SS 2012 20.6.2012 In-class Exercises to the Lecture Logics Sheet 5 Jun.-Prof. Dr. Roland Meyer Discussion on 21./22.6.2012

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SS 2012 20.6.2012 In-class Exercises to the Lecture Logics

Sheet 5

Jun.-Prof. Dr. Roland Meyer Discussion on 21./22.6.2012

Exercise 5.1 [MSO on words]

Let Σ be an alphabet and Φ be the set of formulae in second order predicate logic in which only the following predicates occur:

• p

a

, a unary predicate, for each a P Σ,

• , a binary predicate, and

• suc, a binary predicate.

To each word w P Σ , we associate the interpretation I

^w

p D

^w

, I

_c^w

, I

_v^w

q , where

• D

^w

t 1, . . . , | w |u is the set of positions in w,

• for each a I

^w

p p

_a

q is the set of positions that contain an a,

• I

^w

p x y q 1 if and only if I

^w

p x q I

^w

p y q ,

• I

^w

p suc px, yqq 1 if and only if I

^w

pyq I

^w

pxq 1.

For example, let Σ t a, b u and w ab. Then D

^w

t 1, 2 u , I

^w

p p

_a

q t 1 u and I

^w

p p

_b

q t 2 u . In this situation, we have

I

^w

( D x D y p x y ^ p

a

p x qq ^ D x D y D z p x y ^ y z q . The language defined by A P Φ is

L p A q t w P Σ | I

^w

( A u .

The logic defined this way is also known as Monadic Second Order Logic on words.

a) Present a formula A with L p A q t a u t b u , Σ t a, b u . b) Present a formula A with LpAq Σ

tau Σ

tbu , Σ ta, bu .

c) Suppose you have symbols r, a, and i in Σ, where r, a, and i represent a request, an acknowledge, and an internal event, respectively. Present a formula A such that L p A q is the set of words such that: after each request event, at least one acknowledge event follows eventually.

Exercise 5.2 [MSO on graphs]

a) Describe, similar to Exercise 5.1,

• a collection of predicates,

• an interpretation for each graph,

such that you can define sets of graphs using formulae.

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b) Present a formula that defines precisely the set of connected graphs.

Exercise 5.3 [Undecidability]

A context-free grammar is called linear if in each rule, the right-hand side contains at most one occurrence of a nonterminal symbol. Show that the following problem is undecidable: Given linear context-free grammars G

₁

and G

₂

, is L p G

₁

q X L p G

₂

SS 2012 20.6.2012 In-class Exercises to the Lecture Logics Sheet 5 Jun.-Prof. Dr. Roland Meyer Discussion on 21./22.6.2012

SS 2012 20.6.2012 In-class Exercises to the Lecture Logics

Sheet 5

Jun.-Prof. Dr. Roland Meyer Discussion on 21./22.6.2012

Exercise 5.1 [MSO on words]

Let Σ be an alphabet and Φ be the set of formulae in second order predicate logic in which only the following predicates occur:

• p

, a unary predicate, for each a P Σ,

• , a binary predicate, and

• suc, a binary predicate.

To each word w P Σ , we associate the interpretation I

p D

, I

, I

q , where

• D

t 1, . . . , | w |u is the set of positions in w,

• for each a I

p p

q is the set of positions that contain an a,

• I

p x y q 1 if and only if I

p x q I

p y q ,

• I

p suc px, yqq 1 if and only if I

pyq I

pxq 1.

For example, let Σ t a, b u and w ab. Then D

t 1, 2 u , I

p p

q t 1 u and I

p p

q t 2 u . In this situation, we have

I

( D x D y p x y ^ p

p x qq ^ D x D y D z p x y ^ y z q . The language defined by A P Φ is

L p A q t w P Σ | I

( A u .

The logic defined this way is also known as Monadic Second Order Logic on words.

a) Present a formula A with L p A q t a u t b u , Σ t a, b u . b) Present a formula A with LpAq Σ

tau Σ

tbu , Σ ta, bu .

c) Suppose you have symbols r, a, and i in Σ, where r, a, and i represent a request, an acknowledge, and an internal event, respectively. Present a formula A such that L p A q is the set of words such that: after each request event, at least one acknowledge event follows eventually.

Exercise 5.2 [MSO on graphs]

a) Describe, similar to Exercise 5.1,

• a collection of predicates,

• an interpretation for each graph,

such that you can define sets of graphs using formulae.

b) Present a formula that defines precisely the set of connected graphs.

Exercise 5.3 [Undecidability]

A context-free grammar is called linear if in each rule, the right-hand side contains at most one occurrence of a nonterminal symbol. Show that the following problem is undecidable: Given linear context-free grammars G

and G

, is L p G

q X L p G

q H ? Exercise 5.4 [Formulae in predicate logic]

a) Let A @ x D yp p x, y q and B D y @ xP p x, y q . Which of these formulas is deducible from the other? Are they equivalent?

b) Is the formula @ xp p x q Ñ D xp p x q a tautology?