Evaluation of the Implementation of an Abstract Interpretation - - PowerPoint PPT Presentation

Evaluation of the Implementation of an Abstract Interpretation Algorithm using Tabled CLP

ICLP’19 – Las Cruces, September 22, 2019

Joaqu´ ın Arias1,2 Manuel Carro1,2

1IMDEA Software Institute, 2Universidad Polit´

ecnica de Madrid

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Introduction

• Abstract interpretation is a theory for the approximation of the semantics of programs.
• PLAI is an efficient abstract interpreter for logic programs implemented in CiaoPP

.

• Goal: improve the implementation of PLAI using Tabled CLP:
• Reduce complexity and size.
• Increase performance.

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Introduction

• Abstract interpretation is a theory for the approximation of the semantics of programs.
• PLAI is an efficient abstract interpreter for logic programs implemented in CiaoPP

.

• Goal: improve the implementation of PLAI using Tabled CLP:
• Reduce complexity and size.
• Increase performance.

Main points

• Tabling engine drives fixpoint.
• Fair evaluation of tabling impact:
• Preserve interfaces with rest of CiaoPP (abstract domains, etc).
• Using abstract domains available in CiaoPP

.

• Analysing benchmarks from the CiaoPP distribution.
• Entailment and LUB computation through constraint interface.

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Related Work

• Tabling and abstract interpretation implementations share underlying principles:
• Extension Tables [Dietrich 1987] for tabling.
• Memo Tables with Dependency Tracking [Warren et al. 1988] for abstract interpretation.

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Related Work

• Tabling and abstract interpretation implementations share underlying principles:
• Extension Tables [Dietrich 1987] for tabling.
• Memo Tables with Dependency Tracking [Warren et al. 1988] for abstract interpretation.
• Implementation of Abstract Interpreters using tabling:
• Initial proposals, no implementation/performance details.

[Kanamori and Kawamura 1993; Warren 1999]

• Some implementations used as benchmarks to compare tabling scheduling.

[Demoen and Sagonas 1998; Freire et al. 2001]

• Advanced techniques (e.g., answer merging) to improve their implementation.

[Schrijvers et al. 2008; Chico de Guzm´ an et al. 2012; Swift and Warren 2010]

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Related Work

• Tabling and abstract interpretation implementations share underlying principles:
• Extension Tables [Dietrich 1987] for tabling.
• Memo Tables with Dependency Tracking [Warren et al. 1988] for abstract interpretation.
• Implementation of Abstract Interpreters using tabling:
• Initial proposals, no implementation/performance details.

[Kanamori and Kawamura 1993; Warren 1999]

• Some implementations used as benchmarks to compare tabling scheduling.

[Demoen and Sagonas 1998; Freire et al. 2001]

• Advanced techniques (e.g., answer merging) to improve their implementation.

[Schrijvers et al. 2008; Chico de Guzm´ an et al. 2012; Swift and Warren 2010]

• To the best of our knowledge, only one tabled implementation [Janssens and Sagonas 1998] evaluated against

non-tabled state-of-the-art Abs. Int. AMAI [Janssens et al. 1995] & PLAI [Muthukumar and Hermenegildo 1992].

• Conclusion: ”PLAI is the most efficient”.

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Background

• Mod(ular) TCLP

.

[PPDP’16 & TPLP’19].

• Flexible, simple interface to use arbitrary constraint domains in tabled execution.
• Three operations required: projection, entailment, application.
• More particular calls (w.r.t. ⊑) suspend and reuse answers.

p(X), X #> 1 ⊑ p(X), X #> 0

• More particular answers are discarded/removed.

X #= 2 ⊑ X #> 0 ∧ X #< 5 madrid institute for advanced studies in software development technologies

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Background

• Mod(ular) TCLP

.

[PPDP’16 & TPLP’19].

• Flexible, simple interface to use arbitrary constraint domains in tabled execution.
• Three operations required: projection, entailment, application.
• More particular calls (w.r.t. ⊑) suspend and reuse answers.

p(X), X #> 1 ⊑ p(X), X #> 0

• More particular answers are discarded/removed.

X #= 2 ⊑ X #> 0 ∧ X #< 5

• Aggregate-TCLP

.

[PADL ’19].

• Incremental computation of join of elements in a lattice.

Related to [Swift and Warren 2010; Guo and Gupta 2008; Zhou et al. 2010].

• Call entailment checking.

Detection of semantically equivalent calls ⇒ termination.

• Implemented on top of Mod TCLP

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Abstract Interpretation [Cousot and Cousot 1977; Bruynooghe 1991]

• Analyzes a finite # of abstract executions:
• An abstract substitution (denoted Xlist) approximates an infinite # of concrete substitutions.
• It is defined on an abstract domain.
• Given a clause of the form h :-

p1, ..., pm:

• βentry h βexit

abstract entry and exit substitution of h.

