Applications

3.3.1 Beyond First-order

Here follow some properties that cannot be expressed in rst-order terms.

3.29 Denition.

A formula'='(x) in one free variable xdenes the set fa 2Aj Aj='a]g in A a formula '='(xy) in two free variables

xy denes the relationf(ab)2AAjAj='ab]g inA.

For instance, the formulax < y^:9z(x < z^z < y) denes the (successor)

relationn+ 1 =m in the model!= (N<).

3.30 Example.

The set of even natural numbers is not denable in! = (N<).

Proof. Suppose that ' does dene this set. I.e., ! j= 'n] holds i n is

even. Then the sentence

8x8y(x < y^:9z(x < z^z < y)!('(x)$:'(z)))

holds in !. Since ! !+ (cf. Lemma 3.14.1 page 26), this sentence

immediate successor ofn. Then we have that!+j='n] i!+j=:'m].

Consider the automorphismhof!+for whichh(n) =m. A contradiction follows using Lemma 2.4. a

A setX N is co-nite ifN ;X is nite.

For every natural number n2N it is possible to write down a formula

n=n(x) expressing that (an element assigned to)xhas exactly npre- decessors. Thus, ! j= nm] holds i m =n. It follows that a nite set

A N can be dened in ! by the disjunction W

n2An its complement N ;Ais dened by the negation of this formula.

Therefore, all nite and co-nite sets of natural numbers are denable in!. Conversely:

3.31 Proposition.

Every set denable in ! = (N<) is either nite or

co-nite.

Exercises

55

Prove Proposition 3.31.

3.32 Denition.

Letnb(xy) be the formula

(x < y^:9z(x < z^z < y))_(y < x^:9z(y < z^z < x))

expressing that x and y are neighbours in the ordering <. If S is the relation dened by nb in A = (A<), then (AS) is the neighbour model

corresponding toA notation: Anb.

For instance, the relation of!nb_{is dened by} jn;mj= 1.

56

Show the following: 1. An

+1

B ) AnbnBnb,

2. AB ) AnbBnb.

57

The universe of the modelCm isf1:::mgon which the relationRis

dened byiRj:ji;jj= 1_(i= 1^j=m)_(i=m^j= 1). (Visualize

this model by drawing 1:::m on a circle.) Show that if m 2n, then Sy(Cmnbn).

Hint. After the rst two moves, the game reduces to one on successor structures of linear orderings. After a rst move a2 Cm, \cut"Cm in a

(by which ais doubled into elementsa0 anda00). This operation produces

the structure (

m

+

1

nba0a00), in whicha0 anda00have become endpoints.

Similarly,OP_{, after a rst moveb}2, is cut open, producing

((!+!?₎nb_b0b00):

Now, use Lemma 3.12.2 and Exercise 56.

58

Consider the successor relationS on! dened bynSm:n+ 1 =m.

Hint. S is dened in ! by a modication suc of the formula nb, i.e.: (NS) = !suc. Now suppose that LO(xy) would be a denition of the

required type. Then the sentences

8x8y(x6=y!LO(xy)_LO(yx))

and

8x8y(LO(xy)!:LO(yx))

would hold in (NS). Hence they would hold in (!++)suc. Deduce a

contradiction.

3.3.2 Second-order

In a (monadic) second-order language you can write down everything you could write down before, but now you are also allowed to use relation vari- ablesXYZ:::occurring in atomic formulasX(t1:::tn) and quantiers

over these variables. The meaning of this new form of expressions in the context of a modelAwith a universeAis the following: the new variables

stand for relations onA and the second-order quantiers 8X and9Y are

to be read as: \for every relation X (of the appropriate arity) onA", and \for some relationX onA," respectively. Finally, an atom X(t1:::tn) is

read as \the tuplet1:::tn is in the relationX".

In a similar way, function variables could be quantied over.

More formally, the concept of a second-order formula is most easily obtained by modifying Denition 1.2, page 3, viewing some non-logical symbol as a variable:

3.33 Second-order Formulas.

The class of second-order formulas is obtained by adding the following clause to Denition 1.2 (page 3).

4. If'is anL-formula and 2La relation symbol, then8 'and9 '

areL;f g-formulas.

If second-order quantication is permitted only over unary relation vari- ables, the resulting formulas are called monadic second-order. Such a for- mula is universal if it has the form8

r'

where

r

is a sequence of relation

variables and'is rst-order it is monadic universal if these relation vari- ables are unary. #1

1and # 1

1(mon) are the classes of universal, and monadic

universal second-order formulas, respectively their existential counterparts $1

1and $ 1

1(mon) are dened similarly. (The superscript 1 refers to second

order | a 0 would refer to rst order! |, the subscript counts quantier blocks in the prex, # says the rst block consists of universal, $ says it consists of existential quantiers.)

Although the semantics of this formalism is pretty obvious, here follows the clause that says how to read a universal second-order quantier. In this clause, is an A-assignment for the free rst-order variables in ' in the

r

in the expansion (AR).

