9/18/2000copyright Brian Williams1 Propositional Logic Brian C. Williams 16.412J/6.834J October 10,...
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Transcript of 9/18/2000copyright Brian Williams1 Propositional Logic Brian C. Williams 16.412J/6.834J October 10,...
9/18/2000 copyright Brian Williams 1
Propositional Logic
Brian C. Williams16.412J/6.834J October 10, 2001
9/18/2000 copyright Brian Williams 2
AssignmentsReadings:• Chapter 6 of AIMA• “Generating hard satisfiability problems” by Selman,
Mitchell and Levesque.• Optional Advanced:
• “Finding Hard Instances of the Satisfiability Problem: A Survey” by Cook and Mitchell.
Handout:• Guidelines for Project
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AssignmentHomework:• Project proposal due Wed, October 17th.• Complete draft of lecture slides Wed, Oct.
17th and pass to another team for feedback.
• Sign up for presentation time.
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Outline
• Why propositional inference?• Propositional logic• Satisfiability by enumeration• Summary
5copyright Brian Williams9/18/2000
Monitors
Autonomous Agents: What is missing?
Command dispatchFault protectionAttitude control
Mission Goal Scenario
Self-commandingSelf-commandingSelf-diagnosingSelf-diagnosingSelf-repairingSelf-repairing
RECOVERY
PLANNIN
G
EXECUTION
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senseP(s)
WORLD
observations actions
AGENT Diagnostic Agent:
• Monitors & Diagnoses
• Repairs & Avoids
• Probes and Tests
Plant
act
Reasons about hidden state
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What is Fault Diagnosis?• Given a system with symptomatic behavior,
find diagnoses that eliminate symptoms.
6
6
12 SymptomM1
M2
M3
A1
A2
A
BCD
E
3
223
3
F
G
X
Y
Z
10
12
Constraint Suspension
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AI Conventional Wisdom
• “[For reactive systems] proving theorems is out of the question” [Agre & Chapman 87]
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Propositional Satisfiability
In recent years, we’ve seen substantial progress in propositional reasoning and search methods.
Boolean satisfiability testing: 1990: 100 variables / 200 clauses (constraints) 1998: 10,000 - 100,000 vars / 10^6 clauses Novel applications: e.g. diagnosis, planning, software / circuit testing, machine learning, and protein folding
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Methodology for Fast Inference
Combinatorial Task Propositional Propositional
Logic Encoding Satisfiability Solver
Decode Results
Shift work to “encoding phase’’, use fast, off-the-shelf SAT solver and tools.
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Outline
• Why propositional inference?• Propositional logic• Satisfiability by enumeration• Summary
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Logic in General
• Logics• 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., truth of a sentence in a world.
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Propositional Logic:Syntax• Proposition
• Statement that is true or false• (valve v1)• (= voltage high)
• Propositional sentence (S)• S ::= proposition |• (NOT S) |• (OR S1 ... Sn) |• (AND S1 ... Sn)
• Defined Constructs• (implies S1 S2) => ((not S1) OR S2)• (IFF S1 S2) => (AND (IMPLIES S1 S2)(IMPLIES S2 S1))• . . .
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Engine Example: propositional logic model(mode(E1) = ok implies (thrust(E1) = on exactly when flow(V1) = on and flow(V2) = on)) and (mode(E1) = ok or mode(E1) = unknown) and not (mode(E1) = ok and mode(E1) = unknown)
E1
V1 V2
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Propositional Logic: SemanticsA model assigns true/false to every proposition symbol P i
• A = True, B = False, C = False
Truth determined by applying boolean operators to model:• Not S is True iff S is False
• S1 and S2 is True iff S1 is True and S2 is True
• S1 or S2 is True iff S1 is True or S2 is True
• S1 implies S2 is True iff S1 is False or S2 is True
• S1 iff S2 is True iff S1 implies S2 is True or S2 implies S1 is True
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Entailment and ModelsGiven knowledge base KB, and sentence
KB entails iff is true in all models where KB is true
Example KB = “Giants won”, “Reds won” = “Either the Giants won or the Reds won”
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Models and Entailment• m is a model of sentence if is true given m.• M() denotes the set of all models of sentence
Entailment• KB entails iff M(KB) subset M()
M()
M(KB)
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Outline
• Why propositional inference?• Propositional logic• Satisfiability by enumeration• Summary
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Propositional Inference: Enumeration Method
KB = (A or C) and (B or not C) = A or BDoes KB entail ?
