objectives learn the definitions of trees, lattices, and graphs learn about state and problem spaces...
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ObjectivesLearn the definitions of trees, lattices, and graphsLearn about state and problem spacesLearn about AND-OR trees and goalsExplore different methods and rules of inferenceLearn the characteristics of first-order predicate
logic and logic systems
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ObjectivesDiscuss the resolution rule of inference,
resolution systems, and deductionCompare shallow and causal reasoningHow to apply resolution to first-order predicate
logicLearn the meaning of forward and backward
chaining
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ObjectivesExplore additional methods of inference
Learn the meaning of Metaknowledge
Explore the Markov decision process
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TreesA tree is a hierarchical data structure consisting
of:Nodes – store informationBranches – connect the nodes
The top node is the root, occupying the highest hierarchy.
The leaves are at the bottom, occupying the lowest hierarcy.
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TreesEvery 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.
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Figure 3.1 Binary Tree
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GraphsGraphs 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.
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GraphsA 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.
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Figure 3.2 Simple Graphs
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Making DecisionsTrees / 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.
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Binary Decision TreesEvery 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.
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Decision Tree Example
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State and Problem SpacesA 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.
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Finite State MachineA 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.
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Using FSM to Solve Problems
Characterizing ill-structured problems – one having uncertainties.
Well-formed problems:
Explicit problem, goal, and operations are knownDeterministic – we are sure of the next state when an
operator is applied to a state.The problem space is bounded.The states are discrete.
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Figure 3.5 State Diagram for a Soft Drink Vending Machine Accepting Quarters (Q) and Nickels (N)
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AND-OR Trees and Goals1990s, 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.
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Types of LogicDeduction – 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.
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Types of LogicAbduction – reasoning back from a true condition to the
premises that may have caused the conditionDefault – absence of specific knowledgeAutoepistemic – self-knowledge…The color of the sky as
it appears to you.Nonmonotonic – New evidence may invalidate previous
knowledgeAnalogy – inferring conclusions based on similarities
with other situations ANNCommonsense knowledge – A combination of all based
on our experience
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Deductive LogicArgument – 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
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Syllogisms vs. RulesSyllogism:
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.
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Categorical SyllogismPremises and conclusions are defined using
categorical statements of the form:
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Categorical Syllogisms
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Categorical Syllogisms
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Proving the Validity of Syllogistic Arguments Using Venn Diagrams
1. If a class is empty, it is shaded.2. Universal statements, A and E are always drawn
before particular ones.3. If a class has at least one member, mark it with an *.4. If a statement does not specify in which of two
adjacent classes an object exists, place an * on the line between the classes.
5. If an area has been shaded, no * can be put in it.
Review pages 131 to 135
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Proving the Validity of Syllogistic Arguments Using Venn Diagrams
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Invalid
Valid
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Proving the Validity of Syllogistic Arguments Using Venn Diagrams
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Valid
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Rules of InferenceVenn 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
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Direct Reasoning Modus Ponens
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Truth Table Modus Ponens
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Some Rules of Inference
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(Modus Ponens)
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Rules of Inference
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The Modus Meanings
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The Conditional and Its Variants
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Requirements of a Formal System
1. An alphabet of symbols
2. A set of finite strings of these symbols, the wffs.
3. Axioms, the definitions of the system.
4. 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.
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Requirements of a FS Continued
5. Completeness – every wff can either be proved or refuted.
6. The system must be sound – every theorem is a logically valid wff.
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Logic SystemsA 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
sentencesInference rules: for constructing semantically
equivalent but syntactically different sentences
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WFF and Wang’s Propositional Theorem Proofer
Well Formed Formula for Propositional CalculusWang’s Propositional Theorem Proofer
Handouts are provided in the class
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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.
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Predicate LogicThe following types of symbols are
allowed in predicate logic:TermsPredicatesConnectivesQuantifiers
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Predicate LogicTerms: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
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Predicate LogicPredicates:Predicate symbols: represent true/false-type
relations between objects. Objects are represented by constant symbols.
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Predicate LogicConnectives:ConjunctionDisjunctionNegationImplicationEquivalence… (same as for propositional calculus)
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Predicate LogicQuantifiers:valid for variable symbolsExistential quantifier: “There exists at least
one value for x from its domain.”Universal quantifier: “For all x in its domain.”
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First-Order LogicFirst-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.
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PROLOGProgramming in Logic
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PROLOG: Horn Clauses
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PROLOG: Facts
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Knowledge Representation
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PROLOG: Architecture
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Family Example: Facts
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Family Example: Rules
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PROLOG Sample Dialogue
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PROLOG Sample Inference
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PROLOG Sample Inference
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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.
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ChainingChain – a group of multiple inferences that
connect a problem with its solutionA 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.
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Causal Forward Chaining
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Backward Chaining
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Some Characteristics of Forward and Backward Chaining
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Figure 3.14 Types of Inference
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MetaknowledgeThe 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.
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