Incremental and Modular Context-Sensitive Analysis

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Incremental and Modular Context-Sensitive Analysis

Isabel Garcia-Contreras ^1,2 Jose F. Morales ^1,2 Manuel V. Hermenegildo ^1,2

1 IMDEA Software Institute

2 T. U. Madrid (UPM)

HCVS: Horn Clauses for Verification and Synthesis — March 28th, 2021

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Introduction

Context: Analyzing/Verifying software projects during development in order to:

• Detect and report of bugs as early as possible (e.g., on-the-fly, at commit, …).

• Optimize code and libraries globally for the program being developed.

Problem: Context-sensitive analysis can be quite precise but also expensive, specially for interactive uses.

Make it incremental! – successive changes during development are often comparatively small and localized.

So far, in abstract interpretation, this was achieved by:

• Fine-grain (clause-level) incremental analysis for non-modular programs.

[SAS’96, TOPLAS’00]

• Coarse-grain (module-level) analysis aimed at reducing memory consumption . [ENTCS’00, LOPSTR’01]

We propose an extension of the modular algorithm to react to module changes , and

a way to combine it with fine-grain incrementality .

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Motivation - (Incremental) Static On-the-fly verification

2

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General idea

1. Take “snapshots” of the program sources

(e.g., at each editor save/pause while developing, each commit, ...).

2. Detect the changes w.r.t. the previous snapshot.

3. Reanalyze :

• Annotate and remove potentially outdated information .

• (Re-)Analyze incrementally (only the parts needed) module by module until an intermodular fixpoint is reached again.

→ Recheck assertions/Reoptimize.

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Analysis Basics

Abstract Interpretation

• Simulates the execution of the program using an abstract domain D α , simpler than the contrete one.

• Guarantees:

• Analysis termination, provided that D α meets some conditions.

• Results are safe approximations of the concrete semantics.

We use Prolog syntax for Horn Clauses Head _k :- B _k,1 , . . . , B _k,n _k

1 list([]). % fact

2 list([X|Xs]) :- % rule head

3 list(Xs). % rule body

4

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Concrete (Top-down) Semantics – AND trees

1 par([], P, P).

2 par([C|Cs], P0, P) :-

3 xor(C, P0, P1),

4 par(Cs, P1, P).

5 xor(0,0,0).

6 xor(0,1,1).

7 xor(1,0,1).

8 xor(1,1,0).

AND tree of :- par([0, 1], 0,P).:

true par([0, 1], 0,P) }query par([C|Cs],P0,P) :- }head

C = 0, Cs = [1],

P0 = 0

xor(C,P0,P1), xor(0, 0, 0).

C = 0, Cs = [1],

P0 = 0, P1 = 0

par( Cs, P1, P).

par([C|Cs], P1, P) :-

C = 1, Cs = [],

P0 = 0

xor(C,P0,P1), xor(1, 0, 1).

C = 1, Cs = [],

P0 = 0, P1 = 1

par(Cs, P1, P).

par([], P , P).

C = 1, Cs = [],

P0 = 0, P1 = 1 P = 1 C = 0, Cs = [1],

P0 = 0, P1 = 0, P = 1 P = 1

2,1 2,2

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PLAI Analysis output

A PLAI [NACLP’89] analysis graph has a set of nodes hA, λ ^c i 7→ λ ^s for every potentially reachable predicate, where:

• A is an atom, the predicate identifier.

• λ ^c is an abstract call to A.

• λ ^s is the abstract answer for A and λ ^c if it succeeds.

Example

1 par([], P, P).

2 par([C|Cs], P0, P) :-

3 xor(C, P0, P1),

4 par(Cs, P1, P).

5 6 xor(0,0,0).

7 xor(0,1,1).

8 xor(1,0,1).

9 xor(1,1,0).

Example nodes:

hpar(L, P0, P), >i 7→ (P0/ bit , P/ _bit )

For any call to par that succeeds, P0 and P are either 1 or 0.

hpar(L, P0, P), (P0/−)i 7→ ⊥

If par is called with P0 a negative number, it always fails.

Edges : hP, λi i,j − −→ ^λp

λr hQ, λ ^′ i, calling P with λ causes Q to be called with λ ^′ .

Analysis is interprocedural, multivariant, and context sensitive. ⁶

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Analysis graph – example

Initial query ⟨par(M, X, P), (X/z)⟩

Entry: :- par(M,0,P).

1 par([], P, P).

2 par([C|Cs], P0, P) :-

3 xor(C, P0, P1),

4 par(Cs, P1, P).

