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Automated Reasoning

Matt Whipple and Brian Vees. Automated Reasoning. Overview. What is automated reasoning? Properties of inference procedures Theorem prover Diagnosis with first principles Logic circuit design and validation Program verification and validation Conclusion. What is Automated Reasoning?.

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Automated Reasoning

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  1. Matt Whipple and Brian Vees Automated Reasoning

  2. Overview • What is automated reasoning? • Properties of inference procedures • Theorem prover • Diagnosis with first principles • Logic circuit design and validation • Program verification and validation • Conclusion

  3. What is Automated Reasoning? • “…the attempt to prove statements with a computer in a law-like way.”

  4. Properties of Inference Procedures An implementation of automated reasoning can have these main properties: • Soundness • Completeness • Decidability

  5. Most theorem provers: • Resolution principle • Sound but not complete or decidable Theorem Prover Take… P(x) R(x) R(x) T(x) P(x) -T(x) Theorem Prover Example: Prove: P(x) R(x) R(x) T(x) P(x) T(x) …and determine whether or not this group of statements can be satisfied. If it can, the theorem is false. If it can’t, the theorem is true.

  6. Diagnosis with First Principles Basic idea: Diagnose a device with reasoning based off of how the device actually works. Benefits: • No knowledge needed from an expert like with heuristic classification • Only requires a detailed description of how the device works

  7. Diagnosis with First Principles Example Example (Reiter’s theory): Diagnosis is based on: • the system description (SD) of a device with a finite set of • system componentsand a set of • observations (OBS) (symptoms) *SD & OBS are finite sets of sentences in first-order predicate logic.

  8. Diagnosis with First Principles Example Example (Reiter’s theory): A diagnosis for the set ( SD, COMPONENTS, OBS ) is a set of faulty components. A component is part of this faulty set only if assuming it is non-faulty creates a contradiction with the device description and its symptoms. The idea - form a set that consists of members of each faulty set.

  9. Reiter’s Theory Example Consider a ceiling fan with 4 components: 1) A pull chain controlling a light 2) A light bulb 3) A pull chain controlling a fan 4) Fan w/ motor Then, the system description (SD) might have statements like: L(x) = light x is on Lp(x) = light x’s pull chain is “on” F(x) = fan x is on Fp(x) = fan x’s pull chain is “on” Lp(x)  L(x) -Lp(x)  -L(x) Fp(x)  F(x) -Fp(x)  -F(x)

  10. Ceiling Fan/Light Ex. Cont. … And the symptoms (observations) might be: Lp(x) -L(x) Pull chain is pulled and light is not on. -Fp(x) -F(x) It is easy to see that the faulty component set: * ( SD, COMPONENTS, OBS ) will consist of: * ( light bulb, pull chain that controls light bulb )

  11. Logic Circuit Design and Validation How automated Reasoning can help design and validate logic circuits i.e., turning circuit specifications in terms of ANDs, ORs, and NOTs to produce circuits using NAND gates

  12. Circuit Design Example Converting from various gates to purely NAND o1 = or(and(i1, i2), not (i3)) o2 = and(not(i3),i2) not(x) g nand(x,x) or( x, y) g nand(not( x),not(y)) and(x,y) g not(nand(x,y)) Simplified… nand(nand(x,x),nand(x,x)) gx. Demodulation

  13. Demodulation • The substitutuing or rewriting of one term by an equivalent one • Applied when the first clause unifies with the term we are attempting to rewrite

  14. Program Verification and Validation • Traditionally, program correctness is discovered by testing out a wide range of values • This method does not prove 100% correctness • Automated reasoning can formally prove a program’s correctness

  15. Program Correctness • A program’s correctness can be proved if all inputs satisfying the input assumptions yield results satisfying the exiting requirements • It is the programmer’s job to come up with complete specifications for this procedure

  16. Symbolic Execution • Instead of supplying the normal inputs to a program (e.g. numbers) one supplies symbols representing arbitrary values IF x<O y=3*z ELSE y=2*x Rather than using numbers as inputs, instead symbols would be used (x<0 and x>=0).

  17. Proving Program Correctness Using Automated Reasoning • Demodulation rules are specified, which in turn define how each type of statement affects the program state • The demodulators then go to work attempting to prove that all exit conditions are met • If this happens, the program has been proved correct

  18. Benefits of Using Automated Reasoning • No ambiguity as far as program correctness is concerned • Can be used to solve problems that have a tedious and repititous proof procedure

  19. Conclusion • What is automated reasoning? • Properties of inference procedures • Theorem prover • Diagnosis with first principles • Logic circuit design and validation • Program verification and validation

  20. Questions ?

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