Notes

Outline

Finite Model Theory Lecture 5 Turing Machines and Finite Models

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

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

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

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

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 fM s.t. M halts iff fM has a finite model

Proof of Trakhtenbrot’s Theorem

Proof of Trakhtenbrot’s Theorem Details: M = (Q, D, d, q0, Qa)  states Q,  initial state q0 2 Q,  accepting states Qa µ Q  D = {0, 1} = tape alphabet  d = transitions What is a computation C of M ?

Proof of Trakhtenbrot’s Theorem The formula fM will say the following: < is a linear order T0(p, t) = the tape holds 0 at position p and time t T1(p, t) = similar Hq(p, t) = the machine is in state q at time t, and the head is over position p of the tape [write fM in class]

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

Fagin’s Theorem Define 9 SO to be formulas of the form: Define 8 SO to be formulas of the form:

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

Examples 3-colorability: f    = 8 x.(A(x) Ç B(x) Ç C(x)) yA = 8 x.8 y.(R(x,y) ) : (A(x) Æ A(y))) yB = 8 x.8 y.(R(x,y) ) : (B(x) Æ B(y))) yC = 8 x.8 y.(R(x,y) ) : (C(x) Æ C(y)))

Examples Hamiltonean path: f = says that  < is a total linear order y = 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 !

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

Fagin’s Theorem Need encodings of finite structures: Let A = (A, R1A, …, RpA) where A = {a1, …, an} Fix an order a1 < a2 < … < an where each enc(RiA) is a string of length nk in {0,1}* [what meaning ? See book pp. 88]

Proof  9SO µ NP is obvious [why ?] NP µ 9SO requires us to model a Turing Machine M running in NP with a formula fM: like in Trakhtenbrot’s theorem, with some additional complications

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

Fagin’s Theorem and P¹ NP We have NP = 9 SO,  coNP = 8 SO If we prove 9 SO ¹ 8 SO then P ¹ NP Definition.  MSO = Monadic SO (restrict quantifiers to unary relations) Definition.  Monadic NP = 9MSO; Monadic coNP = 8MSO [give examples of NP-complete problems in monadic NP]

Fagin’s Theorem and P¹ NP Theorem Monadic NP ¹ Monadic coNP Proof. Step 1: Graph connectivity is in 8 MSO Step 2: Graph connectivity is not in 9 MSO