Chapter 3: Methods of Inference

Objectives Learn the definitions of trees, lattices, and graphs Learn about state and problem spaces Learn about AND-OR trees and goals Explore different methods and rules of inference Learn the characteristics of first-order predicate logic and logic systems

Objectives Discuss the resolution rule of inference, resolution systems, and deduction Compare shallow and causal reasoning How to apply resolution to first-order predicate logic Learn the meaning of forward and backward chaining

Objectives Explore additional methods of inference Learn the meaning of Metaknowledge Explore the Markov decision process

Trees A tree is a hierarchical data structure consisting of: Nodes – store information Branches – connect the nodes The top node is the root, occupying the highest hierarchy. The leaves are at the bottom, occupying the lowest hierarcy.

Trees Every node, except the root, has exactly one parent. Every node may give rise to zero or more child nodes. A binary tree restricts the number of children per node to a maximum of two. Degenerate trees have only a single pathway from root to its one leaf.

Figure 3.1 Binary Tree

Graphs Graphs are sometimes called a network or net. A graph can have zero or more links between nodes – there is no distinction between parent and child. Sometimes links have weights – weighted graph; or, arrows – directed graph. Simple graphs have no loops – links that come back onto the node itself.

Graphs A circuit (cycle) is a path through the graph beginning and ending with the same node. Acyclic graphs have no cycles. Connected graphs have links to all the nodes. Digraphs are graphs with directed links. Lattice is a directed acyclic graph. A Degenerate tree is a tree with only a single path from the root to its one leaf.

Figure 3.2 Simple Graphs

Making Decisions Trees / lattices are useful for classifying objects in a hierarchical nature. Trees / lattices are useful for making decisions. We refer to trees / lattices as structures. Decision trees are useful for representing and reasoning about knowledge.

Binary Decision Trees Every question takes us down one level in the tree. A binary decision tree having N nodes: All leaves will be answers. All internal nodes are questions. There will be a maximum of 2N answers for N questions. Decision trees can be self learning. Decision trees can be translated into production rules.

Decision Tree Example

State and Problem Spaces A state space can be used to define an object’s behavior. Different states refer to characteristics that define the status of the object. A state space shows the transitions an object can make in going from one state to another.

Finite State Machine A FSM is a diagram describing the finite number of states of a machine. At any one time, the machine is in one particular state. The machine accepts input and progresses to the next state. FSMs are often used in compilers and validity checking programs.

Using FSM to Solve Problems Characterizing ill-structured problems – one having uncertainties. Well-formed problems: Explicit problem, goal, and operations are known Deterministic – we are sure of the next state when an operator is applied to a state. The problem space is bounded. The states are discrete.

Figure 3.5 State Diagram for a Soft Drink Vending Machine Accepting Quarters (Q) and Nickels (N)

AND-OR Trees and Goals 1990s, PROLOG was used for commercial applications in business and industry. PROLOG uses backward chaining to divide problems into smaller problems and then solves them. AND-OR trees also use backward chaining. AND-OR-NOT lattices use logic gates to describe problems.

Types of Logic Deduction – reasoning where conclusions must follow from premises (general to specific) Induction – inference is from the specific case to the general Intuition – no proven theory-Recognizing a pattern(unconsciously) ANN Heuristics – rules of thumb based on experience Generate and test – trial and error – often used to reach efficiency.

Types of Logic Abduction – reasoning back from a true condition to the premises that may have caused the condition Default – absence of specific knowledge Autoepistemic – self-knowledge…The color of the sky as it appears to you. Nonmonotonic – New evidence may invalidate previous knowledge Analogy – inferring conclusions based on similarities with other situations  ANN Commonsense knowledge – A combination of all based on our experience

Deductive Logic Argument – group of statements where the last is justified on the basis of the previous ones Deductive logic can determine the validity of an argument. Syllogism – has two premises and one conclusion Deductive argument – conclusions reached by following true premises must themselves be true

Syllogisms vs. Rules Syllogism: IF-THEN rule: All basketball players are tall. Jason is a basketball player. Jason is tall. IF-THEN rule: IF All basketball players are tall and Jason is a basketball player THEN Jason is tall.