• λi pi λi+1

abstract call and success substitution of call pi.

βentryh :- λ1p1,λ2..., λmpm λm+1. βexit

• PLAI has a domain-independent interface.
• PLAI constructs a finite abstract tree top-down:
• Start with the λcall for a query.
• Follow top-down execution.
• Fixpoint algorithm used to terminate on recursive calls.

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Construction of the Abstract Tree

λcall q1 λsuccess. βentry q :- λ1p1, λ2..., λmpm λm+1. βexit

... ...

β ′

entry q′ :- λ ′ 1p′

1, λ ′

2 ..., λ ′ mp′

m

λ ′

m+1. β ′ exit

⊔       

β ′′

exit

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Construction of the Abstract Tree

λcall q1 λsuccess. βentry q :- λ1p1, λ2..., λmpm λm+1. βexit

... ...

β ′

entry q′ :- λ ′ 1p′

1, λ ′

2 ..., λ ′ mp′

m

λ ′

m+1. β ′ exit

⊔       

β ′′

exit

⊔

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Construction of the Abstract Tree

It loops in presence of recursion.

λcall q1 βentry q :- λ1p1, λ2..., λmpm λm+1. βexit

... ...

β ′

entry q′ :- λ ′ 1 ..., λcall q1 (loop)

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PLAI’s Fixpoint Algorithm

λcall q1

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PLAI’s Fixpoint Algorithm

λcall q1 βentry q :- λ1p1,λ2 ..., λmpm λm+1. βexit

non-recursive clauses

... ⊔

• βaprox.

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PLAI’s Fixpoint Algorithm

λcall q1 βentry q :- λ1p1,λ2 ..., λmpm λm+1. βexit

non-recursive clauses

... ⊔

• βaprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ ′

• aprox. p′

m

λ ′

m+1.

β ′

exit

recursive clauses

... ⊔               

β ′

aprox.

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PLAI’s Fixpoint Algorithm

λcall q1 βentry q :- λ1p1,λ2 ..., λmpm λm+1. βexit

non-recursive clauses

... ⊔

• βaprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ ′

• aprox. p′

m

λ ′

m+1.

β ′

exit

recursive clauses

... ⊔               

β ′

aprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ ′′

• aprox. p′

m

λ ′

m+1.

...

recursive clauses

⊔                       

β k−1

aprox.

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PLAI’s Fixpoint Algorithm

λcall q1 βentry q :- λ1p1,λ2 ..., λmpm λm+1. βexit

non-recursive clauses

... ⊔

• βaprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ ′

• aprox. p′

m

λ ′

m+1.

β ′

exit

recursive clauses

... ⊔               

β ′

aprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ ′′

• aprox. p′

m

λ ′

m+1.

...

recursive clauses

⊔                       

β k−1

aprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ k

• aprox. p′

m

λ k

m+1.

β k

exit

recursive clauses

⊔                                 

β k

• aprox. ≡ β k−1

aprox.

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PLAI’s Fixpoint Algorithm

λcall q1 λsuccess. βentry q :- λ1p1,λ2 ..., λmpm λm+1. βexit

non-recursive clauses

... ⊔

• βaprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ ′

• aprox. p′

m

λ ′

m+1.

β ′

exit

recursive clauses

... ⊔               

β ′

aprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ ′′

• aprox. p′

m

λ ′

m+1.

...

recursive clauses

⊔                       

β k−1

aprox.

β ′

entry q′ :- λ ′ 1 ..., λcall q1,λ k

• aprox. p′

m

λ k

m+1.

β k

exit

recursive clauses

⊔                                 

β k

• aprox. ≡ β k−1

aprox.

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PLAI’s Fixpoint Algorithm using TCLP

• Tabling suspends equivalent calls.
• Entailment is used:
• To discard subsumed abstract substitutions for a given predicate.
• To detect when a fixpoint has been reached.
• The LUB of abstract substitutions is computed incrementally.

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PLAI’s Fixpoint Algorithm using TCLP

λcall q1 βentry q :- λ1 ..., λcall q1

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PLAI’s Fixpoint Algorithm using TCLP

λcall q1 βentry q :- λ1 ..., λcall q1 β ′

entry q′ :- λ ′ 1p1,λ ′ 2 ..., λ ′ mpm λ ′ m+1.

βexit

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PLAI’s Fixpoint Algorithm using TCLP

λcall q1 βentry q :- λ1 ..., λcall q1 β ′

entry q′ :- λ ′ 1p1,λ ′ 2 ..., λ ′ mpm λ ′ m+1.

βexit

recursive clause non-recursive clause

β ′

exitβexit

⊔       

β ′′

exit

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PLAI’s Fixpoint Algorithm using TCLP

λcall q1 βentry q :- λ1 ..., λcall q1 β ′

entry q′ :- λ ′ 1p1,λ ′ 2 ..., λ ′ mpm λ ′ m+1.