5 6 xor(0,0,0).

7 xor(0,1,1).

8 xor(1,0,1).

9 xor(1,1,0).

⊤

bit

z o

⊥

hpar(M,X,P), ( X/ z ) i 7→

(X/ z ,P/ bit )

hxor(C,P0,P1), (P0/ z ) i 7→

( C/ bit , P0/ z , P1/ bit )

hpar(M,X,P), X/ bit i 7→

( X/ bit , P/ bit )

hxor(C,P0,P1), ( P0/ bit ) i 7→

( C/ bit , P0/ bit , P1/ bit )

2, 1 2, 2

2, 1

2, 2

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Modular CHC Programs

Strict module system

• Modules define an interface of exported and imported predicates.

• Non-exported predicates cannot be seen or used in other modules.

Modular program

1 :- module(main, [main/2]).

2 3 :- use_module(bitops, [xor/3]).

4 5 main(L,P) :-

6 par(L,0,P).

7 8 par([], P, P).

9 par([C|Cs], P0, P) :-

10 xor(C, P0, P1),

11 par(Cs, P1, P).

1 :- module(bitops, [xor/3]).

2 3 xor(0,0,0).

4 xor(0,1,1).

5 xor(1,0,1).

6 xor(1,1,0).

8

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Graphs for Incremental and Modular Analysis

hmain(M,P),

>i 7→

(P/ bit )

hpar(M,X,P), ( X/z)i 7→

( X/z, P/ bit )

hxor(C,P0,P1), (P0/z)i 7→

(C/ bit , P0/z, P1/ bit )

hpar(M,X,P), X/ bit i 7→

(X/ bit ,P/ bit )

hxor(C,P0,P1), (P0/ bit ) i 7→

(C/ bit , P0/ bit , P1/ bit ) 1, 1

2, 1

2, 2

2, 1

2, 2

bitops main

We have:

• A global analysis graph G : call dependencies among imported/exported predicates.

• A local analysis graph L M

per module M: limited to

the predicates used in M.

