayesian analysis, Bayesian networks, and Markov Chains, which have proved better than logic-based systems in modeling uncertainty and predicting probabilistic outcomes.
Completing this unit should take you approximately 8 hours.
Propositional or Boolean logic is a simple (but limited) notation to describe the knowledge associated with different problem domains. Because the notation is limited, describing complex systems using propositional logic can take time and effort. As we review, we will refer to principles such as modus ponens (forward and backward chaining) and resolution over propositional logic.
The simplest language for logical reasoning is propositional logic (PL), also called boolean logic. The range of allowable constructs in PL includes symbols representing some fact or event in the world (which can be true or false) and logical connectives like not, and, or, and implications (also called "if-then statements").
PL can work as a simple language to describe facts (also known as truths or axioms) about the domain of discourse. PL can be used to describe aspects of various systems and domains. Reasoning in PL happens through inference procedures. This is, in general, a very complex topic. Inferences can be drawn by repeated use of modus ponens rules to deduce new facts that must be true if the antecedents of the rules are true. As you read, focus on the general ideas and examples. If you are interested, you can go into the technical details on a second, deeper reading.
First-order logic is a powerful (but fairly complex) notation to describe the knowledge associated with different problem domains. FOL is a far better notation than propositional logic to describe systems, but it can be more challenging to follow or understand. Using first-order logic, we will review inference principles such as modus ponens (forward and backward chaining) and resolution.
FOL is often called the language of computer science. It has far more powerful constructs than PL and is more expressive. Unlike PL, FOL describes objects and their inter-relationships and incorporates the concept of quantifiers. Quantifiers allow you to express properties shared (or not shared) by sets of objects.
In general, FOL is a complex topic. For now, focus on the high-level concepts. Like PL, FOL is also used to describe the details of a system or domain. But, because it has quantifiers and relationships, the hypotheses can be more compactly stated as "well formed formulae (wffs)" or sentences in FOL. Inference in FOL is a very complex subject. Because the notation is more powerful, proving things to be true (or false) in FOL is computationally intensive. Deductions are made in FOL using various principles like modus ponens and resolution.
An inherent part of intelligence is being able to handle uncertainty effectively. Specifically, we will discuss the framework of conditional probability and use Bayes' theorem as the foundation to model the influence of variables on outcomes. Using Bayes' rule, we can probabilistically predict the strategy to use.
Probability theory provides a robust and well-understood platform to handle uncertainty. In addition to "prior" probability, it is also useful to master conditional probability to sharpen our ability to reason about uncertain events. Can you explain how conditional probability works and how to analyze the likelihood of events with some apparent dependence on one another?
One of the common ways to use conditional probability is through Bayes' Theorem. The definition of conditional probability is used in Bayes' Theorem to render inferences in many situations where events are causally linked.
A Bayesian network can model causal relationships probabilistically. Given certain evidence, Bayesian analysis can probabilistically predict the explanation for the evidence. Markov chains and hidden Markov chains are formal ways to model uncertainty in dynamic systems that change state in specific ways.
The Bayesian Network is an easy-to-understand graphical notation representing the conditional inter-dependence of variables within a system. This simple graphical formalism can leverage conditional probability distributions to describe relationships between variables in a system. How can Bayesian networks compute the probabilities of specific events given other facts? Humans also use this kind of reasoning to render decisions in uncertain environments.
Markov chains are one of the most common formalisms to describe event probabilities within a system where the next state is determined only by the current state but not by how the current state was achieved.
In hidden Markov chains, the system's behavior depends on latent (or hidden) variables. This has a lot of applications in contemporary AI. For now, focus on grasping the high-level themes and ideas. If the subject interests you, you can dive deeper into technical details. The examples are particularly instructive.