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Advanced Automated Reasoning: Techniques in Propositional and First-Order Logic

This document explores sophisticated automated reasoning methods, focusing on Propositional and First-Order Logic. It discusses algorithms like DPLL(T) for propositional reasoning and the application of superposition in first-order logic. Key topics include the limits of finite representations, the use of logic in hardware verification, and the modeling of complex systems such as the Dutch soccer league. With insights into model checking for industrial processors and the expressiveness of logic frameworks, this work serves as a comprehensive resource for understanding advanced reasoning techniques.

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Advanced Automated Reasoning: Techniques in Propositional and First-Order Logic

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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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