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test

warplan

robot_operations prolog_sys

Id162 test1/0

Id163 plans/2 C1 L1 Id160 test2/0

Id161 plans/2 C1 L1

Id158 test3/0

Id81 plans/2 C1 L1

Id77 output/1 C2 L5

Id18 time/1 C2 L3 Id95 plan/4

C2 L2 Id82 not/1

C1 L1

Id78 output1/1

C1 L1 Id19 statistics/2

C1 L1

Id96 solve/5 C2 L1 Id136 plan/4

C1 L2 C2 L5 C2 L3 C2 L2

C1 L1 C1 L1 C2 L2 C2 L5 C2 L3

Id12 imposs/1

Id126 achieve/5 C2 L3

Id45 preserves/2 C1 L5

Id50 consistent/2 C1 L3 Id127 retrace/3

C2 L2 Id117 plan/4

C1 L4

Id122 preserved/2 C2 L4

Id49 can/2 C1 L2 Id118 solve/5

C3 L2

Id22 always/1 C1 L1 Id112 and/3

C2 L2

Id119 holds/2 C2 L1

Id43 add/2 C3 L1 Id149 achieve/5

Id71 retrace/3

C2 L2C1 L5

Id142 plan/4

C1 L4

Id33 preserved/2

C2 L4

Id147 preserved/2 C2 L1 C1 L3

C1 L2 Id143 solve/5

C3 L2

C1 L1

Id144 holds/2 C2 L1

Id39 and/3 C2 L2

C3 L1

C2 L5 C2 L3

C2 L2 C1 L1

C2 L2 Id97 always/1

C1 L1 Id116 achieve/5

C3 L2

Id104 holds/2 C2 L1

Id115 add/2 C3 L1 C1 L2

Id137 solve/5 C2 L1

C2 L2

Id138 holds/2 C2 L1 C1 L1

C3 L1 Id46 preserved/2

C2 L4

C2 L2C1 L5 C1 L4

C1 L3

C1 L2

Id47 mkground/3 C1 L1

Id48 del/2 C1 L2 C2 L4

C2 L3

Id61 retrace/3 C2 L2

Id60 preserves/2 C1 L5 Id153 plan/4 C1 L4

Id152 consistent/2 C1 L3

C1 L2

Id66 conjoin/3 C1 L3

Id62 retrace1/4 C1 L2

C1 L1 C1 L2

C1 L1 Id154 solve/5 C2 L1 Id155 plan/4

C1 L2

Id14 implied/2 C1 L4 Id13 not/1

C1 L3

C1 L2 Id148 mkground/3

C1 L1 C1 L2

C1 L1 C3 L2 Id56 and/3

C2 L2 C1 L1 C2 L1

C3 L1 Id156 solve/5

C2 L1 Id68 plan/4

C1 L2

Id69 solve/5 C1 L1

C2 L2 C1 L5

C1 L4

C1 L3 C1 L1 C2 L3

C2 L4 C2 L1

C1 L2 Id76 conjoin/3

C1 L3

C1 L1 C1 L1

C1 L2 C2 L1

Id54 plan/4 C1 L2

C1 L2

Id55 solve/5 C2 L1

Id34 del/2 C1 L2

C1 L1

C1 L2 C1 L3

C1 L2

Id52 implied/2 C1 L4

Id70 achieve/5

C3 L2 C2 L2

Id29 holds/2

C2 L1 C1 L1

C3 L1 C2 L2C1 L5

C2 L4 C1 L4 Id67 achieve/5

C2 L3

C1 L2

Id5 elem/2 C1 L1

Id57 equiv/2 C1 L2 C2 L3

C2 L2

Id31 given/2 C3 L1

Id30 add/2 C1 L1 Id26 always/1

C3 L1

Id23 connects1/3 C2 L1 C1 L2

C2 L1

C3 L2 C2 L1C2 L2 C1 L1

C3 L1

C2 L3 C2 L1

C2 L2

Id146 given/2 C3 L1

Id145 add/2 C1 L1

Id15 elem/2 C2 L1 Id3 my_call/1 C1 L1

