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Automated Reasoning SS08
Christoph Weidenbach
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Content
Logic
Calculus
Algorithms
First-Order Logic Theories
SUP(T)
Coupling
First-Order Logic
Superposition
Indexing, Sharing, Filtering
Propositional Logic Theories
DPLL(T)
Coupling
Linear Arithmetic
Propositional Logic
DPLL
2-Watch, Learning
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Propositional Logic
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Hardware

• Industrial Processor Verification 14-cycle
  Model Checking
• 1Mio Variables, 10 Mio Literals, 4 Mio Clauses
• 3 hours run time (2004)

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SUDOKU
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Summary

• propositional logic is suitable to represent
  finite domains
• software restrict all variables to finite
  domain
• hardware restrict number of cycles
• suitable to test problems with thousands of
  variables
• Limits infinite domains or calculations,
  i.e., mathematical structure

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Propositional Logic T
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Dutch Soccer League

• if Eindhoven and Amsterdam play on the same day
  the TV income is x
• If Eindhoven and Amsterdam play on two different
  days, the income is 2x
• if a team plays on Wednesday champions league it
  doesnt play on Friday
• there are at most 3 plays on Friday
• .. in sum several thousand constraints over LP
  and Boolean variables
• League is modelled by the Barcelogic tool

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Transition Systems
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Summary

• propositional logic T can also represent
  aspects of infinite theories
• for the meta algorithm for the theory often
  nice properties are needed
• bottleneck often the solver for T
• Limits quantification and free structures
  beyond boolean combinations

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First-Order Logic
All professors love squeezing all students. Chris
is a professor.
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SUDOKU
9 5
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2, 3, 4
1, 4, 6, 8
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SUDOKU
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SUDOKU
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1 2 3 4 5 6 7 8 9
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SUDOKU
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LAN Router
Sent(epacket(incoming-net, router-mac, src-mac,
e-ip, ippacket(ip-src, ip-dst, ip-proto,
ip-data))))RouteEntry(route(router,dst-netmask,ds
t-net-addr,outgoing-net)) ipand(ip-dst,dst-netmask
) dst-net-addr Sent(epacket(outgoing-net,
dst-mac, src-mac, e-ip, ippacket(ip-src, ip-dst,
ip-proto, ip-data)))
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Summary

• first-order logic can model freely defined
  infinite theories
• inductive theories are out of scope
• incredible expressiveness
• full quantification
• Limits undecidable (take serious), some
  important theories can not be (finitely)
  represented

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First-Order Logic T
All professors above 50 love squeezing all
students. Chris is a professor below 50.
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Hybrid Automata
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Summary

• first-order logic T can also model aspects
  involving inductive theories
• adequately represent many aspects of software,
  hardware
• incredible incredible expressiveness
• quantification over theory variables potentially
  limited
• Limits very undecidable (take serious), in
  general not compact, and any calculus not
  complete

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The End Lets Start
Logic
Calculus
Algorithms
First-Order Logic Theories
SUP(T)
Coupling
First-Order Logic
Superposition
Indexing, Sharing, Filtering
Propositional Logic Theories
DPLL(T)
Coupling
Linear Arithmetic
Propositional Logic
DPLL
2-Watch, Learning