Decision Procedures for Recursive Data Structures

8/9/2019 Decision Procedures for Recursive Data Structures

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Motivation Problem-Outline The structures Recursive Data Structures The Integrated Theory

Decision Procedures for Recursive Data Structures

Andreas Sander

Decision Procedures for Logical Theories, SS 2010

16.06.2010
http://find/http://goback/


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Motivation

Intuition

A recursive data structure is (partially) composed of smaller orsimpler instances of the same structure.


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Motivation

Intuition


Example: cons(1,cons(2,cons(3,nil)))


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Motivation

Intuition



Main Goal

The main goal is to define decision procedures for recursive data

structures.

M i i P bl O li Th R i D S Th I d Th


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Motivation

Intuition



Main Goal

The main goal is to define decision procedures for recursive data

structures.Whats about:|cons(1,cons(2,cons(3,nil)))| > 3 ?

M ti ti P bl O tli Th t t R i D t St t Th I t t d Th


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Problem-OutlineWhat we have and what we want

Theory of recursive data structures Th(A)

Motivation Problem Outline The structures Recursive Data Structures The Integrated Theory


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Presburger Arithmetic Th(AZ)



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Length Function

|.| : Z



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Length Function

|.| : Z

GoalGet a decision procedure for the combination of Th( A), Th(AZ)and |.| : Z



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Problem-OutlineApplicability of Nelson & Oppen

We want to combine this with Nelsons and Oppens method:





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g y


We want to combine this with Nelsons and Oppens method:



Length Function|.| : Z



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g y


We want can not combine this with Nelsons and Oppens method:



Length Function|.| : Z



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g y

Problem-OutlineThe Solution

Solution

Extension of Nelsons and Oppens original method!



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Solution


Overview of this talk:



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Solution



Presburger Arithmetic

Decision Procedure



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Solution




Decision Procedure

Theory of recursive data structure

Decision Procedure



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Solution




Decision Procedure


Decision ProcedureExtension of Nelsons and Oppens method in the presence oflength |.|



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first-order theory



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first-order theory

corresponding structure AZ : Z; 0, s,+,,


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first-order theory



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first-order theory



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first-order theory



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first-order theory



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first-order theory



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first-order theory



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first-order theory



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Recursive Data Structures

A : ; A, C,S, T consists of



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a data domain



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a data domain

a set of atoms A (e.g. A = {a, b, c,...})



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a data domain


a finite set of constructors C (e.g. C = {,,,...})



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a data domain



a finite set of selectors S (e.g. if has arity k > 0:

{s1 , ..., sk } S)



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a data domain



a finite set of selectors S (e.g. if has arity k > 0:

{s1 , ..., sk } S)

a finite set of testers T (e.g. Is(x) is true iff x is an -term.)


A f A A (LISP l )


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Axiomatization ofA = AList (LISP list structure)

A Example AList


A i i i f A A (LISP li )


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A Example AListt(x) = x cons (x, y) = x


A i i i f A A (LISP li )


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A Example AList

a = (x1, ...,xar()) a = cons(x, y)


A i ti ti f A A (LISP li t t t )


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A Example AList

(x1, ..., xar()) = (y1, ..., yar()) cons(x, y) = cons(z, t)

1iar() xi = yi x = z y = t


A i ti ti f A A (LISP li t t t )


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A Example AList

IsA(x) C Is(x) IsA(x) Iscons(x)


Axiomatization of A A (LISP list structure)


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A Example AList

si

(x) = y z((z) = x y = zi) cdr(cons(x, y)) = y(z((z) = x) x = y) IsA(x) {car, cdr}+(x) = x



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Goal

Check satisfiability of

q1 = r1 ... qk = rk s1 = t1 ... sl = tl

in Th(A).



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Goal


q1 = r1 ... qk = rk s1 = t1 ... sl = tl

in Th(A).

Idea

Use a Directed Acyclic Graph (DAG) for those formulae.


Term Representation


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Term-Representation

Term t can be represented by a tree Tt, s.t.


