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Recall: tpFOk(A, a) = the
set of all FOk formulas that are true at (A, a) |
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First question: given a, b 2 Am, do
they have the same type ? |
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Notation: a ¼FOk b |
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Let k be larger then all m’s below (e.g. k=10) |
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Which implies what ?
(a,b,c) ¼
(a’,b’,c’) (a,c) ¼ (a’,c’)
(a,b) ¼
(a’,b’) (a,a,b) ¼
(a’,a’,b’) |
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Conclusion: if m · k, we may take m = k |
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Theorem
There exists an IFP formula f(x, y) s.t. 8 a, b 2 Ak, a ¼
b iff A ² f(a, b) |
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Proof
Will compute the negation, a À b as an IFP formula (should be not ¼) |
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Let a1(x), …, as(x) be all
quantifier free types with k variables (i.e. in FOk[0]) |
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y0(x,
y) = Çi ¹ j (ai(x) Æ aj(y)) |
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y(R, x, y)
= |
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What does IFP(y)(x, y) say ? |
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The n’th unfolding says that the spoiler can win
the pebble game after at most n moves, if starting at x, y |
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The IFP says that the spoiler wins if starting
at x, y |
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An order on the FOk types of A is a
total preorder a ¹ b s.t. a ¼ b iff a ¹ b and b ¹ a |
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There are many possible orders of types… |
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Theorem There exists an IFP formula f(x, y) that
computes an order on types |
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Proof [
in class ] |
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Given A 2 STRUCT[s] and a formula f in some
logic with iteration, we can compute f in two steps: |
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First, compute a “canonical” structure Ck(A)
= A/¼k over s’ |
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s’ =
<, U, U1, …, Up, S1, …, Sk,
P1, …, Pt |
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Where: < is order on types, Ui(a1,
…, ak) iff Ri(a1, …, am) (for m
· k); the others will be explained |
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Second, compute some modified formula f0
on Ck(A) |
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Let’s construct f0, and discover what
we need in s’ |
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f: xi = xj f0: 9 y.(Pp(x,y)
Æ U(y))
where p(1) = i, p(2) = j |
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f: R(xi1,
…, xim) f0:
9 y.(Pp(x,y) Æ Ui(y)) |
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: f : f0 |
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f1
Æ f2 f10
Æ f20 |
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9 xi
f 9 y.(Si(x,y) Æ f0(y)) |
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Theorem
PTIME=PSPACE ) IFP = PFP |
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Proof.
Supposes PTIME = PSPACE.
Consider a PFP formula f. It
can be expressed in two stages: first compute a canonical structure, using
IFP, then compute f0 (still a PFP) on the canonical
structure. The latter is PSPACE
problem, hence in PTIME, and, since it is ordered, f0 can be
expressed as IFP. |
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Extends this theorem to other forms of
iterations and other complexity classes |
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Slightly harder question: |
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Given (A, a), derive a formula f(x) s.t. forall
(B, b):
B ² f(b)
iff tpFOk(B,b) = tpFOk(A,a) |
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Theorem f can be expressed in FOk |
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Proof in the book |
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Corollary Every formula in Lk1w
is equivalent to:
Çi 2 N fi
where f0,
f1, … 2 FOk |
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Comments on loose Generic Machines in class |
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