Id9 mkgroundlist/3 C3 L2 Id41 equiv/2

C1 L2

Id32 range/3 C3 L1

Id25 range/3 C2 L3 Id113 equiv/2

C1 L2

C2 L3 C2 L2

Id140 given/2

C3 L1

Id139 add/2 C1 L1 Id107 range/3

C3 L1

Id100 range/3 C2 L3 Id101 always/1 C3 L1

Id98 connects1/3 C2 L1

C1 L5 C1 L3

C2 L3

C1 L4

C1 L3 Id129 retrace1/4 C1 L2

Id128 can/2 C1 L1 C2 L1

C1 L2

Id134 plan/4 C1 L2

C1 L2

Id135 solve/5 C2 L1 C1 L1

C1 L1

Id123 del/2 C1 L2 C2 L2

C2 L1 C1 L1

C3 L1 C2 L2 C1 L5

C1 L4

C2 L4 C2 L3

C1 L3

C1 L2 C2 L2

C2 L1 C1 L1

C3 L2

C3 L1

C1 L1 C3 L1

Id132 equiv/2 C2 L2 Id131 equiv/2

C1 L2

Id130 add/2 C1 L1 Id114 not/1

C1 L1 C1 L1

C2 L3 C2 L1

C2 L2

Id121 given/2 C3 L1

Id120 add/2 C1 L1

C3 L2 C3 L1 C1 L1

C2 L3 C2 L2

Id106 given/2 C3 L1

Id105 add/2 C1 L1 C3 L1

C2 L3 Id103 connects1/3

C2 L1 Id102 connects1/3 C1 L1

Id99 range/3 C1 L1

C2 L3 C1 L1 C1 L1

C3 L2 Id17 not/1 C1 L1 Id53 plan/4

C1 L2 Id21 solve/5

C2 L2 C1 L5

C2 L4 C1 L3

C1 L4

C2 L3

C1 L2 Id1 plans/2

C2 L5C2 L3

Id20 plan/4 C2 L2

Id2 not/1 C1 L1

C1 L1

C1 L2 C2 L1

C2 L1 C1 L1

C2 L2 C3 L2

C3 L1

C2 L2C1 L5 C2 L4

C1 L4 C2 L3

C1 L3

C1 L2 C1 L1

C3 L1

Id75 equiv/2 C2 L2

C2 L1 Id73 equiv/2

C1 L2 Id63 add/2

C1 L1

Id74 not/1 C1 L1

C1 L1

C2 L1 C1 L1

C5 L1 C4 L1 Id7 nonequiv/2

C2 L1 Id4 intersect/2

C1 L1

C1 L4 C1 L3

C1 L2 C1 L1

C1 L1

C2 L1 C3 L1

C1 L2

C1 L1

Id58 not/1 C1 L1 C1 L1

C1 L1

C3 L2 C1 L1

C1 L2

C2 L1 C1 L2

C2 L1 C2 L1

C3 L2 Id42 not/1

C1 L1

C3 L1

Id24 range/3 C1 L1

C2 L3 C1 L1 C1 L1

C1 L1

C1 L1 C1 L1

Id6 elem/2 C1 L2 C2 L1

Id35 moved/2 C8 L1 Id36 del/2 C2 L1

Id79 can/2

Id38 del/2 C2 L1

Id37 moved/2 C8 L1 C2 L1 C7 L1

C7 L1 C2 L1 C6 L1 test10

test20 test40 test30

del0 add0

moved0 can0 plans0

Id114 not/1

Id83 my_call/1

Id89 mkground/3 Id88 mkgroundlist/3

C3 L2 Id91 not/1

C1 L1Id90 consistent/2 C1 L3

Id87 mkground/3 C1 L1 Id92 implied/2

C1 L4 Id94 not/1 C1 L1

Id40 elem/2 Id114 not/1 C1 L1

C2 L1

C4 L1 C5 L1

Id86 nonequiv/2 C2 L1

Id84 intersect/2 C1 L1

C3 L2

Id85 elem/2

C2 L1 C1 L2

C2 L1 C2 L1

C1 L1

C1 L1 C1 L2 C1 L1

C1 L2

lib planner

Snapshot of Analysis Graphs

Changes detected!

planner.pl

100 %%

101 - explore(P,Map,[P|Map]) :-

102 - safe(P).

