Expressiveness and Complexity of Crosscut Languages

04/19/23 FOAL 2005 1

Expressiveness and Complexity of Crosscut Languages

Karl Lieberherr, Jeffrey Palm and Ravi Sundaram

Northeastern UniversityFOAL 2005 presentation

04/19/23 FOAL 2005 2

Goal

• Crosscut Languages are important in AOP– Encapsulate crosscuts– Delimit aspects

• Study them abstractly using expressions on graphs: lower bounds and upper bounds

• Assumption: know entire call or class graph

• Of interest to: AOSD language designers and tool builders

04/19/23 FOAL 2005 3

Are algorithmic results of any use to AOSD tool builders/users?

• YES!– Positive results: Fast algorithms lead to faster

tools.– Negative results: Indicate that we need to use

different kinds of algorithms.

04/19/23 FOAL 2005 4

Surprise

• Deciding pointcut satisfiability of an AspectJ pointcut using call, cflow and || and && on a Java program that only contains method calls (no conditionals) is NP-complete.

• pointcut satisfiability: Is there an execution of the program so that the pointcut selects at least one join point.

04/19/23 FOAL 2005 5

Insights

• AspectJ pointcuts and Demeter traversals have same expressiveness: Integration.

• Enhanced regular expressions on graphs and their instances are foundation for both.

• Enhanced regular expression evaluation on instances may be exponentially faster if graph (meta information) is used.

04/19/23 FOAL 2005 6

Canonical Crosscut Language

Enhanced Regular Expressions

AspectJ Pointcuts Traversal Strategies subset of XPath

04/19/23 FOAL 2005 7

Some PARC-Northeastern History about Crosscut Languages:

Enhanced Regular Expressions (ERE)>From [email protected] Thu Aug 31 13:33:57 1995 >To [email protected] (cc to Gregor, Crista, Boaz Patt-Shamir and Jens Palsberg et al.)Subject: Re: Boolean and Regular We seem to be converging, but I still think that enhanced regular

expressions can express all of the operators. Here is the enhanced regular expression language from a while back:

Atomic expressions: A The empty traversal at class Alnk A link of type lnk ("any" is a special case of any link type)

For combining expressions, the usual regular expression crowd: . concatenation \cap intersection \cup union * repetition not negation

04/19/23 FOAL 2005 8

My responseFrom lieber Thu Aug 31 13:51:57 1995From: Karl Lieberherr <lieber>To: [email protected], [email protected]: Re: Boolean and RegularCc: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]

Hi John:

yes, we agree. The operators of what I called Boolean algebra operatorsare just as well counted as regular expression operators.

I like your integration; have to think more about how expressive it is.

-- KarlCLAIM: ERE are a good foundation for crosscut languages.Confirmed by de Moor / Suedholt / Krishnamurti etc.

04/19/23 FOAL 2005 9

Enhanced Regular Expressions

• ERE = regular expressions (primitive, concatenation, union, star) with – complement/negation– nodes and edges (can eliminate need for

edges by introducing a node for each edge)

04/19/23 FOAL 2005 10

Same Lamping message continued:Demeter in ERE

[A,B] A.any*.B through edges any*.lnk.any* bypassing edges not(any*.lnk.any*) through vertices any*.A.any* bypassing vertices not(any*.A.any*) d1 join d2 [d1].[d2] d1 merge d2 [d1] \cup [d2] d1 intersect d2 [d1] \cap [d2] not d1 not([d1])

04/19/23 FOAL 2005 11

Using ERE for AspectJ

AspectJ

k (a primitive)

cflow(k)

&&

||

!

ERE

main any* k

main any* k any*

\cap

\cup

!

04/19/23 FOAL 2005 12

We continue the study of crosscut languages

• and show that AspectJ pointcuts are equivalent to Demeter strategies and vice versa if you abstract from the unimportant details.

• we show the correspondence by direct translations in both directions (rather than using ERE).