βexit

recursive clause non-recursive clause

β ′

exitβexit

⊔       

β ′′

exit

λ ′′

exit p′

m

λ ′′

m+1.

β ′′′

exitβ ′′ exit

⊔                 

β ′v

exit

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PLAI’s Fixpoint Algorithm using TCLP

λcall q1 βentry q :- λ1 ..., λcall q1 β ′

entry q′ :- λ ′ 1p1,λ ′ 2 ..., λ ′ mpm λ ′ m+1.

βexit

recursive clause non-recursive clause

β ′

exitβexit

⊔       

β ′′

exit

λ ′′

exit p′

m

λ ′′

m+1.

β ′′′

exitβ ′′ exit

⊔                 

β ′v

exit

λ ′v

exit ...

... ⊔                         

β k

exit

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PLAI’s Fixpoint Algorithm using TCLP

λcall q1 βentry q :- λ1 ..., λcall q1 β ′

entry q′ :- λ ′ 1p1,λ ′ 2 ..., λ ′ mpm λ ′ m+1.

βexit

recursive clause non-recursive clause

β ′

exitβexit

⊔       

β ′′

exit

λ ′′

exit p′

m

λ ′′

m+1.

β ′′′

exitβ ′′ exit

⊔                 

β ′v

exit

λ ′v

exit ...

... ⊔                         

β k

exit

λ k

exit p′

m

λ k

m+1.

β k+1

exit ⊑ β k exit

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PLAI’s Fixpoint Algorithm using TCLP

λcall q1 λsuccess. βentry q :- λ1 ..., λcall q1 β ′

entry q′ :- λ ′ 1p1,λ ′ 2 ..., λ ′ mpm λ ′ m+1.

βexit

recursive clause non-recursive clause

β ′

exitβexit

⊔       

β ′′

exit

λ ′′

exit p′

m

λ ′′

m+1.

β ′′′

exitβ ′′ exit

⊔                 

β ′v

exit

λ ′v

exit ...

... ⊔                         

β k

exit

λ k

exit p′

m

λ k

m+1.

β k+1

exit ⊑ β k exit

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TCLP Implementation: Internals

• PLAI reimplementation: uses the lattice-based aggregate framework.
• Abstract substitutions connected with Mod TCLP

, using attributed variables [8].

• Mod TCLP sees abstract substitutions as constraints.
• However, abstract substitutions are manipulated instead of constraints.

(transparently to the engine implementation.)

• abs_lub: implementation of constraint interface that operates on abstract domains.
• abs_lub is used to:
• Determine suspension of calls.
• Detect termination of fixpoint.
• Aggregate (e.g., compute LUB) abstract substitutions.
• Project and apply resulting abstract substitution.

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Fixpoint Code

call to success/7 using Mod TCLP

.

1

:- use_package(tclp_aggregate).

2

:- table call_to_success_fixpoint(_, _, abst_lub).

3

call_to_success(SgKey, Call, Proj, Sg, Sv, AbsInt, Succ) :-

4

call_to_success_fixpoint(SgKey, Sg, st(Sv, Call, Proj, AbsInt, Prime) ),

5

each_extend(Sg, Prime, AbsInt, Sv, Call, Succ).

6

call_to_success_fixpoint(SgKey, Sg, st(Sv, Call, Proj, AbsInt, Prime) ) :-

7

trans_clause(SgKey, _, Clause),

8

do_nr_cl(Clause, Sg, Sv, Call, Proj, AbsInt, Prime).

9

call_to_success_fixpoint(SgKey, Sg, st(Sv, _Call, Proj, AbsInt, Prime) ) :-

10

\+ trans_clause(SgKey, _, _),

11

apply_trusted0(Proj, SgKey, Sg, Sv, AbsInt, _ClId, Prime).

Note: do_nr_cl_clause/7 calls call_to_success/7 .

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PLAI’s code using Prolog

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PLAI’s code using TCLP

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abst_lub Implementation

Interface to Constraint Solver Implementing Domain Operations

1

call_entail(abst_lub, st(Sv, _, EntryA, Abs, _),

2

st(Sv, _, EntryB, Abs, _) ) :- identical_abstract(Abs, EntryA, EntryB).

3

answer_entail(abst_lub, st(Sv, _, _, Abs, ExitA),

4

st(Sv, _, _, Abs, ExitB) ) :- less_or_equal(Abs, ExitA, ExitB).

5

answer_join(abst_lub, st(Sv, _, _, Abs, ExitA),

6

st(Sv, _, _, Abs, ExitB),

7

st(Sv, _, _, Abs, Join ) ) :- compute_lub(Abs, [ExitA, ExitB], Join).

