1. Automated Reasoning SS08 Christoph Weidenbach TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A

2. Christoph Weidenbach Automated Reasoning SS08 Content Propositional LogicDPLL2-Watch, Learning Propositional Logic + Theories DPLL(T)Coupling First-Order LogicSuperposition Indexing, Sharing, Filtering First-Order Logic + Theories SUP(T)Coupling Linear Arithmetic LogicCalculusAlgorithms

3. Christoph Weidenbach Automated Reasoning SS08 Propositional Logic & & 1 1 P Q R

4. 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)

5. Christoph Weidenbach Automated Reasoning SS08 SUDOKU 1 2 3 4 5 6 7 8 9 123456789123456789

6. 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

7. Christoph Weidenbach Automated Reasoning SS08 Propositional Logic + T

8. 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

9. Christoph Weidenbach Automated Reasoning SS08 Transition Systems

10. 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

11. Christoph Weidenbach Automated Reasoning SS08 First-Order Logic All professors love squeezing all students. Chris is a professor.

12. Christoph Weidenbach Automated Reasoning SS08 SUDOKU 2, 3, 4 9595 1, 4, 6, 8 7

13. Christoph Weidenbach Automated Reasoning SS08 SUDOKU

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15. Christoph Weidenbach Automated Reasoning SS08 SUDOKU 7 1 2 3 4 5 6 7 8 9 123456789123456789

16. 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)))

17. 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

18. Christoph Weidenbach Automated Reasoning SS08 First-Order Logic + T All professors above 50 love squeezing all students. Chris is a professor below 50.

19. Christoph Weidenbach Automated Reasoning SS08 Hybrid Automata

20. 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

21. Christoph Weidenbach Automated Reasoning SS08 The End: Let’s Start Propositional LogicDPLL2-Watch, Learning Propositional Logic + Theories DPLL(T)Coupling First-Order LogicSuperposition Indexing, Sharing, Filtering First-Order Logic + Theories SUP(T)Coupling Linear Arithmetic LogicCalculusAlgorithms