Automated Reasoning SS08

Christoph Weidenbach

Christoph Weidenbach Automated Reasoning SS08

Content

Propositional Logic DPLL 2-Watch, Learning

Propositional Logic+

TheoriesDPLL(T) Coupling

First-Order Logic SuperpositionIndexing, Sharing,

Filtering

First-Order Logic+

TheoriesSUP(T) Coupling

LinearArithmetic

Logic Calculus Algorithms

Christoph Weidenbach Automated Reasoning SS08

Propositional Logic

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&

1

1PQ

R

Christoph Weidenbach Automated Reasoning SS08

Hardware

• Industrial Processor Verification: 14-cycle Model Checking

• 1Mio Variables, 10 Mio Literals, 4 Mio Clauses

• 3 hours run time (2004)

Christoph Weidenbach Automated Reasoning SS08

SUDOKU

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Christoph Weidenbach Automated Reasoning SS08

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

Christoph Weidenbach Automated Reasoning SS08

Propositional Logic + T

Christoph Weidenbach Automated Reasoning SS08

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 doesn’t 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

Christoph Weidenbach Automated Reasoning SS08

Transition Systems

Christoph Weidenbach Automated Reasoning SS08

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

Christoph Weidenbach Automated Reasoning SS08

First-Order Logic

All professors love squeezing all students.Chris is a professor.

Christoph Weidenbach Automated Reasoning SS08

SUDOKU

2, 3, 4

95

1, 4, 6, 8

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Christoph Weidenbach Automated Reasoning SS08

SUDOKU

Christoph Weidenbach Automated Reasoning SS08

SUDOKU

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Christoph Weidenbach Automated Reasoning SS08

SUDOKU

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Christoph Weidenbach Automated Reasoning SS08

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,dst-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)))

Christoph Weidenbach Automated Reasoning SS08

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

Christoph Weidenbach Automated Reasoning SS08

First-Order Logic + T

All professors above 50 love squeezing all students.Chris is a professor below 50.

Christoph Weidenbach Automated Reasoning SS08

Hybrid Automata

Christoph Weidenbach Automated Reasoning SS08

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

Christoph Weidenbach Automated Reasoning SS08

The End: Let’s Start

Propositional Logic DPLL 2-Watch, Learning

Propositional Logic+

TheoriesDPLL(T) Coupling

First-Order Logic SuperpositionIndexing, Sharing,

Filtering

First-Order Logic+

TheoriesSUP(T) Coupling

LinearArithmetic

Logic Calculus Algorithms