04/19/23 FOAL 2005 13

Examples first

• Show two programs and their graph abstractions

04/19/23 FOAL 2005 14

class Example { // AspectJ program public static void main(String[] s) {x1(); nx1();} static void x1() { x2(); nx2(); } static void x2() { x3(); nx3(); } static void x3() { target(); } static void nx1() { x2(); nx2(); } static void nx2() { x3(); nx3(); } static void nx3() { target(); } static void target() {}}aspect Aspect { pointcut p1(): cflow(call (void x1())) || cflow(call (void nx2())) || cflow(call (void x3())); pointcut p2() : cflow(call (void nx1())) || cflow(call (void x2())); pointcut p3() : cflow(call (void x1())); pointcut p4() : cflow(call (void nx3())); pointcut all(): p1() && p2() && p3() && p4(); before(): all() && !within(Aspect) { System.out.println(thisJoinPoint); }}

x1

main

nx1

x2 nx2

x3 nx3

targetmain x1 x2 x3 target nx3 target…

Meta graph=Call graph

Instance treeCall tree

Selected by all()

04/19/23 FOAL 2005 15

class Main { // Java Program with DJ X1 x1; Nx1 nx1; public static void main(String[] s) { ClassGraph cg = new ClassGraph(); Main m = new Main(); String strategy = "intersect(" + // union is expressed by concatenation of edges "{Main -> X1 X1 -> Target " + "Main ->Nx2 Nx2 -> Target " + "Main -> X3 X3 -> Target}," + "{Main -> Nx1 Nx1 -> Target " + "Main -> X2 X2 -> Target}," + "{Main -> X1 X1 -> Target}," + "{Main -> Nx3 Nx3 -> Target})“;cg.traverse(m, // m is the complete tree with 8 leaves strategy, new Visitor(){ public void start (){System.out.println(" start traversal");} public void finish (){System.out.println(" finish traversal");} void before (Target host){System.out.print(host + ' ');} void before (Nx3 host) {System.out.print(host + ' ');} void before (X2 host) {System.out.print(host + ' ');} void before (X1 host) {System.out.print(host + ' ');} });}}class X1 { X2 x2; Nx2 nx2; } class Nx1 { X2 x2; Nx2 nx2; }class X2 { X3 x3; Nx3 nx3; } class Nx2 { X3 x3; Nx3 nx3; }class X3 { Target t; } class Nx3 { Target t; }class Target {}

X1

Main

Nx1

X2 Nx2

X3 Nx3

TargetMain X1 X2 X3 Target Nx3 Target…

Meta graph=Class graph

Instance treeObject tree

Selected by strategy

04/19/23 FOAL 2005 16

Regular Expressions on Graphs

• Questions: Given graph G and reg. exp. r:– Is there a path in G satisfying r? (SAT)– Do all paths in G that satisfy r contain n in G?

(ALWAYS)

• Questions: Given graph G and reg. exps r1 and r2: – Is the set of paths in G satisfying r1 a subset of

the set of paths satisfying r2? (IMPL)

• What has this to do with AOSD?

ALL PROBLEMSARE POLYNOMIAL

Generalizes regular expressions on strings:sentences must be node paths in graphs. Work by Tarjan and Mendelzon/Wood.

04/19/23 FOAL 2005 17

Enhanced Regular Expressions on Graphs

• Questions: Given G and enh. reg. exp. r:– Is there a path in G satisfying r? (SAT)– Do all paths in G that satisfy r contain n in G? (ALWAYS)

• Questions: Given G and enh. reg. exps. r1 and r2: – Is the set of paths in G satisfying r1 a subset of the set of

paths satisfying r2? (IMPL)

• Ok, related to Demeter but how does AspectJ come in?

ALL PROBLEMSBECOME NP-COMPLETE

04/19/23 FOAL 2005 18

Crosscut Language SAJ

S ::= a set of nodes

k | set of nodes having label k

flow(S) | set of nodes reachable from S

S | S | union

S & S | intersection

!S complement

base language

04/19/23 FOAL 2005 19

Crosscut language SD

D ::= a set of paths

[A,B] | paths from A to B

D . D | concatenation of paths

D | D | union of paths

D & D | intersection of paths

!D complement of paths

base language

04/19/23 FOAL 2005 20

Crosscut Language

Graph• Path set• Defines set of

instance trees

Instance trees• Subtree or its leaves• Conform to a graph

(expansion)

04/19/23 FOAL 2005 21

Instance trees

Meaning of a crosscut language expression– Without meta graph

• Cannot look ahead: before we enter a join point we want to know whether it is selected based on information on the path back to the root: target node semantics.