8

apply_answer(abst_lub, st(Sv, _, _, _, Join),

9

st(Sv, _, _, _, Var ) ) :- Var = Join.

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Evaluation: Code Complexity of TCLP vs. Prolog

• TCLP implementation notably simpler.
• Original implementation needs to keep track of dependencies among data, program points where

analysis was suspended/resumed, ...

• Using e.g. external database, numerous parameters, special cases.
• Original code:

call_to_success/13 has 8 clauses. proj_to_prime_nr/9 has 6 clauses... ...it starts the fixpoint.

• TCLP code:

call_to_success/7 has 1 clause. call_to_success_fixpoint/3 has 2 clauses... ...it is evaluated under tabling.

• TCLP does not need to pre-classify the non-recursive clauses.
• The size of code with Mod TCLP is one-third the original size.

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Evaluation: Performance of TCLP vs. Prolog

+ Fixpoint handled at a lower level (tabling). − Internal data structures not optimal for TCLP (redundancy, copying overhead).

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Evaluation: Performance of TCLP vs. Prolog

+ Fixpoint handled at a lower level (tabling). − Internal data structures not optimal for TCLP (redundancy, copying overhead).

fi b f a l t a i a k l b

• y

e r p v q u e e n s u b s t p v g a b r i e l r d t

• k

m m a t f h a n

• i

r e v f l i n a p p e n d r e v l i n p r e fi x r e v f p v p l a n s u b l i s t a p p r e v e r s e fl a t t e n p a l i n d r

• f

a c t r

• t

a t e m a x t r e e z e b r a b r

• w

s e 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Speedup of TCLP w.r.t. Prolog ( Prologrun−time

TCLPrun−time )

Groundness Domain

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Evaluation: Performance of TCLP vs. Prolog

+ Fixpoint handled at a lower level (tabling). − Internal data structures not optimal for TCLP (redundancy, copying overhead).

f a c t p v q u e e n m m a t f m m a t r i x p r e fi x r e v f r e v f l i n r e v e r s e r e v l i n r

• t

a t e p v p g a p p e n d s u b l i s t a p p z e b r a 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Speedup of TCLP w.r.t. Prolog ( Prologrun−time

TCLPrun−time )

Sharing & Freeness [12] Domain

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Conclusions

• The TCLP tabling engine keeps tracks of dependencies, suspensions, restarts:
• Code size: one-third.
• Code simpler.
• Easier maintainability.
• Faster in majority of cases (up of 1.60×).

+ Evaluated with several standard benchmarks in two relevant abstract domains.

• This implementation would benefit from changes in the CiaoPP internal representation.

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Conclusions

• The TCLP tabling engine keeps tracks of dependencies, suspensions, restarts:
• Code size: one-third.
• Code simpler.
• Easier maintainability.
• Faster in majority of cases (up of 1.60×).

+ Evaluated with several standard benchmarks in two relevant abstract domains.

• This implementation would benefit from changes in the CiaoPP internal representation.

Future work

• Explore additional abstract domains.
• Study scheduling strategies to favor propagation in constraint system.
• Adapt abstract substitution representation / CiaoPP interface.

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Conclusions

• The TCLP tabling engine keeps tracks of dependencies, suspensions, restarts:
• Code size: one-third.
• Code simpler.
• Easier maintainability.
• Faster in majority of cases (up of 1.60×).

+ Evaluated with several standard benchmarks in two relevant abstract domains.

• This implementation would benefit from changes in the CiaoPP internal representation.

Future work

• Explore additional abstract domains.
• Study scheduling strategies to favor propagation in constraint system.
• Adapt abstract substitution representation / CiaoPP interface.

THANKS!

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Bibliography I

[1] Bruynooghe, M. (1991). A Practical Framework for the Abstract Interpretation of Logic Programs. Journal of Logic Programming, 10:91–124. [2] Chico de Guzm´ an, P ., Carro, M., Hermenegildo, M. V., and Stuckey, P . (2012). A General Implementation Framework for Tabled CLP. In Int’l. Symposium on Functional and Logic Programming (FLOPS’12), number 7294 in LNCS, pages 104–119. Springer Verlag. [3] Cousot, P . and Cousot, R. (1977). Abstract Interpretation: a Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints. In ACM Symposium on Principles of Programming Languages (POPL ’77), pages 238–252. ACM Press. [4] Demoen, B. and Sagonas, K. (1998). CAT: The Copying Approach to Tabling. In Programming Language Implementation and Logic Programming, volume 1490 of Lecture Notes in Computer Science, pages 21–35. Springer-Verlag. [5] Dietrich, S. W. (1987). Extension Tables: Memo Relations in Logic Programming. In Fourth IEEE Symposium on Logic Programming, pages 264–272. madrid institute for advanced studies in software development technologies

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