Aj=8

r'

], for allR A, (AR)j=']:

We leave it to the reader to write down the other necessary clauses extend- ing Denition 1.7.

A property (of models, or of elements in a given model) that is both #1 1and $ 1 1(respectively, # 1 1(mon) and $ 1

1(mon)) denable is said to be & 1 1

(respectively, &1

1(mon)).

A class of nite models (suitably coded as sequences of symbols) is in NP if membership in the class is

N

on-deterministically Turing machine decidable in

P

olynomial time. Cf. the discussion in A.12. The following result explains the relationship with second-order denability.

3.34 Theorem.

On the class of nite models: $1 1= NP.

a

Below, we deal especially with monadic second-order sentences that are universal, i.e., have the form8X'where'uses the unary relation variable

X in atomsX(t), but it does not quantify over such second-order variables itself.

Instead of X(t), we often write t2X. Notations such as 8x2X '

and 9x2X ' are abbreviations for 8x(x2X ! ') and 9x(x2X^'),

respectively. Using such sentences you can express a lot of things you cannot express in rst-order terms.

3.35 Examples.

1. Compare 3.30. The set of even natural numbers can be dened in!using a monadic second-order formula. An example of such a second-order denition is8X('(X)!x2X), where ' is the formula

9y2X:9z(z < y))^

8y8z(y < z^:9u(y < u^u < z)!(y2X $z62X)):

(\The least element is in X, and from an element and its immediate suc- cessor exactly one is inX.") The formula ' is satised by the set of even natural numbers only. As a consequence,!j= 'n] holds i nis even:

()) If ! j= 8X(' ! x 2 X)n], then in particular we have that

! j= (' ! x 2 X)nE], where E is the set of even natural numbers.

However,E satises the left-hand side ' of this implication. Therefore,E also satises the right-hand side, i.e.,n2E,nis even.

(() Ifnis even andA N is a set satisfying ', then obviouslyA=E.

Indeed, the rst conjunct of ' says that 0 2A, the second conjunct has

the eect that 162A, next, that 22A, 362A, etc. Therefore,n2A.

2. The orderings! and!+cannot be distinguished by means of a rst- order sentence: they are elementarily equivalent. However, consider the following monadic second-order sentence:

This sentence (the Principle of Strong Induction) expresses a fundamental property of!, but is false in!+. (Consider the initial of type! of this model as value forX.)

3. The elementary equivalent orderings (Q) and (R) are distinguished

by the monadic second-order Principle of the Least Upper Bound (\every non-empty set of reals that has an upper bound also has a least upper bound"): 8X9x(x2X)^9y8x2X(xy)! 9y(8x2X(xy)^8y 0( 8x2X(xy 0) !yy 0))]:

(By the way, this sentence also serves to distinguish! from!+.)

Exercises

59

Look up the formula ' in Example 3.35.1. Verify that the formula

9X('^x2X) also denes the set of even natural numbers in!.

60

Cf. Exercise 58. '(xy) is the monadic second-order formula

9X(8z(z2X $(

r

(xz)_9z 0(z0

2X^

r

(z 0z))))

^y2X):

1. Show that ' denes the usual ordering<ofN in the successor struc-

ture!suc.

2. Show that the usual ordering ofZ is not dened by ' in suc.

3. Produce a monadic second-order formula that denes the ordering in both structures !suc and suc. (Can you nd one that is #1

1(mon)

and one that is $1

1(mon)?)

61

Formulate and prove a counterpart of Lemma 2.4 (page 12) for monadic second-order languages.

A graph is a model (GS), where S is a binary relation on G that is symmetric. A graph is called connected if for all ab2Gthere is a nite

sequencea1=a:::an =bsuch that for alli, 1

i < n: aiSai +1.

The notion of connectedness is #1

1(mon) on the class of graphs: G is

connected i for all ab 2 G and all U G: if a 2 U and U is closed

under neighbourship, then b 2 U. However, the connected graph !nb is

elementarily equivalent to the graph (!+)nb, which is not connected. Thus, the rst claim of the next proposition.

3.36 Proposition.

Connectedness is not rst-order denable on the class of graphs. In fact, it is not$1

1(mon) on the class of nite graphs.

62

Show that no rst-order sentence exists such that for all nite graphs

A: Aj= iAis connected.

Hint. Use models of the formCm and Cm+Cm, respectively (cf. Exer- cise 57).

Connectedness is $1

strict partial orderingwith a least element, such that ifyis an immediate -successor ofx, then xRy.

By Konig's Lemma B.7 (page 115), a nitely branching tree is nite i all of its branches are nite. Thus, niteness is #1

1(mon) on the class of

nitely branching trees. However:

3.37 Proposition.

Finiteness is not$1

1(mon) on the class of binary trees. a

Finally, here is an example of a partition argument.

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