Logic

Outline
• Knowledge-based agents
• Logic in general
• Propositional (Boolean) logic
• Equivalence, validity, satisfiability

Knowledge bases
• Knowledge base = set of sentences in a formal language
• Declarative approach to building an agent (or other system):
– Tell it what it needs to know
• Then it can Ask itself what to do - answers should follow from the
KB
• Agents can be viewed at the knowledge level
i.e., what they know, regardless of how implemented
• Or at the implementation level
– i.e., data structures in KB and algorithms that manipulate them
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A simple knowledge-based agent
• The agent must be able to:
– Represent states, actions, etc.
– Incorporate new percepts
– Deduce appropriate actions
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Update internal representations of the world
– Deduce hidden properties of the world
–

Logic in general
• Logics are formal languages for representing information
such that conclusions can be drawn
• Syntax defines the sentences in the language
• Semantics define the "meaning" of sentences;
– i.e., define truth of a sentence in a world
• E.g., the language of arithmetic
– x+2 ≥ y is true iff the number x+2 is no less than the number y
– x+2 ≥ y is true in a world where x = 7, y = 1
– x+2 ≥ y is false in a world where x = 0, y = 6
–
–
»x+2 ≥ y is a sentence; x2+y > {} is not
a sentence

Propositional logic: Syntax
• Propositional logic is the simplest logic – illustrates
basic ideas
• The proposition symbols P1, P2 etc are sentences
– If S is a sentence, ØS is a sentence (negation)
– If S1 and S2 are sentences, S1 Ù S2 is a sentence (conjunction)
– If S1 and S2 are sentences, S1 Ú S2 is a sentence (disjunction)
– If S1 and S2 are sentences, S1 Þ S2 is a sentence (implication)
– If S1 and S2 are sentences, S1 Û S2 is a sentence (biconditional)
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Propositional logic: Semantics
Each model specifies true/false for each proposition symbol
E.g. P1,2 P2,2 P3,1
false true false
With these symbols, 8 possible models, can be enumerated automatically.
Rules for evaluating truth with respect to a model m:
ØS is true iff S is false
S1 Ù S2 is true iff S1 is true and S2 is true
S1 Ú S2 is true iff S1is true or S2 is true
S1 Þ S2 is true iff S1 is false or S2 is true
i.e., is false iff S1 is true and S2 is false
S1 Û S2 is true iff S1ÞS2 is true andS2ÞS1 is true
Simple recursive process evaluates an arbitrary sentence, e.g.,
ØP1,2 Ù (P2,2 Ú P3,1) = true Ù (true Ú false) = true Ù true = true

Wumpus world sentences
Let Pi,j be true if there is a pit in [i, j].
Let Bi,j be true if there is a breeze in [i, j].
Ø P1,1
ØB1,1
B2,1
• "Pits cause breezes in adjacent squares"
B1,1 Û (P1,2 Ú P2,1)
B2,1 Û (P1,1 Ú P2,2 Ú P3,1)
»

Logical equivalence
• Two sentences are logically equivalent} iff true in same
models: α ≡ ß iff α╞ β and β╞ α
•
•

Validity and satisfiability
A sentence is valid if it is true in all models,
e.g., True, A ÚØA, A Þ A, (A Ù (A Þ B)) Þ B
Validity is connected to inference via the Deduction Theorem:
KB ╞ α if and only if (KB Þ α) is valid
A sentence is satisfiable if it is true in some model
e.g., AÚ B, C
A sentence is unsatisfiable if it is true in no models
e.g., AÙØA
Satisfiability is connected to inference via the following:
KB ╞ α if and only if (KB ÙØα) is unsatisfiable
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Summary
• Logical agents apply inference to a knowledge base to derive new
information and make decisions
• Basic concepts of logic:
– syntax: formal structure of sentences
– semantics: truth of sentences wrt models
– entailment: necessary truth of one sentence given another
– inference: deriving sentences from other sentences
– soundness: derivations produce only entailed sentences
– completeness: derivations can produce all entailed sentences
• Wumpus world requires the ability to represent partial and negated
information, reason by cases, etc.
• Resolution is complete for propositional logic
Forward, backward chaining are linear-time, complete for Horn
clauses
• Propositional logic lacks expressive power
•
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Logic

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