monadic second-order logic - tata institute of …pandya/grad/aut06/lect3.pdf0.5 setg r a y0 0.5...

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Monadic Second-Order LogicAutomata: Theory and Practice

Paritosh K. Pandya

(TIFR, Mumbai, India)

Unversity of Trento

10-24 May 2005

Trento’2005 – p. 1/31

First-order logic

Language for describing properties of MathematicalStructures.

Precise and rigorous.Rules and techniques for sound inference.

Structure = (Domain, Constants, Functions, Relations)Domain

�

Set of values.Constants Some specific values with names.Functions Operation

� � � �Relations Operation

� � � �� .

Trento’2005 – p. 2/31

Arithmetic

Natural Numbers �

Peano Arithmetic:

�� Presburger Arithmetic:

� ��

Successor Arithmetic:

��

Real Arithmetic

��

Signature (Symbol set)

� � � !� "�Signature gives symbols that can be used in Formulae.

Example:

�$# %&(' )* � �� +� � � � � and�$,- '. /10 -2 ' - � �� +� � � .

Trento’2005 – p. 3/31

First Order Logic (2)

3 ��

First Order Logic with Equality over signature�

Example:

�# %&(' ) * � �� . Following are3 ��

formulae.465 467�8 � � 5 9 � � 7� : 5 � 7 ; �465 467 46<8 5 � 7 � <� � 5 � 7 � 5 � <465 467�8 5 � 7 : = <8 5 � < 9 < � 7�465 8 5� �> � � 5�

Trento’2005 – p. 4/31


3 ��


Example:

�# %&(' ) * � �� . Following are3 ��

formulae.465 467�8 � � 5 9 � � 7� : 5 � 7 ; �465 467 46<8 5 � 7 � <� � 5 � 7 � 5 � <465 467�8 5 � 7 : = <8 5 � < 9 < � 7�465 8 5� �> � � 5�

Syntax

� @? � 8 8 8� �� A ? � @B A C? 9 CB A D C A = 5 8 C

where term is� A E ? � 8 8 8� �� .

Trento’2005 – p. 4/31


3 ��


Example:

�# %&(' ) * � �� . Following are3 ��

formulae.465 467�8 � � 5 9 � � 7� : 5 � 7 ; �465 467 46<8 5 � 7 � <� � 5 � 7 � 5 � <465 467�8 5 � 7 : = <8 5 � < 9 < � 7�465 8 5� �> � � 5�

Syntax

� @? � 8 8 8� �� A ? � @B A C? 9 CB A D C A = 5 8 C

where term is� A E ? � 8 8 8� �� .�

structure

F

hasDomain

A F AE G

and

� Gas interpretation of

E

and

�

Trento’2005 – p. 4/31

Semantics

Notation:

C 5? � 5B � 8 8 8� 5�� means

H�I � � C� J K 5? � 8 8 8� 5� L

F� ? � 8 8 8� � � A� C 5? � 8 8 8� 5��

Defined as usual. (omitted.)

Example:

� � �� A� 4678 5� 7> 5 � 7�Sentence is formula without free variables.Example:

� A� 465 8 5 � 5 � � �.

Trento’2005 – p. 5/31

First-order logic of Finite Words

Finite word

� � M ? B 8 8 8 ON 8 8 8 ��P ?Q�R � � S 8 8 8 T 8 8 8 UWV �

XZY 5�

At position 5the letter is .� 5� 7�

Position 7is next to position 5.5 � 7Position 5is before position 7.

Example Every is followed by[4 5 8 XZY 5� : = 78 � 5� 7� 9 X / 7�

Trento’2005 – p. 6/31

Finite Word Structure

Specifications F.O. Sentences with equality over�]\ #- * .

Signature

�\ #- * � �� XZY � Y ^ _� .

A word ` a b c

gives the word structure

Given sentence

C

over

�\ #- * , we can evaluate` A� C

Langauge of a sentence

C

3 C� dfe g� K ` a h c A ` A� C L

Trento’2005 – p. 7/31

Finite Word Structure

Specifications F.O. Sentences with equality over�]\ #- * .

Signature

�\ #- * � �� XZY � Y ^ _� .

