Expressiveness and Complexity of Crosscut Languages

1
Expressiveness and Complexity of Crosscut
Languages

• Karl Lieberherr, Jeffrey Palm and Ravi Sundaram
• Northeastern University
• FOAL 2005 presentation

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

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.

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.

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.

6
Canonical Crosscut Language
7
Some PARC-Northeastern History about Crosscut
LanguagesEnhanced Regular Expressions (ERE)

• gtFrom lamping_at_parc.xerox.com Thu Aug 31 133357
  1995
• gtTo lieber_at_ccs.neu.edu
• (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 A
• lnk 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

8
My response

• From lieber Thu Aug 31 135157 1995
• From Karl Lieberherr ltliebergt
• To lamping_at_parc.xerox.com, lieber_at_ccs.neu.edu
• Subject Re Boolean and Regular
• Cc Gregor_at_parc.xerox.com, boaz_at_ccs.neu.edu,
  crista_at_ccs.neu.edu,
• huersch_at_ccs.neu.edu, ivbaev_at_ccs.neu.edu,
  palsberg_at_theory.lcs.mit.edu,
• salil_at_ccs.neu.edu, seiter_at_ccs.neu.edu,
  yangl_at_ccs.neu.edu
• Hi John
• yes, we agree. The operators of what I called
  Boolean algebra operators
• are just as well counted as regular expression
  operators.
• I like your integration have to think more about
  how expressive it is.
• -- Karl

CLAIM ERE are a good foundation for crosscut
languages. Confirmed by de Moor / Suedholt /
Krishnamurti etc.
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)

10
Same Lamping message continuedDemeter 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)

11
Using ERE for AspectJ

• AspectJ
• k (a primitive)
• cflow(k)
• !

• ERE
• main any k
• main any k any
• \cap
• \cup
• !

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).

13
Examples first

• Show two programs and their graph abstractions

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(thisJoinPo
int)
Meta graph Call graph
main
nx1
x1
x2
nx2
x3
nx3
main x1 x2 x3 target
nx3 target
target
Instance tree Call tree
Selected by all()
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 -gt X1 X1 -gt Target "
"Main -gtNx2 Nx2 -gt Target "
"Main -gt X3 X3 -gt Target," "Main
-gt Nx1 Nx1 -gt Target " "Main -gt X2
X2 -gt Target," "Main -gt X1
X1 -gt Target," "Main -gt Nx3 Nx3 -gt
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
Meta graph Class graph
Main
Nx1
X1
X2
Nx2
X3
Nx3
Main X1 X2 X3 Target
Nx3 Target
Target
Instance tree Object tree
Selected by strategy
16
Regular Expressions on Graphs
ALL PROBLEMS ARE POLYNOMIAL

• 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?

Generalizes regular expressions on
strings sentences must be node paths in graphs.
Work by Tarjan and Mendelzon/Wood.
17
Enhanced Regular Expressions on Graphs
ALL PROBLEMS BECOME NP-COMPLETE

• 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?

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
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
20
Crosscut Language

• Graph
• Path set
• Defines set of instance trees

• Instance trees
• Subtree or its leaves
• Conform to a graph (expansion)

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.

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

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.

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
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

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.

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.
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)

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 graph Alph(G)
set of node labels of G Union over all k in S and
all elements of Alph(G)
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)
Meta graph
main
nx1
x1
x2
nx2
x3
nx3
main x1 x2 x3 target
nx3 target
target
Instance tree
Selected by all()
APPROXIMATION
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.

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.

33
Same results for 5 problems

• We dont know yet how to unify all the proofs.
• So we prove the results separately.

34
Results (Problem)
Problem SD SAJ
- P P
NP-complete NP-complete
! NP-complete NP-complete
35
Results (Problem)

• Results(Select-Sat)
• Results(Not Select-Impl)
• Results(Select-First)
• Results(Not Select-Always)
• Results(Not Select-Never)

36
Implementation

• SD
• AP Library
• DJ
• DAJ

37
Future Work

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

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.

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)
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.

41
An idea by Gregor

• add a new primitive pointcut to AspectJ
  traversal(D).
• cflow(call (void class(traversal(A-gtB)).
  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.

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.

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 pointcuts and Demeter traversals. This
is basically what Gregor Kiczales suggested a few
years agohe called it traversal(D), instead of
nodes(D).
44
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

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
46
SAT is there a path in G satisfying r?
results identical for class graphs
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)
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
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.

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.

50
Polynomial Translations ( any mistakes?)
translate row to column
N no, unless PNP NN no Y yes
RE ERE SG ESG SAJ SD SAJB SDB
RE Y Y Y N N
ERE NN N
SG Y
ESG NN N
SAJ Y Y Y N N
SD Y Y Y N N
SAJB Y Y Y Y Y
SDB Y Y Y Y NN
51
Crosscut language SDk

• D a set of paths
• A,Bk paths from A to B of length k
• A,Bk bypassing A1, ignore A1,
• D . D concatenation of paths
• D D union of paths
• D D intersection of paths
• !D complement of paths

base languageSDB
see work on poly lingual systems
52
Crosscut language SD

• D a set of paths
• A,B paths from A to B
• A,B bypassing A1, ignore A1,
• D . D concatenation of paths
• D D union of paths
• D D intersection of paths
• !D complement of paths

base languageSDB
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
  PNP (consider SAT).

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.

55

• Puntingam non regular process types
• Some context-free, context-sensitive
• FSM with counters the same?
• Reussner

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 lt 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?

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)

58
(No Transcript)
59
Quality of model

• Meta graph defines set of instances
• Precisely
• Class graph
• Too many
• Call graph
• Pcflow what traversals do use meta information
• Problem with interior nodes

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
61
Guenter Kniesel

• Add meta variables

62
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
• ?v meta variable

base language
63
Crosscut language SD

• D a set of paths
• A,B paths from A to B
• A,B bypassing A1, ignore A1,
• D . D concatenation of paths
• D D union of paths
• D D intersection of paths
• !D complement of paths
• ?v meta variable

base languageSDB
64

• The meta variables are used to introduce
  dependencies.
• (flow(k1) ?v1) (flow(k2) ?v1) k3
• From A via ?v to C bypassing ?v to C

65
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