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Finite Model Theory Lecture 5

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

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Finite Model Theory Lecture 5

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  1. Finite Model TheoryLecture 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. f is finitely satisfiable if there exists a finite model A s.t. A²f Theorem [Trakhtenbrot] Suppose s has at least one relation symbol of arity ¸ 2. Then it is undecidable whether a sentence f is finitely satisfiable.

  4. 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|).

  5. Consequence 2 Definition. A sentence f is finitely valid, ²finf, if for all finite models A, A²f Question: is there a complete proof system `, i.e. such that `f iff ²finf ? 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 fM s.t. M halts iff fM has a finite model

  7. Proof of Trakhtenbrot’s Theorem M halts: 9 C. (C is a halting computation of M) 9 A. A²fM fM has a finite model: Hence a model A should stand for a computation of M

  8. Proof of Trakhtenbrot’s Theorem Details: • M = (Q, D, d, q0, Qa) • states Q, initial state q02 Q, accepting states Qaµ Q • D = {0, 1} = tape alphabet • d = transitions • What is a computation C of M ?

  9. Proof of Trakhtenbrot’s Theorem s = {<, T0(¢,¢), T1(¢, ¢), (Hq(¢, ¢))q 2 Q} 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] A²fM 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.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 ???] Q1 R1 … Qm Rm. f

  11. Fagin’s Theorem • Define 9 SO to be formulas of the form: • Define 8 SO to be formulas of the form: 9 R1 … 9 Rm. f 8 R1 … 8 Rm. f

  12. 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

  13. 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))) 9 A.9 B.9 C.(fÆyAÆyBÆyC)

  14. Examples Hamiltonean path:f = says that < is a total linear ordery = 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 <. (fÆy)

  15. Fagin’s Theorem Theorem [Fagin] 9SO captures precisely NP More precisely: • For every f29SO, the problem of checking whether A ²f is in NP • For any property of models, P, s.t. checking A 2 P is in NP, there exists a formula f29SO s.t. A ²f iff A 2 P

  16. 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] enc(A) = 0n¢ 1 ¢ enc(R1A) ¢¢¢ enc(RpA)

  17. Proof • 9SO µ NP is obvious [why ?] • NP µ9SO requires us to model a Turing Machine M running in NP with a formula fM: like in Trakhtenbrot’s theorem, with some additional complications 9 <. 9 T0. 9 T19 Hq1 … 9 Hqm. fM [what are the arities of T0, T1, (Hq)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 P¹ NP • We have NP = 9 SO, coNP = 8 SO • If we prove 9 SO ¹8 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]

  20. 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

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