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# EECS 662 Blog

Errata March 22

After a disastrous lecture, I thought I would create a blog post clarifying some concepts that were discussed. Hopefully this will help and not hurt…

Given predicates `A` and `B`, if `A=>B` we say that `A` is stronger than `B` and `B` is weaker than `A`. If `A=>B` and `B=>A`, then `A` and `B` are isomorphic. This is the definition that I started with before things went downhill.

If `A` is stronger than `B`, there may be things that are derivable from `A` that are not derivable from `B`. However, there is nothing in `B` or derivable from `B` that is not also derivable from `A`.

A quick example from propositional logic: `X/\Y` is stronger than `X` and stronger than `X\/Y`. `X/\Y` is also stronger than `Y/\X` and visa versa.

Now consider `P` and `P'` predicates over state. If `forall st, P st => P' st`, then `P` is stronger than `P'` and `P->>P'` means that `P` is stronger than `P'`. The confusion erupts when we think about what this means with respect to programs. Whether we are weakening or strengthening has depends on whether we start with `P` and choose `P'` or `P'` and choose `P`. If we are reasoning backwards from `Q`, which is typically the case, we will frequently derive a precondition for `c` that gives us `Q`. I think of this as `P'`. More specifically, if we want `Q` to result from `c`, then find a `P'` that in conjunction with the Hoare rules for `c` makes `Q` true. If we use the Hoare logic backwards over `c`, `P'` is the weakest precondition that makes `Q` true.

The consequence rule allows strengthening `P'` to some `P` such that:

`P->>P' /\ {P'}c{Q}`

Because `P` implies `P'` we can be assured that when `P` is true, `P'` is also true. Thus we can have the following rule:

`P->>P' -> {P'}c{Q} -> {P}c{Q}`

which is called the precondition consequence rule.

In summary, you can safely strengthen the precondition to get a precondition that better suits the proof structure.

So why did we get wrapped around the axel in class? It would help if I transcribed from the book a bit better, but the real reason is what strengthening means. When `P` is stronger than `P'` it admits fewer states and all those states satisfy `P'`. With respect to `c`, that means simply we know states satisfying `P'` will result in `Q`, but we’re taking the subset of `P'` that satisfy `P`. Effectively, we’re not allowing states we know satisfy `P'`, the weakest precondition. If we weaken `P'` all bets are off because `P'` is weakest.

Burrito King

To make things official, class on February 25 will start at 11:30 in the Nichols Hall lobby where we will carpool to The Burrito King. If it’s a nice day, we’ll stay there and do some Coq proofs at Burrito King. If it’s not a nice day, we’ll come back to Nichols, hang out, eat burritos, and do proofs. Pretty much geek heaven as far as I’m concerned.

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