103 %%

lib.pl

41 %%

42 + add(Node,Graph) :-

43 + %% implementation

44 + %% implementation

45 %%

10

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test

warplan

Id162 test1/0

Id161 plans/2 C1 L1

Id158 test3/0

Id81 plans/2 C1 L1

Id77 output/1 C2 L5

C2 L2 Id82 not/1

C1 L1

Id78 output1/1

C1 L1

C1 L2 C2 L5 C2 L3 C2 L2

C1 L1 C1 L1 C2 L2 C2 L5 C2 L3

Id12 imposs/1

C2 L2 Id117 plan/4

C1 L4

C3 L2

C2 L2

Id119 holds/2 C2 L1

Id71 retrace/3

C2 L2C1 L5

Id142 plan/4

C1 L4

Id33 preserved/2

C2 L4

C1 L2 Id143 solve/5

C3 L2

C1 L1

Id144 holds/2 C2 L1

Id39 and/3 C2 L2

C3 L1

C2 L5 C2 L3

C2 L2 C1 L1

C2 L2 Id97 always/1

C3 L2

Id104 holds/2 C2 L1

Id115 add/2 C3 L1 C1 L2

Id137 solve/5 C2 L1

C2 L2

C2 L4

C2 L2C1 L5 C1 L4

C1 L3

C1 L2

C2 L3

C1 L2

C1 L1 C1 L2

C1 L2

C1 L3

C1 L1 C1 L2

C2 L2 C1 L1 C2 L1

C3 L1 Id156 solve/5

C2 L1 Id68 plan/4

C1 L2

Id69 solve/5 C1 L1

C2 L2 C1 L5

C1 L4

C1 L3 C1 L1 C2 L3

C2 L4 C2 L1

C1 L3

C1 L1 C1 L1

C1 L2 C2 L1

Id54 plan/4 C1 L2

C1 L2

Id55 solve/5 C2 L1

Id34 del/2 C1 L2

C1 L1

C1 L2 C1 L3

C1 L2

Id70 achieve/5

C3 L2 C2 L2

Id29 holds/2

C2 L1 C1 L1

C3 L1 C2 L2C1 L5

C2 L3

C1 L2

Id5 elem/2 C1 L1

C2 L2

Id31 given/2 C3 L1

C3 L1

C2 L1

C3 L2 C2 L1C2 L2 C1 L1

C3 L1

C2 L3 C2 L1

C2 L2

Id146 given/2 C3 L1

Id145 add/2 C1 L1

C1 L2

Id32 range/3 C3 L1

C1 L2

C2 L3 C2 L2

Id140 given/2

C3 L1

C1 L5 C1 L3

C2 L3

C1 L4

Id128 can/2 C1 L1 C2 L1

C1 L2

Id134 plan/4 C1 L2

C1 L2

C1 L1

Id123 del/2 C1 L2 C2 L2

C2 L1 C1 L1

C3 L1 C2 L2 C1 L5

C1 L4

C2 L4 C2 L3

C1 L3

C1 L2 C2 L2

C2 L1 C1 L1

C3 L2

C3 L1

C1 L1 C3 L1

C1 L2

C1 L1 C1 L1

C2 L3 C2 L1

C2 L2

Id121 given/2 C3 L1

Id120 add/2 C1 L1

C3 L2 C3 L1 C1 L1

C2 L3 C2 L2

Id106 given/2 C3 L1

Id105 add/2 C1 L1 C3 L1

Id99 range/3 C1 L1

C2 L3 C1 L1 C1 L1

C1 L2 Id21 solve/5

C2 L2 C1 L5

C2 L4 C1 L3

C1 L4

C2 L3

C1 L2 Id1 plans/2

C2 L5C2 L3

Id20 plan/4 C2 L2

Id2 not/1 C1 L1

C1 L1

C1 L2 C2 L1

C2 L1 C1 L1

C2 L2 C3 L2

C3 L1

C2 L2C1 L5 C2 L4

C1 L4 C2 L3

C1 L3

C1 L2 C1 L1

C3 L1

Id75 equiv/2 C2 L2

C2 L1 Id73 equiv/2

C1 L2 Id63 add/2

C1 L1

Id74 not/1 C1 L1

C1 L1

C2 L1 C1 L1

C1 L1

C1 L4 C1 L3

C1 L2 C1 L1

C1 L1

C2 L1 C3 L1

C1 L2

C1 L1

C3 L2 C1 L1

C1 L2

C2 L1 C1 L2

C2 L1 C2 L1

C3 L2 Id42 not/1

C1 L1

C3 L1

Id24 range/3 C1 L1

C2 L3 C1 L1 C1 L1

C1 L1

C1 L1 C1 L1

Id79 can/2

Id38 del/2 C2 L1

del0 add0

moved0 can0 plans0

add

Id83 my_call/1

C3 L2 Id91 not/1

C1 L1Id90 consistent/2 C1 L3

Id40 elem/2 Id114 not/1

C1 L1

C2 L1

C4 L1 C5 L1

C3 L2

Id85 elem/2

C2 L1 C1 L2

C2 L1 C2 L1

C1 L1

C1 L1 C1 L2 C1 L1

C1 L2

delete

lib planner

Snapshot of Analysis Graphs

