Entailment with Conditional Equality Constraints

1
Entailment with Conditional Equality Constraints

• Zhendong Su Alex Aiken
• University of California, Berkeley

2
Constraint Based Analysis

• Basic Idea

program source
constraint generation
constraints
constraint resolution
analysis results
3
Constraint Simplification

• Removing redundant constraints from a system.
• Example (using unification constraints)
• C xy, yz, xz
• xy and yz implies xz, get C xy, yz
• How about if we are only interested in the
  variables x and y?

4
Constraint Entailment

• Entailment is a decision problem
• justify a potential candidate for simplification
• shed light on the hardness of simplification
• Simple Entailment C1 ? C2
• holds if every solution of C1 is a solution of
  C2.
• Example xy,yz ? xz
• Restricted Entailment C1 ?V C2
• holds if for every solution of C1 there exists a
  solution of C2 s.t. they agree on V.
• Example xy ?x,y xy,yz,xz

5
How to Use Entailment?

• Simplify C
• Simplify C with respect to V

C C for each ci ? C if(C\ci ? ci)
C C\ci
C C for each ci ? C if(C\ci ?V C)
C C\ci
6
Related Work

• Theoretical issues
• subtyping entailment
• (C1 ? ? ? ?) Henglein and Rehof LICS97,
  ICALP98
• set constraint entailment
• (C1 ?V C2) Flanagan and Felleisen PLDI97
• atomic set constraint entailment
• (C1 ? ? ? ?) Niehren, Mueller, and Talbot
  LICS99
• polymorphic type simplification
• (??.?\C) Aiken, Wimmers, and Palsberg TACS97

7
Related Work (cont.)

• Practical simplifications
• polymorphic type simplification
• Fahndrich and Aiken SC96
• Pottier ICFP96
• Marlow and Wadler (Erlang) ICFP97
• set constraint simplification (C1 ?V C2)
• Flanagan and Felleisen PLDI97
• constraint solving
• Fahndrich, Foster, Su, and Aiken PLDI98
• Su, Fahndrich, and Aiken POPL00

8
Contributions

• Give the first interesting class with efficiently
  decidable entailment problem.
• Use novel techniques in constructing the
  algorithm.
• Provide a natural boundary between tractable and
  intractable constraint theories.

9
Conditional Unification

• simple types ? ? ? ? ?1??2
• ground types simple types without variables
• constraints ?1 ?2 ? ? ?
• valuation ? ? ? ?
• ? variables ? ground types
• satisfaction
• ? ? ?1 ?2 ? ?(?1) ?(?2)
• ? ? ? ? ? ? ?(?) ? or ?(?) ?(?)

10
Simple Entailment

• Theorem 1. C1 ? C2 is decidable in polynomial
  time.
• Basic idea to decide C ? ? ?
• ? ? ? ? ? implies ? ? ?
• ? ? ? and ? ? ? implies ? ?
• ? ? ? ? ? and ? ? ? implies ? ?
• apply unification and congruence closure to check
  if ? ?

11
Restricted Entailment

• Theorem 2. C1 ?V C2 is decidable in polynomial
  time.
• The most interesting and difficult result of the
  paper.
• The key idea transform the constraints so that
  we only need to consider at most two conditional
  constraints at a time.

12
An Example
?
?
?
?
?
?
t1
t2
t3
t
C2
C1
Does C1 ? C2 ?
NO
Does C1 ??, ?, ? C2 ?
YES
13
Outline of the Algorithm

• Introduce closed systems, for which it is
  sufficient to consider only pairs of conditional
  constraints
• Entailment with pair constraints can be decided
  in polynomial time
• Reduce entailment to entailment in terms of
  closed systems to get quadratic of entailments
  of pair constraints

14
Closed Systems
?
?
?
?
?
?
?
?
?
?
t1
t2
?
?
t3
t4
t1
t2
?
?
?1
?1
t
t
Example
Closed system for example
15
Closed Systems (cont.)

• A property it suffices to consider pairs of
  conditional constraints for the solutions of a
  closed system w.r.t. to a set of variables V.

?
?
?
?
?
?
t3
t4
t1
t2
?
?
?1
?1
t
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Pair Constraint Entailment

• Lemma 1. C1 ?V C2 can be decidable in polynomial
  time if C2 consists only unification constraints
  .
• Lemma 2. C1 ?V C2 can be decidable in polynomial
  time if C2 consists at most two conditional
  constraints .

17
Completion

• Lemma 3. Through a completion procedure, C1 ?V C2
  can be reduced to C1 ?V C2, where C2 is a
  closed system.
• See paper for details.

18
Putting Things Together

• Main Theorem C1 ?V C2 can be decide in
  polynomial time.
• proof sketch By Lemma 3, we reduce to entailment
  in terms of closed systems. We then consider
  quadratic of entailment of pair constraints,
  where each one can be decided in polynomial time
  by Lemma 2.

19
An Extension

• A natural extension for comparison
• add ? ? (?1 ?2)
• intuitively means either ? ? or ?1 ?2
• ? ? ? ? ? ? (? ?)
• Theorem 3. The entailment problem C1 ?V C2
  is coNP-complete (reduction from 3SAT).

20
The Reduction

• 3SAT
• ? C1 ? ? Cm
• Ci x ? y ? z
• associate x with ?x and ?x
• meaning x is true iff ?x?? and ?x?

?
? ? (?1 ?2)
Use
for
?2
?1
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The Reduction (cont.)

• For each boolean variable x
• For each clause Ci x ? y ? z

?x
?x
C1 part 1
?
?
t
? x
?y
?z
C1 part 2
?
t
s
?
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The Reduction (cont.)

• To ensure at most one of ?x and ?x is ?
• ? is not satisfiable iff C1 ?V C2 (?x, ?x in V)

?x1
?x1
?xn
?xn
...
?xn
?x1
t1
...
?xn
tn
?x1
...
?
s1
sn-1
?
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Summary

• Polynomial time algorithms for entailment
• Algorithms can be used in practice to scale
  analyses based on conditional equality
  constraints
• An extension to be used as a natural boundary
  between tractable and intractable constraint
  theories

24
Open Problems

• Many open problems in this area
• Two long standing ones are
• Non-structural subtype entailment
• Subtyping polymorphically constrained types
• Hardness of simplification means good heuristics
  are very valuable