A word ` a b c

gives the word structure` � i � j `� � � \ � � \ � X \Y � Y ^ _�where for ` � M B 8 8 8 �P ? ,i � j `� � K �� 8 8 8� UV � L

.� \ k � l �

iff

l � k � �X \Y k �

iff � &

Given sentence

C

over�\ #- * , we can evaluate` A� C

Langauge of a sentence

C

3 C� dfe g� K ` a h c A ` A� C L

Trento’2005 – p. 7/31

Examples

Let

b � K � [�� L

.

Ek I R 5� de g� D = 78 � 7� 5�

. Similarly,

� R 5�.

C dfe g� = 5 8 Ek I R 5� 9 XZY 5� 9= 78 � 5� 7� 9 � R 7� 9 X / 7� � �Then, [ A� C [� mA� C no mA� C

All words s.t. between an and the next

[

there are atleast two�465 4678 XZY 5� 9 5 � 7 9 X / 7� 9 46<8 5 � < � 7 : D X / 7� �

: = <� = <S 8 5 �� <� � <S �� 7 9 Xqp <� � 9 Xqp <S� �

Trento’2005 – p. 8/31

Monadic Second Order Logic ( )

First order logic extended with

Set variables

r? � rB � 8 8 8. These are also calledmonadic predicates.

construct

r& �

. Some people write a r& .

Quantification over set variables:= r& 8 C and

4r& 8 C.Semantics Let

s& J A F A F� s? � 8 8 8� sN � ? � 8 8 8� � � A� C r? � 8 8 8� rN � 5? � 8 8 8� 5��

Example 7is upper bound of set

s � � K � � t� u L � � � � A� 4 5 8 s 5� � 5 v 7

Trento’2005 – p. 9/31

MSO over words

Examples:

Even length finite words over

h

.= r 8 465 8 Ek I R 5� : r 5� � 9 4 78 � 5� 7� 9 r 5� : D r 7� � 9 4 78 � 5� 7� 9 D r 5� : r 7� � 9 � R 5� : D r 5� �

Exercise Infinite words where every even position hasletter .

Question (a) What is the class of (finite word) languagesdefinable using

�w? (b) What is the class of � languages

definable using�w

?

Trento’2005 – p. 10/31

Buchi Theorems

Theorem 1 (Buchi60, Elgot61) A language (fromb c

) isrecognisable by a finite state automaton if and only if it is�w ��

definable, and both conversions are effective.

Theorem 2 (Buchi61) A language (fromb x

) is recognisableby a Nondeterministic Buchi Automataon if and only if it is�w ��

definable, and both conversions are effective.

Trento’2005 – p. 11/31

Proof of Theorem 1 (NFA to )

Let

h � X� by� z� {� !

with

X � K �� 8 8 8� T L

.Consider ` � M 8 8 8 �P ? , run | �� 8 8 8 | U� .r& contains positions

l

where | l � � k

.}

is

& ~�� 465 D r& 5� 9 r � 5� � � 9 465 8 Ek I R 5� : > & ^ � r& 5� � 9 465� 78 � 5� 7� :> � &�� Y � � � ^ � r& 5� 9 XZY 5� 9 r � 7� � � 9 465 8 � R 5� : > � ^ � r � 5� � 9 XZY 5� �

Claim accepts ` iff` A� = rM 8 8 8 rN 8 }

Trento’2005 – p. 12/31

Logic �

Simplified

� w

with no first-order variables.

Atomic Formulaer J �

, set

r

is subset of

��k U� r�

set

r

has exactly one element.�� r� ��

sets

r� �

are singleton andelement in

�

is successor of element in

r

.r J XZY . Subset

We can translate

�wto

�w M .Example:

465 8 XZY 5� : = 7�8 � 5� 7� 9 � 7� �

becomes:

4r 8 �k U� r� 9 r J XZY := � 8 �k U � �� 9 �� r� �� 9 � J ��

Trento’2005 – p. 13/31

Models of �

A model for

C r? � 8 8 8� r �� is

`� s? � 8 8 8� s � � .

Example

4r 8 �k U� r� 9 r J XZY := � 8 �k U� �� 9 �� r� �� 9 � J ��Model

[ [� K �� t� � L�

Example

4�� 8 � { � " �� 9 � J rM 9 � J X /9 �� : J rM �

Model

[ [ [ [ � K �� S� u� � L �

Trento’2005 – p. 14/31

Continued

Representation

` � s? � 8 8 8� s � � as word overb � � b1� K �� L �

.

