CS195Y Lecture 29

4/27/16

Announcement 1: There are only two lectures left! On Friday we will finish up Natural Deduction and do a demo of a language called Datalog. Next Wednesday, we will talk more about Completeness.

Announcement 2: We will be sending out a survey during reading week for Tim’s research about Alloy. You will get half a homework’s worth of extra credit for completing it, and if half the class completes the survey, Tim will bake us all cookies!

Announcement 3: Project signups will go out today. Projects are due electronically on 5/9

Natural Deduction

Notation:

A and B denote formulas
A1,...,AN ⊨ B: For any I, if I satisfies A1,...,AN then I satisfies B (where I is an instance)
A1,...,AN ⊢ B: There exists a proof of B from A1,...,AN
       Implicitly, this can only exist in the context of some proof system. For the rest of class, assume we have the context of the set of axioms we set up last class.

Q: What is I?
A: I is an instance. So for a boolean formula, an instance would be an assignment to all of the variables.

Soundness and Completeness

A1,...,AN ⊨ B ---> A1,...,AN ⊢ B (completeness)
If semantically, A1,...,AN entails B, then there exists a proof that shows that.
.
A1,...,AN ⊢ B ---> A1,...,AN ⊨ B (soundness)
If there is a proof of B from A1,...,AN, then A1,...,AN entails B

Example 1:

p --> q, not q ⊢ not p
_______________________
1: p --> q          premise
2: not q            premise
3:               p            assume p
4:               q            implies elimination(3,1)
5:               ⊥            not elimination(4,2)
6: not p            not introduction(3-5)

Example 2:

⊢ (p and q) --> (q and p)
_________________________
1:              p and q             assume (p and q)
2:                 p                and elim1(1)
3:                 q                and elim2(1)
4:              q and p             and introduction(3,2)
5: p and q --> q and p    implication introduction(1-4)

Q: What is “completeness” in Alloy verses “completeness” in Natural Deduction?
A: In Alloy, we’re looking for an instance. In Natural Deduction, we’re looking for a proof.
Q: What is the relation between proof and instance?
A: Looking for an instance is kind of like looking for a counterexample, and looking for a proof is sort of like looking for whether or not an instance exists. Alloy looks for finite instances. All proofs are finite, but counterexamples to things we’re trying to prove may be infinite.

Compactness

Let T be a set of formulas.
Then T is satifiable if and only if every finite subset of T is satisfiable.

Note that T itself may or may not be finite.
Contrapositive: T is unsatisfiable if there exists a finite subset of T that is unsatisfiable.

Q: Why is that true?
A: All proofs are finite. So you can only use a finite subset of the premises in T. Then if we find a proof that a subset is unsatisfiable, we can conclude that the whole thing is unsatisfiable. From that, we can construct the compactness theorem.