– With meta graph• Can look ahead in meta graph: before we enter a join point we

want to know whether it is selected based on information on the path back to the root and if there is a possibility for success based on meta information: may use path set semantics. Include inner nodes, not just target nodes.

• Of course, we can always restrict semantics to target nodes.• May give exponential speedup.

04/19/23 FOAL 2005 22

Instance trees

AspectJ• Execution tree• Traversed anyway by

Java virtual machine• Can cut exponentially

the size of the tree where we pay attention to events

Demeter• Object tree• Traverse only what is

needed• Can cut exponentially

the tree to be traversed

04/19/23 FOAL 2005 23

Exponential improvement

• There is a sequence of crosscut expression/ meta graph/ instance triples (Qn; Dn; Pn) such that Pn conforms to Dn, |Qn| = O(n), |Dn| = O(n), and |Pn| = o(2n), and so that the naive evaluation will pay attention to o(2n) nodes in Pn while the meta-information-based evaluation will pay attention to O(n) nodes in Pn.

04/19/23 FOAL 2005 24

Expressions on GraphsExpressions on Instances

• Questions: Given graph G and r: Exists J sat G:– Is there a path in J satisfying r? (SAT)– For a given node m in G: Do all paths in J that

satisfy r contain a node n in J with Label(n) = m? (ALWAYS)

• Questions: Given G and r1 and r2: Exists J sat G:– Is the set of paths in J satisfying r1 a subset of

the set of paths satisfying r2? (IMPL)push down to instances

04/19/23 FOAL 2005 25

Connections between SAJ and SD

SAJ• selects set of nodes

in tree (but there is a unique path from root to each node)

• set expression flavor

SD• selects set of paths in

tree• regular expression

flavor

04/19/23 FOAL 2005 26

Equivalence of node sets and path sets

In a rooted tree, such as an instance tree, there is a one-to-one correspondence between nodes, and, paths from the root, because there is a unique path from the root to each node.

We say a set of paths P is equivalent to a set of nodes N if for each n in N there is a path p in P that starts at the root and ends at n and similarly for each p in P it is the case that p starts at the root and ends in a node n in N.

04/19/23 FOAL 2005 27

Theorem 1

• A selector expression in SD (SAJ) can be transformed into an expression in SAJ (SD) in polynomial-time, such that for all meta graphs and instance trees the set of paths (nodes) selected by the SD (SAJ) selector is equivalent to the set of nodes (paths) selected by the SAJ (SD) selector.

Motivation for theorem: SD and SAJ have identical complexity results.

04/19/23 FOAL 2005 28

Proof: T: SD to SAJ

SD

T([A,B])

T(D1.D2)

T(D1 | D2)

T(D1 & D2)

!D

SAJ

flow(A) & B

flow(T(D1)) & T(D2)

T(D1) | T(D2)

T(D1) & T(D2)

!T(D)

04/19/23 FOAL 2005 29

Proof: T: SAJ to SDfor a graph G

SAJ

T(k)

T(flow(S))

T(S1 | S2)

T(S1 & S2)

T(!S)

SD

[Start(G),k]

| [(Start(G),k].[k,Alph(G)]

T(S1) | T(S2)

T(S1) & T(S2)

!T(S)

Start(G): distinguished root of graphAlph(G): set of node labels of GUnion over all k in S and all elements of Alph(G)

04/19/23 FOAL 2005 30

class Example { // AspectJ program public static void main(String[] s) {x1(); nx1();} static void x1() {if (false) x2(); nx2(); } static void x2() { if (false) x3(); nx3(); } static void x3() { if (false) target(); } static void nx1() {if (false) x2(); nx2(); } static void nx2() {if (false) x3(); nx3(); } static void nx3() {if (false) target(); } static void target() {}}aspect Aspect { pointcut p1(): cflow(call (void x1())) || cflow(call (void nx2())) || cflow(call (void x3())); pointcut p2() : cflow(call (void nx1())) || cflow(call (void x2())); pointcut p3() : cflow(call (void x1())); pointcut p4() : cflow(call (void nx3())); pointcut all(): p1() && p2() && p3() && p4(); before(): all() && !within(Aspect) { System.out.println(thisJoinPoint); }}

x1

main

nx1

x2 nx2

x3 nx3

targetmain x1 x2 x3 target nx3 target…

Meta graph

Instance tree

Selected by all()

APPROXIMATION

04/19/23 FOAL 2005 31

Computational Properties

• Select-Sat: Given a selector p and a meta graph G, is there an instance tree for G for which p selects a non-empty set of nodes.

