Automated Reasoning Matt Whipple and Brian Vees

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

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

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

Theorem Prover Example: Prove: P(x) R(x) R(x) T(x) P(x) T(x) Most theorem provers: Resolution principleResolution principle Sound but not complete or decidableSound but not complete or decidable Take… P(x) R(x) R(x) T(x) P(x) -T(x) Theorem Prover …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.

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 No knowledge needed from an expert like with heuristic classification Only requires a detailed description of how the device works Only requires a detailed description of how the device works

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 the system description (SD) of a device with a finite set of system components and a set of system components and a set of observations (OBS) (symptoms) observations (OBS) (symptoms) *SD & OBS are finite sets of sentences in first-order predicate logic.

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.

Reiter’s Theory Example Consider a ceiling fan with 4 components: 1) A pull chain controlling a light2) A light bulb 3) A pull chain controlling a fan4) 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)

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 )

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 Logic Circuit Design and Validation

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

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

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

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

Symbolic Execution Instead of supplying the normal inputs to a program (e.g. numbers) one supplies symbols representing arbitrary values 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).

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

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

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

Questions ?