Categorical Syllogism Premises and conclusions are defined using categorical statements of the form:

Categorical Syllogisms

Categorical Syllogisms

Proving the Validity of Syllogistic Arguments Using Venn Diagrams If a class is empty, it is shaded. Universal statements, A and E are always drawn before particular ones. If a class has at least one member, mark it with an *. If a statement does not specify in which of two adjacent classes an object exists, place an * on the line between the classes. If an area has been shaded, no * can be put in it. Review pages 131 to 135

Proving the Validity of Syllogistic Arguments Using Venn Diagrams Invalid Valid

Proving the Validity of Syllogistic Arguments Using Venn Diagrams

Rules of Inference Venn diagrams are insufficient for complex arguments. Syllogisms address only a small portion of the possible logical statements. Propositional logic offers another means of describing arguments  Variables

Direct Reasoning Modus Ponens

Truth Table Modus Ponens

Some Rules of Inference (Modus Ponens)

Rules of Inference

The Modus Meanings

The Conditional and Its Variants

Requirements of a Formal System An alphabet of symbols A set of finite strings of these symbols, the wffs. Axioms, the definitions of the system. Rules of inference, which enable a wff to be deduced as the conclusion of a finite set of other wffs – axioms or other theorems of the logic system.

Requirements of a FS Continued Completeness – every wff can either be proved or refuted. The system must be sound – every theorem is a logically valid wff.

Logic Systems A logic system consists of four parts: Alphabet: a set of basic symbols from which more complex sentences are made. Syntax: a set of rules or operators for constructing expressions (sentences). Semantics: for defining the meaning of the sentences Inference rules: for constructing semantically equivalent but syntactically different sentences

WFF and Wang’s Propositional Theorem Proofer Well Formed Formula for Propositional Calculus Wang’s Propositional Theorem Proofer Handouts are provided in the class

Predicate Logic Syllogistic logic can be completely described by predicate logic. The Rule of Universal Instantiation states that an individual may be substituted for a universe. Compared to propositional logic, it has predicates, universal and existential quantifies.

Predicate Logic The following types of symbols are allowed in predicate logic: Terms Predicates Connectives Quantifiers

Predicate Logic Terms: Constant symbols: symbols, expressions, or entities which do not change during execution (e.g., true / false) Variable symbols: represent entities that can change during execution Function symbols: represent functions which process input values on a predefined list of parameters and obtain resulting values

Predicate Logic Predicates: Predicate symbols: represent true/false-type relations between objects. Objects are represented by constant symbols.

Predicate Logic Connectives: Conjunction Disjunction Negation Implication Equivalence … (same as for propositional calculus)

Predicate Logic Quantifiers: valid for variable symbols Existential quantifier: “There exists at least one value for x from its domain.” Universal quantifier: “For all x in its domain.”

First-Order Logic First-order logic allows quantified variables to refer to objects, but not to predicates or functions. For applying an inference to a set of predicate expressions, the system has to process matches of expressions. The process of matching is called unification.

PROLOG Programming in Logic

PROLOG: Horn Clauses

PROLOG: Facts

Knowledge Representation

PROLOG: Architecture

Family Example: Facts

Family Example: Rules

PROLOG Sample Dialogue

PROLOG Sample Inference

PROLOG Sample Inference

Shallow and Causal Reasoning Experiential knowledge is based on experience. In shallow reasoning, there is little/no causal chain of cause and effect from one rule to another. Advantage of shallow reasoning is ease of programming. Frames are used for causal / deep reasoning. Causal reasoning can be used to construct a model that behaves like the real system.

Chaining Chain – a group of multiple inferences that connect a problem with its solution A chain that is searched / traversed from a problem to its solution is called a forward chain. A chain traversed from a hypothesis back to the facts that support the hypothesis is a backward chain. Problem with backward chaining is find a chain linking the evidence to the hypothesis.

Causal Forward Chaining

Backward Chaining

Some Characteristics of Forward and Backward Chaining

Figure 3.14 Types of Inference

Metaknowledge The Markov decision process (MDP) is a good application to path planning. In the real world, there is always uncertainty, and pure logic is not a good guide when there is uncertainty. A MDP is more realistic in the cases where there is partial or hidden information about the state and parameters, and the need for planning.