• X/Y/Z– X is a problem, e.g., Select-Sat– Z is a language, e.g. SAJ or SD– Y is one of -,&,! representing a version of Z.

• X/-/Z base language of Z. • X/&/Z is base language of Z plus intersection. • X/!/Z is base language of Z plus negation.

04/19/23 FOAL 2005 32

Approximation and Computational Properties

• Not Select-Sat: Given a selector p and a meta graph G, for all instance trees for G selector p selects an empty set of nodes, i.e. p is useless.

• If Not Select-Sat(p,G)/*/SAJ holds then also for the original Java program the selector p (pointcut) is useless.

04/19/23 FOAL 2005 33

Same results for 5 problems

• We don’t know yet how to unify all the proofs.

• So we prove the results separately.

04/19/23 FOAL 2005 34

Results (Problem)

Problem SD SAJ

- P P

& NP-complete NP-complete

! NP-complete NP-complete

04/19/23 FOAL 2005 35

Results (Problem)

• Results(Select-Sat)

• Results(Not Select-Impl)

• Results(Select-First)

• Results(Not Select-Always)

• Results(Not Select-Never)

04/19/23 FOAL 2005 36

Implementation

SD

• AP Library

• DJ

• DAJ

04/19/23 FOAL 2005 37

Future Work

• Complexity of more expressive crosscut languages, e.g., sequences.

04/19/23 FOAL 2005 38

Conclusions

• AspectJ pointcuts and traversal strategies are equivalent and founded on enhanced regular expressions and graphs as discussed in 1995.

• Surprising NP-completeness.

• Exponential improvement is possible if meta information is used.

• Several useful algorithms in paper.

04/19/23 FOAL 2005 39

Graph Theory for AOP

string graph/

instance tree

class graph/

instance tree

reg. exp. Kleene Mendelzon (SIAM Comp. 95, no instance trees)

Palsberg/Xiao/

Lieberherr (TOPLAS 95)

e. reg. exp Kleene PARC/Northeastern (summer 95)

Palsberg/Patt-Shamir/ Lieberherr (96)

Palsberg/Patt-Shamir/Lieberherr (96)

Palm/Sundaram/ Lieberherr (04)

strategy graph

? Patt-Shamir/ Orleans/Lieberherr (97,05)

Patt-Shamir/ Orleans/Lieberherr (97, 05)

Wand/Lieberherr (01)

04/19/23 FOAL 2005 40

Select-Sat

• Select-Sat/&/SAJ is NP-complete

• This is unexpected because we have only primitive pointcuts (e.g., call), cflow, union and intersection. Looks like Satisfiability of a monotone Boolean expression which is polynomial.

04/19/23 FOAL 2005 41

An idea by Gregor

• add a new primitive pointcut to AspectJ: traversal(D).

• cflow(call (void class(traversal({A->B})). foo())) && this(B)– in the cflow of a call to void foo() of a class

between A and B and the currently executing object is of class B.

04/19/23 FOAL 2005 42

Combining SAJ and SD

• Extend SD with [A,*]: all nodes reachable from A

• Replace in SAJ: flow(S) by nodes(D)

• Can simulate flow(S): use [X,*] for each X in S and take the union.

04/19/23 FOAL 2005 43

Crosscut Language SAJ/SD

S ::= a set of nodes

k | set of nodes having label k

nodes(D) | set of nodes selected by D in SD

S | S | union

S & S | intersection

!S complement

SAJ/SD seems interesting. Have both capabilities of AspectJ pointcutsand Demeter traversals.This is basically what Gregor Kiczales suggested a few years ago:he called ittraversal(D), instead of nodes(D).

