1 Finite Model Theory Lecture 5 Turing Machines and Finite Models

2 Outline Trakhtenbrot’s theorem (Chapter 9.1) Fagin’s theorem (Chapter 9.2)

3 Trachtenbrot’s Theorem Definition.  is finitely satisfiable if there exists a finite model A s.t. A ²  Theorem [Trakhtenbrot] Suppose  has at least one relation symbol of arity ¸ 2. Then it is undecidable whether a sentence  is finitely satisfiable.

4 Consequence 1 Corollary. There is no recursive function f such that if  has a finite model then it has a finite model of size at most f(|  |).

5 Consequence 2 Definition. A sentence  is finitely valid, ² fin , if for all finite models A, A ²  Question: is there a complete proof system `, i.e. such that `  iff ² fin  ? Answer: NO ! [why ?]

6 Proof of Trakhtenbrot’s Theorem By reduction from the halting problem: given a Turing machine M, does it halt on the empty tape ? Given M, construct  M s.t. M halts iff  M has a finite model

7 Proof of Trakhtenbrot’s Theorem 9 A. A ²   9 C. (C is a halting computation of M) M halts:  M has a finite model: Hence a model A should stand for a computation of M

8 Proof of Trakhtenbrot’s Theorem Details: M = (Q, , , q 0, Q a ) – states Q, initial state q 0 2 Q, accepting states Q a µ Q –  = {0, 1} = tape alphabet –  = transitions What is a computation C of M ?

9 Proof of Trakhtenbrot’s Theorem The formula  M will say the following: < is a linear order T 0 (p, t) = the tape holds 0 at position p and time t T 1 (p, t) = similar H q (p, t) = the machine is in state q at time t, and the head is over position p of the tape [write  M in class]  = {<, T 0 ( ¢, ¢ ), T 1 ( ¢, ¢ ), (H q ( ¢, ¢ )) q 2 Q } A ²  M iff A represents a valid computation of M

10 Fagin’s Theorem Recall Second Order Logic, SO: –May have formulas of the form 9 R.  or 8 R. , where R is a relation symbol Every SO formula can be written in prenex form like this: where each Q i is either 9 or 8, and  is in FO [WHY ???] Q 1 R 1 … Q m R m. 

11 Fagin’s Theorem Define 9 SO to be formulas of the form: Define 8 SO to be formulas of the form: 9 R 1 … 9 R m.  8 R 1 … 8 R m. 

12 Examples Let  = {R}, i.e. a single binary relation. Finite models are graphs. Express the following in SO –The graph is connected [what 2 nd order quantifiers did we need ?] –The graph is 3-colorable –The graph has a Hamiltonean path

13 Examples 3-colorability:  = 8 x.(A(x) Ç B(x) Ç C(x))  A = 8 x. 8 y.(R(x,y) ) : (A(x) Æ A(y)))  B = 8 x. 8 y.(R(x,y) ) : (B(x) Æ B(y)))  C = 8 x. 8 y.(R(x,y) ) : (C(x) Æ C(y))) 9 A. 9 B. 9 C.(  Æ  A Æ  B Æ  C )

14 Examples Hamiltonean path:  = says that < is a total linear order  = says that if x<y are consecutive, then R(x,y) Little dirty secret of SO (actually 9 SO): we don’t need order, because we can express it ! 9 <. (  Æ  )

15 Fagin’s Theorem Theorem [Fagin] 9 SO captures precisely NP More precisely: For every  2 9 SO, the problem of checking whether A ²  is in NP For any property of models, P, s.t. checking A 2 P is in NP, there exists a formula  2 9 SO s.t. A ²  iff A 2 P

16 Fagin’s Theorem Need encodings of finite structures: Let A = (A, R 1 A, …, R p A ) where A = {a 1, …, a n } Fix an order a 1 < a 2 < … < a n where each enc(R i A ) is a string of length n k in {0,1} * [what meaning ? See book pp. 88] enc( A ) = 0 n ¢ 1 ¢ enc(R 1 A ) ¢ ¢ ¢ enc(R p A )

17 Proof 9 SO µ NP is obvious [why ?] NP µ 9 SO requires us to model a Turing Machine M running in NP with a formula  M : like in Trakhtenbrot’s theorem, with some additional complications 9 <. 9 T 0. 9 T 1 9 H q 1 … 9 H q m.  M [what are the arities of T 0, T 1, (H q ) q 2 Q ?

18 Discussion This is a characterization that does not mention computational resources ! Later, other complexity classes were captured by logics, but for all lower classes Logic need help from order. Descriptive complexity

19 Fagin’s Theorem and P  NP We have NP = 9 SO, coNP = 8 SO If we prove 9 SO  8 SO then P  NP Definition. MSO = Monadic SO (restrict quantifiers to unary relations) Definition. Monadic NP = 9 MSO; Monadic coNP = 8 MSO [give examples of NP-complete problems in monadic NP]

20 Fagin’s Theorem and P  NP Theorem Monadic NP  Monadic coNP Proof. Step 1: Graph connectivity is in 8 MSO Step 2: Graph connectivity is not in 9 MSO