Example Model

[ [� K �� t� � L �

is encoded aspos 0 1 2 3 4word a b a a bP0 1 1 0 1 1Example Model

[ [� K �� L � K �� t L �is encoded as

pos 0 1 2 3word a b a aP0 1 1 0 0P1 1 0 0 1

Trento’2005 – p. 15/31

Proof ( � to Automata)

Given

C r? � 8 8 8� r �� construct

h�� C�

over alphabetb1� K �� L �

. Each letter (is a column vector that) has theform

� � [? � [B � 8 8 8� [ � � . .

Example

r �� 9 4Q 8 r Q� : =6� 8 �� Q� � � 9 � �� .

Construction

h� C�

is by Induction on the structure of

C

.The construction is bottom-up. Automata for subformulaeare combined to make the automaton for whole formula.

Trento’2005 – p. 16/31

Atomic Formulae

h�� r& J r � �h�� k U� r& � �h�� r � � rN � �

h�� r � J XZY � .

Trento’2005 – p. 17/31

Proof (cont)

h� C? 9 CB � � h� C? � � h� CB �h� C? > CB � � h� C? � � h� CB � .h� D C� � h�� C�

= r& 8 C r? � 8 8 8� r ��

Use projection to “erase”

r& in the automaton forC r? � 8 8 8� r �� .i.e.h = r& 8 C r? � 8 8 8� r �� h C r? � 8 8 8� r � � �

Complexity Non-elementary. Every

>

or

= r 8 C addsnondeterminism. Complexity grows by one exponent withevery alternation of

= rand D.

Corollary Any�w ��

formula can be written as

� �w ��

formula.Trento’2005 – p. 18/31

Tool MONA

Validity checker for MSO on Words.

Given

C r? � 8 8 8� r �� constructs automatonh C�

overalphabet

K �� L �

.

Finds a model and counter-model of the formula byfinding accepting/rejecting path.

Finds if formula is valid or unsatisfiable.

Efficient BDD-based representation of transition table.Keeps automata in minimal, deterministic form.

D. Basin, CAV 2001 tutorial:http://www.inf.ethz.ch/personal/basin/teaching/scripts/cav01.pdfN. Klarlund MONA Overview:http://www.clairgrove.com/papers/Mona-tutorial.pdf

Trento’2005 – p. 19/31

MONA Example

Example

4 5 8 � Ek I R 5� : s 5� � 9 � R 5� : X 5� � 9 D � R 5� : D X 5� � �

4

2

XX

XX

3 X1

X0

0 1

XX

11

0X

10

Trento’2005 – p. 20/31

Finite Tree Languages

Tree Domain

i � j � � K �� L c

with conditionsi � j � is finite, 5� a i � j � : 5 a i � j �5� a i � j � � 5� a i � j � .

Tree

�� i � j � � b

.Use

�¡ for all

b

=labelled tries. Tree language

3 J � .

Example All trees such that any subtree with label at itsroot has a node inside labelled

[.

Trento’2005 – p. 21/31

MSO over finite trees

Logic MSO with equality over signature �M � �? � � � XZY � Y ^ � .�M 5� 7�

means node 7is left child of 5.�? 5� 7�

means node 7is right child of 5.5 � 7means 5is a proper ancestor of 7.The individual variables range over tree nodes, i.e.

i � j � .

Example

4 5 8 XZY 5� : = 7�8 5 � 7 9 X / 7� �

Formally

� i � j � � �M � �? � � � XZY � Y ^ � .Then for sentence

C, we have

3 C� � K A A� C L

.

Trento’2005 – p. 22/31

Tree Automaton

A bottom-up tree automaton

h � X� by� ¢� �M � !�where�M a X

is the initial state,

! J X

is the set of final states.¢ J X� b1� X� X

.

A run | on tree

is a map |� i � j � � Xsatisfying

transition relation. Run is accepting if | ¤£ � a !.

Initialize leaf � with symbol to � using

�� M � �M � a ¢�

Label non-leaf � with symbol to � using

� � � � � ? � � B � a ¢�

provided children of � are labelled � ? and � B .

Theorem (Thatcher,Wright) A set of finite trees isrecognizable by finite tree automata iff it is MSO definable.

Trento’2005 – p. 23/31

Proof of Buchi Theorem 2

Exactly follows the proof of Theorem 1, but using BuchiAutomata.

