Expressiveness and Complexity of Crosscut Languages

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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
• Study them abstractly using expressions on
  graphs lower bounds and upper bounds
• Assumption know entire program or class graph
• Of interest to AOSD language designers and tool
  builders and mathematicians.

3
Regular Expressions on Graphs
ALL PROBLEMS ARE POLYNOMIAL

• Questions Given 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)
• Is there a path in G that starts with e and
  satisfies r? (FIRST)
• Questions Given 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.
4
Enhanced Regular Expressions

• ERE regular expressions (primitive,
  concatenation, union, star) with
• complement/negation
• nodes and edges

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Canonical Crosscut Language
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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 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

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Same 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)

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Enhanced Regular Expressions on Graphs
ALL PROBLEMS BECOME NP-COMPLETE

• Questions Given 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)
• Is there a path in G that starts with e and
  satisfies r? (FIRST)
• Questions Given 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)
• Ok, related to Demeter but how does AspectJ come
  in?

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Mathematics for AOP
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SAT is there a path in G satisfying r?
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SAT is there a path in G satisfying r?
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SAT is there a path in G satisfying r?
results identical for class graphs
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Abbreviations
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Polynomial Translations
N no, unless PNP NN no
translate row to column
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(No Transcript)
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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 the foundation for crosscut
languages
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Using ERE for AspectJ

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

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

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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.

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Examples first

• Show two programs and their graph abstractions

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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()
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class Main // Java Program with DJ X1 x1 Nx1
nx1 public static void main(String s)
ClassGraph cg new ClassGraph() Main m new
Main() cg.traverse(m, // m is the complete
tree with 8 leaves "intersect(" // union is
expressed by concatenation of edges
"source Main -gt X1 X1 -gt target Target "
"source Main -gtNx2 Nx2 -gt target Target "
"Main -gt X3 X3 -gt Target,"
"source Main -gt Nx1 Nx1 -gt target Target "
"Main -gt X2 X2 -gt Target,"
"sourceMain -gt X1 X1 -gt target Target,"
"sourceMain -gt Nx3 Nx3 -gt target Target)",
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
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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
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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
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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)

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Expressions on GraphsExpressions on Instances

• Questions Given graph G and r Exists J sat G
• Is there a path in J satisfying r?
• 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?
• Is there a path in J that starts with e and
  satisfies r?
• 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?

push down to instances
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Expressions on instances

• Questions Given graph G, r and instance J
  satisfying G
• Determine the set of edges e from Root(J) so that
  there is a path in J starting with e and
  satisfying r?

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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

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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.

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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.

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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
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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.

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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.

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Same results for 5 problems

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

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Results (Problem)
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Results (Problem)

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

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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.

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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.

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Crosscut Language SAJ/SD

• S a set of nodes
• n(l) set of nodes having label l
• nodes(D) set of nodes selected by D
• S S union
• S S intersection
• !S complement

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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