Logic: Lecture 15, 01 October 2015
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Expressiveness of First-Order Logic
Proofs of inexpressiveness using Lowenheim-Skolem theorem usually
produce infinite counterexamples. What if we are interested in
finite models?
In general, for finite models, we assume our language L = (R,F,C)
is "relational" --- that is, F is empty. We can encode a k-ary
function f by a (k+1)-ary relation r_f such that
f(x1,...,xk) = y iff (x1,x2,...,y) in r_f
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Compactness fails over finite models (Libkin, Proposition 3.2)
Proposition: There is an infinite set of sentences that is not
satisfiable over finite models, such that every finite subset
is satisfiable over finite models
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Inexpressibility of evenness (Libkin, Proposition 3.3)
Proposition: Assume that L is empty. Then EVEN is not
FO-definable.
Note: Prop 3.3 fails if we have R = {<}, for instance. How do we
extend this result to a richer class of finite models?
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The "natural" approach to show that a property P is not satisfiable
over finite models is to exhibit two models A and B that satisfy
the same formulas such that A has property P and B does not.
Unfortunately, this strategy fails because no two non-isomorphic
models satisfy the same FO formulas.
Characteristic formula for finite structures (Libkin, Lemma 3.4)
Lemma: For every finite structure A, there is a sentence Phi_A
such that B |= Phi_A iff B is isomorphic to A.
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A more sophisticated strategy:
- Stratify formulas by structural complexity. More formally,
define quantifier rank of a formula as maximum depth of nested
quantifiers. Let FO[k] denote all formulas of quantifier rank
at most k.
- Find a family of structures A_k and B_k, one for each k, that
agree on all of FO[k] where A has property P and B does not.
- Suppose Psi captures property P. Psi is a finite sentence and
has a fixed quantifier rank m. Consider the structures A_m and
B_m from the previous step. These agree on FO[m], hence agree
on Psi, but B does not have property P. This is a contradiction
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Ehrenfeucht games (Libkin, Section 3.2)
Partial isomorphism of structures (Libkin, Definition 3.5)
- A and B structures with {a1,...,am} and {b1,...,bm} m-tuples
from A and B, respectively
- The substructures defined by {a1,...,am} and {b1,...,bm} are
isomorphic
The game:
- Spoiler and Duplicator
- In each move j
- Spoiler picks aj in A or bj in B
- Duplicator picks a matching element in the other structure
- After k rounds
- Duplicator wins if {a1,...,ak} and {b1,...,bk} are partially
- isomorphic
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Examples:
1. Games on Sets
Let A, B be sets such that |A|, |B| are both at least m. Then
A and B are m-equivalent wrt Ehrenfeucht games.
2. Games on Linear Orders
Let (A,<), (B,<) be linear orders such that |A|, |B| are both
at least 2^m. Then A and B are m-equivalent wrt Ehrenfeucht
games.
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