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test

warplan

Id162 test1/0

Id161 plans/2 C1 L1

Id158 test3/0

Id81 plans/2 C1 L1

Id77 output/1 C2 L5

C2 L2 Id82 not/1

C1 L1

Id78 output1/1

C1 L1

C1 L2 C2 L5 C2 L3 C2 L2

C1 L1 C1 L1 C2 L2 C2 L5 C2 L3

Id12 imposs/1

C2 L2 Id117 plan/4

C1 L4

C3 L2

C2 L2

Id119 holds/2 C2 L1

Id71 retrace/3

C2 L2C1 L5

Id142 plan/4

C1 L4

Id33 preserved/2

C2 L4

C1 L2 Id143 solve/5

C3 L2

C1 L1

Id144 holds/2 C2 L1

Id39 and/3 C2 L2

C3 L1

C2 L5 C2 L3

C2 L2 C1 L1

C2 L2 Id97 always/1

C3 L2

Id104 holds/2 C2 L1

Id115 add/2 C3 L1 C1 L2

Id137 solve/5 C2 L1

C2 L2

C2 L4

C2 L2C1 L5 C1 L4

C1 L3

C1 L2

C2 L3

C1 L2

C1 L1 C1 L2

C1 L2

C1 L3

C1 L1 C1 L2

C2 L2 C1 L1 C2 L1

C3 L1 Id156 solve/5

C2 L1 Id68 plan/4

C1 L2

Id69 solve/5 C1 L1

C2 L2 C1 L5

C1 L4

C1 L3 C1 L1 C2 L3

C2 L4 C2 L1

C1 L3

C1 L1 C1 L1

C1 L2 C2 L1

Id54 plan/4 C1 L2

C1 L2

Id55 solve/5 C2 L1

Id34 del/2 C1 L2

C1 L1

C1 L2 C1 L3

C1 L2

Id70 achieve/5

C3 L2 C2 L2

Id29 holds/2

C2 L1 C1 L1

C3 L1 C2 L2C1 L5

C2 L3

C1 L2

Id5 elem/2 C1 L1

C2 L2

Id31 given/2 C3 L1

C3 L1

C2 L1

C3 L2 C2 L1C2 L2 C1 L1

C3 L1

C2 L3 C2 L1

C2 L2

Id146 given/2 C3 L1

Id145 add/2 C1 L1

C1 L2

Id32 range/3 C3 L1

C1 L2

C2 L3 C2 L2

Id140 given/2

C3 L1

C1 L5 C1 L3

C2 L3

C1 L4

C1 L2

Id109 del/2 C1 L2 Id133 conjoin/3

Id128 can/2 C1 L1 C2 L1

C1 L2

Id134 plan/4 C1 L2

C1 L2

C1 L1

Id123 del/2 C1 L2 C2 L2

C2 L1 C1 L1

C3 L1 C2 L2 C1 L5

C1 L4

C2 L4 C2 L3

C1 L3

C1 L2 C2 L2

C2 L1 C1 L1

C3 L2

C3 L1

C1 L1 C3 L1

C1 L2

C1 L1 C1 L1

C2 L3 C2 L1

C2 L2

Id121 given/2 C3 L1

Id120 add/2 C1 L1

C3 L2 C3 L1 C1 L1

C2 L3 C2 L2

Id106 given/2 C3 L1

Id105 add/2 C1 L1 C3 L1

Id99 range/3 C1 L1

C2 L3 C1 L1 C1 L1

C1 L2 Id21 solve/5

C2 L2 C1 L5

C2 L4 C1 L3

C1 L4

C2 L3

C1 L2 Id1 plans/2

C2 L5C2 L3

Id20 plan/4 C2 L2

Id2 not/1 C1 L1

C1 L1

C1 L2 C2 L1

C2 L1 C1 L1

C2 L2 C3 L2

C3 L1

C2 L2C1 L5 C2 L4

C1 L4 C2 L3

C1 L3

C1 L2 C1 L1

C3 L1

Id75 equiv/2 C2 L2

C2 L1 Id73 equiv/2

C1 L2 Id63 add/2

C1 L1

Id74 not/1 C1 L1

C1 L1

C2 L1 C1 L1

C1 L1

C1 L4 C1 L3

C1 L2 C1 L1

C1 L1

C2 L1 C3 L1

C1 L2

C1 L1

C3 L2 C1 L1

C1 L2

C2 L1 C1 L2

C2 L1 C2 L1

C3 L2 Id42 not/1

C1 L1

C3 L1

Id24 range/3 C1 L1

C2 L3 C1 L1 C1 L1

C1 L1

C1 L1 C1 L1

Id125 del/2 C2 L1

Id124 moved/2 C8 L1

Id111 del/2 C2 L1 C7 L1

Id110 moved/2

C6 L1 C2 L1

C7 L1 C2 L1

C8 L1 Id79 can/2

Id38 del/2 C2 L1

del0 add0

moved0 can0 plans0

recompute delete

lib planner

Snapshot of Analysis Graphs

12

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test

warplan

Id162 test1/0

Id161 plans/2 C1 L1

Id158 test3/0