The theory has been extended to MSO over finite andinfinite trees.

Even MSO over graphs can be defined.

Trento’2005 – p. 24/31

MSO ove Infinite Words

Word ¥ � M ? 8 8 8 a b x

gives structure¥ � i � j � Q ¦6 � � �� XZY �

where

i � j ¥� � �, the set of natural numbers.

Logic

�w b x �

. We can specify properties of omegawords using FOL or MSO sentence

C. This logic is also

called S1S.

� Language

3 x C� de g� K ¥ a h x A ¥ A� C L

Examples

Every is followed by[465 8 XZY 5� : = 78 � 5� 7� 9 X / 7�

occurs infinitely often in the � word.465= 78 5 � 7 9 XZY 7�

.

Trento’2005 – p. 25/31

Logic WS1S

Logic WS1S has same syntax as S1S. Domain is� � K �� S� 8 8 8 L.Set variables take finite subsetes of �.Example

C de g� = r 8 r �� 9 465 8 r 5� : = 78 5 � 7 9 r 7�

.Then

A� §? § C

. However

mA� ¨ §? § C. (Why?)

Models of WS1S formulae are finite words.

WS1S can be decided using Finite State Automata.(Details in Classroom)

We can decide Presburger Arithmetic Sentences usingWS1S. (Details in Classroom.)

Trento’2005 – p. 26/31

Presburger Arithmetic in WS1S

Preburger arithmetic formulae are First Order Logicwith Equality formulae using signature

�� .Multiplication symbol is not alloweed.

The formulae are interpreted overNaturnal Numbers

©

.

Example:

465 8 = 78 5 � � � 7Encoding of Numbers

Encode number U as reverse binary string I [k U U� .Further encode this binary string as finite set � U� � i � U�

giving positions of�

bits in I [k U U� .

Example: I [k U �� . Hence, � U� � i � �� K � � S L

.

Map � U� � i � � K r J %& � © L � ©

is a bijection.

Trento’2005 – p. 27/31

Encoding Presburger Formulae

Encoding Formulae

We can model a variable 5over natural numbers by aset variable

r

.

We define a formula

s 3ª � r� �� and

3 � � � r� ��

such thats 3ª � � U� � i � 5� � � U� � i � 7� � � U� � i � �� 5 � 7� <3 � � � � U� � i � 5� � � U� � i � 7� � � 5 � 7.

The encoding of

s 3ª �is a WS1S formula mimicing the

binary full adder circuit. Define

3 � � �

using

s 3ª �

andequality.

Trento’2005 – p. 28/31

(Cont)

Given Presburger formula

C 5? � 8 8 8� 5�� we can defineWS1S formula

« C r? � 8 8 8� r �� such that©¬� j ? � 8 8 8� j � A�C 5? � 8 8 8� 5� � � ©¬� � U� � i � j ? � � 8 8 8� � U� � i � j �� A�« C r? � 8 8 8� r �� .

Example: For

C� 4 5 8� 5kR R 78 5 � � � 7we have

« C

as4r 8 = � 8 = � 8 s 3ª � r� K � L � �� 9 3 � � � ��

.

Trento’2005 – p. 29/31

Omega Tree Automata

A muller omega-tree automaton

h� X� by� ¢� �M � ! ��withz J X� b1� X� X

.

Trento’2005 – p. 30/31

Omega Tree Automata



.

A run | of

h

on omega-tree

is a map |� K �� L � � b


The run is accepting if every infinite path from root satisfiesthe Muller acceptance condition.

Trento’2005 – p. 30/31

Omega Tree Automata



.

A run | of

h

on omega-tree

is a map |� K �� L � � b


The run is accepting if every infinite path from root satisfiesthe Muller acceptance condition.

Example Construct Muller Omega-tree automaton whichaccepts all trees having at least one path with infinitelymany

[

. (AlphabetK � [ L

).

Trento’2005 – p. 30/31

Rabin Tree Theorem

Difficult Result: Muller-tree Automata Can be Determinisedand Complemented.

Theorem Theory S2S is decidable.

Theorem For any sentence

C

of

�wover omega-trees we

can effectively construct a Muller tree automaton

h C�

suchthat A� C

iff

a 3 h C� �

.

Trento’2005 – p. 31/31

monadic second-order logic - tata institute of …pandya/grad/aut06/lect3.pdf0.5 setg r a y0 0.5...

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