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Automated Reasoning SS08

Automated Reasoning SS08. Christoph Weidenbach. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A. Content. Logic. Calculus. Algorithms. First-Order Logic + Theories. SUP(T). Coupling. First-Order Logic. Superposition. Indexing, Sharing,

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Automated Reasoning SS08

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

  2. 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

  3. P & 1 Q & R 1 Propositional Logic

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

  5. SUDOKU 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

  6. 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. Propositional Logic + T

  8. 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 Dutch Soccer League

  9. Transition Systems

  10. 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. First-Order Logic All professors love squeezing all students. Chris is a professor.

  12. SUDOKU 9 5 7 2, 3, 4 1, 4, 6, 8

  13. SUDOKU

  14. SUDOKU 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 7

  15. SUDOKU 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 7

  16. 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. 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. First-Order Logic + T All professors above 50 love squeezing all students. Chris is a professor below 50.

  19. Hybrid Automata

  20. 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. The End: Let’s 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

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