Id81 plans/2 C1 L1

Id77 output/1 C2 L5

C2 L2 Id82 not/1

C1 L1

Id78 output1/1

C1 L1

C1 L2

Id83 my_call/1 C1 L1

C2 L5 C2 L3 C2 L2

C1 L1 C1 L1 C2 L2 C2 L5 C2 L3

C3 L2 Id91 not/1

C1 L1 Id90 consistent/2 C1 L3

Id12 imposs/1 C1 L2

C2 L2 Id117 plan/4

C1 L4

C3 L2

C2 L2

Id119 holds/2 C2 L1

Id71 retrace/3

C2 L2C1 L5

Id142 plan/4

C1 L4

Id33 preserved/2

C2 L4

C1 L2 Id143 solve/5

C3 L2

C1 L1

Id144 holds/2 C2 L1

Id39 and/3 C2 L2

C3 L1

C2 L5 C2 L3

C2 L2 C1 L1

C2 L2 Id97 always/1

C3 L2

Id104 holds/2 C2 L1

Id115 add/2 C3 L1 C1 L2

Id137 solve/5 C2 L1

C2 L2

C2 L4

C2 L2C1 L5 C1 L4

C1 L3

C1 L2

C2 L3

C1 L2

C1 L1 C1 L2

C1 L2

C1 L3

C1 L1 C1 L2

C2 L2 C1 L1 C2 L1

C3 L1 Id156 solve/5

C2 L1 Id68 plan/4

C1 L2

Id69 solve/5 C1 L1

C2 L2 C1 L5

C1 L4

C1 L3 C1 L1 C2 L3

C2 L4 C2 L1

C1 L3

C1 L1 C1 L1

C1 L2 C2 L1

Id54 plan/4 C1 L2

C1 L2

Id55 solve/5 C2 L1

Id34 del/2 C1 L2

C1 L1

C1 L2 C1 L3

C1 L2

Id70 achieve/5

C3 L2 C2 L2

Id29 holds/2

C2 L1 C1 L1

C3 L1 C2 L2C1 L5

C2 L3

C1 L2

Id5 elem/2 C1 L1

C2 L2

Id31 given/2 C3 L1

C3 L1

C2 L1

C3 L2 C2 L1C2 L2 C1 L1

C3 L1

C2 L3 C2 L1

C2 L2

Id146 given/2 C3 L1

Id145 add/2 C1 L1

Id9 mkgroundlist/3 C3 L2 Id40 elem/2

C1 L1 Id41 equiv/2 C1 L2

C1 L2

Id32 range/3 C3 L1

Id25 range/3 C2 L3 C1 L1

Id113 equiv/2 C1 L2

C2 L3 C2 L2

Id140 given/2

C3 L1

C1 L5 C1 L3

C2 L3

C1 L4

C1 L2

C1 L1

Id128 can/2 C1 L1 C2 L1

C1 L2

Id134 plan/4 C1 L2

C1 L2

C1 L1

Id123 del/2 C1 L2 C2 L2

C2 L1 C1 L1

C3 L1 C2 L2 C1 L5

C1 L4

C2 L4 C2 L3

C1 L3

C1 L2 C2 L2

C2 L1 C1 L1

C3 L2

C3 L1

C1 L1 C3 L1

Id132 equiv/2 C2 L2

C1 L1

C2 L1 C1 L1

C2 L3 C2 L1

C2 L2

Id121 given/2 C3 L1

Id120 add/2 C1 L1

C3 L2 C3 L1 C1 L1

C4 L1 C5 L1

C2 L3 C2 L2

Id106 given/2 C3 L1

Id105 add/2 C1 L1

C3 L2

C3 L1

Id99 range/3 C1 L1

C2 L3 C1 L1 C1 L1

Id85 elem/2

C2 L1 C1 L2

C2 L1 C2 L1

C1 L1

C1 L1 C1 L2 C1 L1

C1 L2

C1 L2 Id21 solve/5

C2 L2 C1 L5

C2 L4 C1 L3

C1 L4

C2 L3

C1 L2 Id1 plans/2

C2 L5C2 L3

Id20 plan/4 C2 L2

Id2 not/1 C1 L1

C1 L1

C1 L2 C2 L1

C2 L1 C1 L1

C2 L2 C3 L2

C2 L2C1 L5 C2 L4

C1 L4 C2 L3

C1 L3

C1 L2 C1 L1

C3 L1

Id75 equiv/2 C2 L2

C2 L1 Id73 equiv/2

C1 L2 Id63 add/2

C1 L1

Id74 not/1 C1 L1

C1 L1

C2 L1 C1 L1

C1 L1

C1 L4 C1 L3

C1 L2 C1 L1

C1 L1

C2 L1 C3 L1

C1 L2

C1 L1

C3 L2 C1 L1

C1 L2

C2 L1 C1 L2

C2 L1 C2 L1

C3 L2 Id42 not/1

C1 L1

C3 L1

Id24 range/3 C1 L1

C2 L3 C1 L1 C1 L1

C1 L1

C1 L1 C1 L1

Id125 del/2 C2 L1

Id124 moved/2 C8 L1

Id111 del/2 C2 L1 C7 L1

Id110 moved/2

C6 L1 C2 L1

C7 L1 C2 L1

C8 L1 Id79 can/2

Id38 del/2 C2 L1

del0 add0

moved0 can0 plans0

lib planner

Snapshot of Analysis Graphs

(15)