A B C A or C B or not C KB
False False False
False False True
False True False
False True True
True False False
True False True
True True False
True True True
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Propositional Inference: Enumeration Method
KB = (A or C) and (B or not C) = A or BDoes KB entail ?
A B C A or C B or not C KB
False False False False True False False
False False True True False False False
False True False False True False True
False True True True True True True
True False False True True True True
True False True True False False True
True True False True True True True
True True True True True True True
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Propositional Satisfiability by Enumeration• Assign true or false to an
unassigned proposition.• Backtrack as soon as a
clause is violated.
Example:• C1: Not A or B• C2: Not C or A• C3: Not B or C
AF T
BF T
CF T
CF T
BF T
C CF TF T
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Backtrack Search ProcedureBT(phi,A) Input: A cnf theory phi, an assignment A to propositions in
phiOutput: A decision of whether phi is satisfiable.1. If a clause is violated return(false);2. Else if all propositions are assigned return(true);3. Else Q = some unassigned proposition in phi;4. Return (BT(phi, A[Q = True]) or 5. BT(phi, A[Q = False])
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Validity and Satisfiability• A sentence is valid if it is true in all models.• A sentence is satisfiable if it is true in some model.• A sentence is unsatisfiable if it is true in no model.
• Validity connects to entailment by the Deduction Theorem:KB entails iff “KB implies ” is valid
• Satisfiability connects to entailment by Reductio Ad Absurdum:KB entails iff “KB and not ” is unsatisfiable.
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Types of Logic• Logics are characterized by their primitives• Ontological commitment: – what exists?
• facts, objects, time, beliefs
• Epistemological commitment: - states of knowledge?• True, false, unknown …
Language Ontological Commitment
Epistemological Commitment
Propositional Logic facts True/false/unknownFirst-order Logic Facts,objects,relations True/false/unknown
Temporal logic Facts,objects,relations,times
True/false/unknown
Probability theory facts Degree of belief 0…1Fuzzy logic Degree of truth Degree of belief 0…1
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Outline
• Why propositional inference?• Propositional logic• Satisfiability by enumeration• Summary
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SummaryLogical agents apply inference to a knowledge base to derive new
information and make decisions.Basic concepts in logic• Syntax: formal structure of sentences.• Semantics: truth of sentence wrt models.• Entailment: necessary truth of a sentence given another.• Inference: deriving sentences from others.• Soundness: derivations produce only entailed sentences.• Completeness: derivations can produce all entailed sentences.• Enumeration method is sound and complete for propositional logic.
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Models and Truth• A model assigns true/false to every proposition symbol
• The truth of a sentence wrt a model. is determined by applying boolean operators (and, or, not) to the model.
• A sentence is valid if it is true in all models.• A sentence is satisfiable if it is true in some model.• A sentence is unsatisfiable if it is true in no model.
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Propositional Satisfiability by Depth First Search• Assign true or false to an
unassigned proposition.• Backtrack as soon as a
clause is violated.• Satisfiable if assignment
is complete.
Example:• C1: Not A or B• C2: Not C or A• C3: Not B or C
AF T
BF T
CF T
CF T
BF T
C CF TF T
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EntailmentKB entails
iff is true in all models where KB is true.
Proving Entailment:• KB entails iff (KB implies ) is valid• KB entails iff (KB and not ) is unsatisfiable.
M()
M(KB)