Term-Representation


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Term-Representation


1 If t is a constant, Tt is a leaf vertex (labeld by t)


Term-Representation


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Term-Representation



2 If t = (t1, ..., tk), then Tt is a tree (root labeled by t),having Tt1, ...,Ttk as its subtrees (ordered).


Term-Representation


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Term Representation



2 If t = (t1, ..., tk), then Tt is a tree (root labeled by t),having Tt1, ...,Ttk as its subtrees (ordered).

A directed acyclic graph (DAG) Gt of t obtained from Tt byfactoring out common subtrees.



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Goal


q1 = r1 ... qk = rk s1 = t1 ... sl = tl

in Th(A).



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Goal


q1 = r1 ... qk = rk s1 = t1 ... sl = tl

in Th(A).

Idea

Use a Directed Acyclic Graph (DAG) for those formulae.

1 Add (qi, ri)(i {1,...k}) to an equivalence relation R on

vertices2 Close R under congruences and unification.


Groundwork


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Groundwork

G = (V,E) a DAG (with ordered edges).


Groundwork


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Let be u V


Groundwork


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Let be u V

outdegree (u)


Groundwork


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Let be u V

outdegree (u)

the ith successor u[i] (i [0; (u)])


Groundwork


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Let be u V

outdegree (u)

the ith successor u[i] (i [0; (u)])

R is an equivalence relation on the vertices of G.


Bidirectional Closure vs. Congruence Closure


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Bidirectional Closure instead of Congruence Closure [Oppen 1980]

For the following decision procedures we will use the bidirectional

closure, instead of the congruence closure!


Bidirectional Closure vs. Congruence Closure


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Bidirectional Closure instead of Congruence Closure [Oppen 1980]

For the following decision procedures we will use the bidirectional

closure, instead of the congruence closure!

Idea

cons(x, y) = cons(z, t) x = z y = t


The Congruence Closure R


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R is the unique minimal extension of R with the followingproperties:

R is an equivalence relation

Two vertices u,v with equal (nonzero) outdegree:

(u, v) R

(u[i], v[i]) R i [1; (u)]


The Unification Closure R


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(u, v) R

(u[i], v[i]) R i [1; (u)]


The Bidirectional Closure R


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Theorem [Oppen 1980]




(u, v) R

(u[i], v[i]) R i [1; (u)]


Oppens Decision Procedure for acyclic A


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Input: : q1 = r1 ... qk = rk s1 = t1 ... sl = tl

Task: Check satisfiability of




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Algorithm 11 Step 1: Construct the DAG G of




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2 Step 2: Compute R of R = {(qi, ri)|1 i k}.




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2 Step 2: Compute R of R = {(qi, ri)|1 i k}.

3 Return UNSATISFIABLE if i(si, ti) R;

Return SATISFIABLE otherwise.


Problem for Cyclic Data Structures


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In our setting

If x is NOT an -term, then si

(x) = x




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In our setting


(x) = x

ComplicationWe dont know a priori wether s(x) is a proper subterm of x.




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In our setting


(x) = x

ComplicationWe dont know a priori wether s(x) is a proper subterm of x.

Solution for Complication

We have to guess the type of all terms occuring inside an selectorfunction before applying the algorithm.


Oppens Decision Procedure for cyclic A


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Input : q1 = r1 ... qk = rk s1 = t1 ... sl = tl





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Algorithm 2

1 Step 1:Guess a type completion

of and simplify selectorterms accordingly.




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Algorithm 2

1 Step 1:Guess a type completion

of and simplify selectorterms accordingly.

2 Step 2: Call Algorithm 1 on


Example Application of Algorithm 2


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Example

cons(y,z) = cons(cdr(x),z) cons(car(x),y) = x

Unification Closure

(u, v) R

(u[i], v[i]) Ri [1; (u)]


The Structure B


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Overview - Where Are We Now?