04/19/23 FOAL 2005 44








04/19/23 FOAL 2005 45

SAT: is there a path in G satisfying r?

graph G/

instance tree

class graph G/

instance tree

reg. exp. r Mendelzon (SIAM J. Comp. 95, no instance trees): polynomial

Palsberg/Xiao/

Lieberherr (TOPLAS 95): polynomial (special case)

e. reg. exp

r

PARC/Northeastern (summer 95)

Palm/Sundaram/ Lieberherr (04): NP-complete

Palm/Sundaram/ Lieberherr (04): NP-complete

strategy graph r

Patt-Shamir/ Orleans/Lieberherr (97,05): polynomial

Patt-Shamir/ Orleans/Lieberherr (97, 05): polynomial

04/19/23 FOAL 2005 46

SAT: is there a path in G satisfying r?

graph G

reg. exp. poly.

e. reg. exp. NPC (add negation)

strat. graph poly.

e. strat. graph NPC (add intersection/negation)

SAJ (AspectJ) NPC

SD (Demeter) NPC

SAJ-base poly. (without intersection)

SD-base poly. (without intersection)

results identical for class graphs

04/19/23 FOAL 2005 47

Abbreviations

Language Abbreviation

regular exp. RE

enhanced regular exp. ERE

strategy graph SG

enhanced strategy graph ESG

SAJ (AspectJ) SAJ

SD (Demeter) SD

SAJ-base SAJB

SD-base SDB

04/19/23 FOAL 2005 48

Polynomial Translations

• We want to know which languages are fundamental. We conjecture that all languages can be translated in polynomial time into ERE. Maybe we also need ESG?

• The translations must preserve the meaning: – same set of nodes or – same set of paths or – set of paths corresponding to a set of nodes or – set of nodes corresponding to a set of paths.

04/19/23 FOAL 2005 49

Motivation for polynomial translations

• If a large number of languages can be translated efficiently to ERE, we only need an efficient implementation for ERE.

• Currently the AP Library uses SG with intersection. If we would add complement, the AP Library would use ESG.

04/19/23 FOAL 2005 50

Polynomial Translations ( any mistakes?)

RE ERE SG ESG SAJ SD SAJB SDB

RE Y Y Y N NERE NN NSG YESG NN NSAJ Y Y Y N NSD Y Y Y N NSAJB Y Y Y Y YSDB Y Y Y Y NN

translate row to column N: no, unless P=NP; NN: no Y: yes

04/19/23 FOAL 2005 51

Crosscut language SDk


[A,B]k | paths from A to B of length = k

[A,B]k bypassing {A1,…} ignore {A1,…}





base language:SDB

see work on poly lingual systems

04/19/23 FOAL 2005 52




[A,B] bypassing {A1,…} ignore {A1,…}





base language:SDB

04/19/23 FOAL 2005 53

Discussion

• some results are trivial: an RE sentence is trivially an ERE sentence.

• an ERE sentence can not be translated in polynomial time to an RE sentence because negation cannot be simulated by union et al.

• An SAJ sentence cannot be translated to an SDB sentence in polynomial time because otherwise P=NP (consider SAT).

04/19/23 FOAL 2005 54

Assignments

• We want to fill in all 64 entries and have a proof for them. This is a good opportunity for a beginning PhD student.

• Yuantai: please can you do the upper triangle.

• Jingsong: please can you do the lower triangle.

04/19/23 FOAL 2005 55

• Puntingam: non regular process types

• Some context-free, context-sensitive

• FSM with counters: the same?– Reussner

04/19/23 FOAL 2005 56

Mario

• Given G and sequence of reg. exps. r1, r2. r1 and r2 are over the same alphabet.– Is there a pair of paths in G satisfying r1 and

r2? Node selected by r1 < Node selected by r2.

– After having visited a node satisfying r1, how can we find all nodes satisfying r2?

– Instance-level dependencies between r1 and r2?

04/19/23 FOAL 2005 57

Instance Tree

J is called an instance tree of graph G, if J is a tree, Root(J)=Start(G) and for each edge e=(u,v) in E(J), there is an edge e’ = (u’, v’) in G so that Label(u)=Label(u’) and Label(v)=Label(v’). J is a rooted tree with edges directed away from the root. (think of Label = Class)

04/19/23 FOAL 2005 58

04/19/23 FOAL 2005 59

Quality of model

• Meta graph defines set of instances– Precisely

• Class graph

– Too many• Call graph• Pcflow: what traversals do: use meta information

04/19/23 FOAL 2005 60

FIRST

• Given a reg. exp. r, a graph G, compute for each node n in G the set of outgoing edges from n that are part of a path p from Start(G) through n to a node so that p satisfies r.

• Polynomial for regular expressions and NP-complete for enhanced regular expressions. see TOPLAS 2004 paper

Documents

Expressiveness and Complexity of Crosscut Languages