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sav07_homework_2 [2007/03/28 14:43] vkuncak |
sav07_homework_2 [2007/04/06 20:15] vkuncak |
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====== Homework 2 ====== | ====== Homework 2 ====== | ||
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F ::= A | (F&F) | (F|F) | ~F | ALL v.F | EX v.F | F ::= A | (F&F) | (F|F) | ~F | ALL v.F | EX v.F | ||
- | A ::= v=v | v + K <= v | + | A ::= v=v | v + K ≤ v |
K ::= ... -2 | -1 | 0 | 1 | 2 | ... | K ::= ... -2 | -1 | 0 | 1 | 2 | ... | ||
v ::= x | y | z | ... | v ::= x | y | z | ... | ||
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For in depth understanding you can also read references in there, in particular the relevant Wilfrid Hodges model theory book sections. | For in depth understanding you can also read references in there, in particular the relevant Wilfrid Hodges model theory book sections. | ||
- | + | **Hint**: You should be able to reduce the problem to reasoning about conjunctions and generate a disjunction over total orders over terms of form v+c. Alternatively, you may be able to use some ideas of [[http://citeseer.ist.psu.edu/71579.html|Fourier-Motzkin elimination]], but it is not necessary for this problem to have an efficient algorithm, only an algorithm that works in principle. | |
==== Verification condition generator ==== | ==== Verification condition generator ==== | ||
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1. Implement a verification condition generator that takes abstract syntax trees of the following form | 1. Implement a verification condition generator that takes abstract syntax trees of the following form | ||
- | S ::= (x=T) | + | S ::= (v=T) |
| assume(F) | | assume(F) | ||
| assert(F) | | assert(F) | ||
| havoc(v,...,v) | | havoc(v,...,v) | ||
| S ; S | | S ; S | ||
- | | while (F) { S } | + | | while [inv F] (F) { S } |
| if (F) { S } else { S } | | if (F) { S } else { S } | ||
- | T ::= T+T | T-T | K*T | v | + | T ::= T+T | T-T | K*T | v | K |
- | A ::= T=T | T < T | + | A ::= T=T | T < T | True | False |
F ::= A | (F&F) | (F|F) | ~F | ALL v.F | EX v.F | F ::= A | (F&F) | (F|F) | ~F | ALL v.F | EX v.F | ||
K ::= 0 | 1 | 2 | ... | K ::= 0 | 1 | 2 | ... | ||
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This notation corresponds to a fragment of the language of the [[wk>Isabelle theorem prover]], as well as the [[Jahob system]]. | This notation corresponds to a fragment of the language of the [[wk>Isabelle theorem prover]], as well as the [[Jahob system]]. | ||
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+ | The notation for while loop indicates that all loops have supplied loop invariants. | ||
You do not need to build a parser for programs and formulas for this exercise, but it may make the testing of your implementation easier. | You do not need to build a parser for programs and formulas for this exercise, but it may make the testing of your implementation easier. | ||
- | 2. Given a statement S, compute wp(S,true) and pretty print it as a string conforming to the above grammar. Test your program on | + | To make sure that you correctly avoid accidental variable capture, one of the many tests for your program should be the following 4 line sequence of commands: |
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+ | x = 0; | ||
+ | y = x + 3; | ||
+ | havoc(x); | ||
+ | assert (y=3); | ||
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+ | 2. Given a statement S, compute wp(S,true) using the part 1, and pretty print it as a string conforming to the above grammar. Test your program on | ||
* two input programs that generate valid formulas | * two input programs that generate valid formulas | ||
* two input programs that generate invalid formulas | * two input programs that generate invalid formulas | ||
Try using the [[wk>Isabelle theorem prover]] or formDecider in the [[Jahob system]] to prove the validity of the printed formulas and describe your experience. | Try using the [[wk>Isabelle theorem prover]] or formDecider in the [[Jahob system]] to prove the validity of the printed formulas and describe your experience. | ||