Decision Procedure


The Structure B


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Decision Procedure


Decision Procedure


The Structure B


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Decision Procedure


Decision Procedure

Extension of Nelsons and Oppens method in the presence oflength |.|


The Structure B


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Decision Procedure


Decision Procedure


The Structure of the Integrated Theory

B = (

A;AZ; |.| :

Z)


The Structure B


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Decision Procedure


Decision Procedure



B = (

A;AZ; |.| :

Z)

For any atom a, |a| = 1


The Structure B


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Decision Procedure


Decision Procedure



B = (

A;AZ; |.| :

Z)

For any atom a, |a| = 1

For a term (t1, ..., tk), |(t1, ..., tk)| =

k

i=1 |ti|.


Why not Nelson & Oppen?


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Why?

We want can not combine this with Nelsons and Oppens method


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Why not Nelson & Oppen?


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Why?

We want can not combine this with Nelsons and Oppens method

Consider BList,

: x = cons(car(y),y) and Z :|x|


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IdeaExplicitly compute the hidden constraints!

Length constraint :


Induced Length Constraint


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Length constraint :

is sound, if satisfying of , || is a satisfyingassignment for .




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Length constraint :

is complete, if whenever is satisfiable, for any satisfyingassignment of there exists a satisfying assignment

of s.t. || = .




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Length constraint :

is induced by , if is sound & complete.


Satisfiability ofB


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Main Theorem

Let be in the form Z . Let be the induced lengthconstraint with respect to .

is satisfiable in B

Z is satisfiable in AZ and is satisfiable in A


Predicates based on the DAG


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How to compute the induced length constraints?

Tree(t) : x1, ..., xn 0 (|t| = (

n

i=1(di 1)xi) + 1)




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Tree(t) : x1, ..., xn 0 (|t| = (

n

i=1(di 1)xi) + 1)

Node(t, t) : |t| =()

i=1 |ti|




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Tree(t) : x1, ..., xn 0 (|t| = (

n

i=1(di 1)xi) + 1)

Node(t, t) : |t| =()

i=1 |ti|

Tree(t) : t(Node(t, t) ()

i=1 Tree(ti))


Constructing Length Constraint


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Algorithm 3 (Construction of in B)

Let be (type-complete) data constraint. G the DAG of andR bidirectional closure.

From initial state = add the following




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|t| = 1 if t is an atom




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|t| = |s|, if (t, s) R




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|t| = |s|, if (t, s) R

Tree(t) if t is an untyped leaf vertex.




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Algorithm 3 (Construction of in B)Let be (type-complete) data constraint. G the DAG of andR bidirectional closure.



|t| = |s|, if (t, s) R


Node

(t, t) if t is an -typed vertex with children t




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Algorithm 3 (Construction of in B)Let be (type-complete) data constraint. G the DAG of andR bidirectional closure.



|t| = |s|, if (t, s) R


Node

(t, t) if t is an -typed vertex with children t

Tree(t) if t is an -typed leaf vertex.


Some Deductions


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Proposition 2

obtained by Algorithm 3 is expressible in a quantifier-freePresburger formula linear in the size of .

Theorem 1

obtained by Algorithm 3 is the induced length constraint of .


Decision Procedure for a Quantifier-Free B


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Input: Z.

1 Guess a type completion

of .

2

Call Algorithm 1 on

.Return FAIL if

is unsatisfiable; continue otherwise.

3 Construct from G

using Algorithm 3.

Return SUCCESS if is satisfiable and Z is satisifiable.

Return FAIL otherwise.


Example Application of Algorithm 4


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Example

x = cons(car(y),y) |cons(car(y),y)| < 2|car(x)|.


Some Final Remarks


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In Theories were quantifiers (,) occurs, we can usequantifier-elimination to get rid of the quantifiers!

Application on theories with finite number of atoms is alsopossible, but its far more complicated!



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Thank you for your attention!


Literature


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Zhang, T.; Sipma H.B.; Manna,Z.:

Decision Procedures for Recursive Data Structures with

Integer Constraints.Springer-Verlag, 2004.

Oppen,D.C:

Reasoning About Recursively Defined Data Structures.Journal of the Association for Computing Machinery (Vol. 27),1980

Nelson,G; Oppen, D.C:

Fast Decision Procedures Based on CongruenceClosure.Journal of the Association for Computing Machinery (Vol. 27),1980

Decision Procedures for Recursive Data Structures

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