test

warplan

Id162 test1/0

Id161 plans/2 C1 L1

Id158 test3/0

Id81 plans/2 C1 L1

Id77 output/1 C2 L5

C2 L2 Id82 not/1

C1 L1

Id78 output1/1

C1 L1

C1 L2

Id83 my_call/1 C1 L1

C2 L5 C2 L3 C2 L2

C1 L1 C1 L1 C2 L2 C2 L5 C2 L3

C3 L2 Id91 not/1

C1 L1 Id90 consistent/2 C1 L3

Id12 imposs/1 C1 L2

C2 L2 Id117 plan/4

C1 L4

C3 L2

C2 L2

Id119 holds/2 C2 L1

Id71 retrace/3

C2 L2C1 L5

Id142 plan/4

C1 L4

Id33 preserved/2

C2 L4

C1 L2 Id143 solve/5

C3 L2

C1 L1

Id144 holds/2 C2 L1

Id39 and/3 C2 L2

C3 L1

C2 L5 C2 L3

C2 L2 C1 L1

C2 L2 Id97 always/1

C3 L2

Id104 holds/2 C2 L1

Id115 add/2 C3 L1 C1 L2

Id137 solve/5 C2 L1

C2 L2

C2 L4

C2 L2C1 L5 C1 L4

C1 L3

C1 L2

C2 L3

C1 L2

C1 L1 C1 L2

C1 L2

C1 L3

C1 L1 C1 L2

C2 L2 C1 L1 C2 L1

C3 L1 Id156 solve/5

C2 L1 Id68 plan/4

C1 L2

Id69 solve/5 C1 L1

C2 L2 C1 L5

C1 L4

C1 L3 C1 L1 C2 L3

C2 L4 C2 L1

C1 L3

C1 L1 C1 L1

C1 L2 C2 L1

Id54 plan/4 C1 L2

C1 L2

Id55 solve/5 C2 L1

Id34 del/2 C1 L2

C1 L1

C1 L2 C1 L3

C1 L2

Id70 achieve/5

C3 L2 C2 L2

Id29 holds/2

C2 L1 C1 L1

C3 L1 C2 L2C1 L5

C2 L3

C1 L2

Id5 elem/2 C1 L1

C2 L2

Id31 given/2 C3 L1

C3 L1

C2 L1

C3 L2 C2 L1C2 L2 C1 L1

C3 L1

C2 L3 C2 L1

C2 L2

Id146 given/2 C3 L1

Id145 add/2 C1 L1

Id9 mkgroundlist/3 C3 L2 Id40 elem/2

C1 L2

Id32 range/3 C3 L1

Id25 range/3 C2 L3 C1 L1

Id113 equiv/2 C1 L2

C2 L3 C2 L2

Id140 given/2

C3 L1

C1 L5 C1 L3

C2 L3

C1 L4

C1 L2

C1 L1

Id128 can/2 C1 L1 C2 L1

C1 L2

Id134 plan/4 C1 L2

C1 L2

C1 L1

Id123 del/2 C1 L2 C2 L2

C2 L1 C1 L1

C3 L1 C2 L2 C1 L5

C1 L4

C2 L4 C2 L3

C1 L3

C1 L2 C2 L2

C2 L1 C1 L1

C3 L2

C3 L1

C1 L1 C3 L1

Id132 equiv/2 C2 L2

C1 L1

C2 L1 C1 L1

C2 L3 C2 L1

C2 L2

Id121 given/2 C3 L1

Id120 add/2 C1 L1

C3 L2 C3 L1 C1 L1

C4 L1 C5 L1

C2 L3 C2 L2

Id106 given/2 C3 L1

Id105 add/2 C1 L1

C3 L2

C3 L1

Id99 range/3 C1 L1

C2 L3 C1 L1 C1 L1

Id85 elem/2

C2 L1 C1 L2

C2 L1 C2 L1

C1 L1

C1 L1 C1 L2 C1 L1

C1 L2

C1 L2 Id21 solve/5

C2 L2 C1 L5

C2 L4 C1 L3

C1 L4

C2 L3

C1 L2 Id1 plans/2

C2 L5C2 L3

Id20 plan/4 C2 L2

Id2 not/1 C1 L1

C1 L1

C1 L2 C2 L1

C2 L1 C1 L1

C2 L2 C3 L2

C2 L2C1 L5 C2 L4

C1 L4 C2 L3

C1 L3

C1 L2 C1 L1

C3 L1

Id75 equiv/2 C2 L2

C2 L1 Id73 equiv/2

C1 L2 Id63 add/2

C1 L1

Id74 not/1 C1 L1

C1 L1

C2 L1 C1 L1

C1 L1

C1 L4 C1 L3

C1 L2 C1 L1

C1 L1

C2 L1 C3 L1

C1 L2

C1 L1

C3 L2 C1 L1

C1 L2

C2 L1 C1 L2

C2 L1 C2 L1

C3 L2 Id42 not/1

C1 L1

C3 L1

Id24 range/3 C1 L1

C2 L3 C1 L1 C1 L1

C1 L1

C1 L1 C1 L1

Id125 del/2 C2 L1

Id124 moved/2 C8 L1

Id111 del/2 C2 L1 C7 L1

Id110 moved/2

C6 L1 C2 L1

C7 L1 C2 L1

C8 L1 Id79 can/2

Id38 del/2 C2 L1

del0 add0

moved0 can0 plans0

lib planner

Snapshot of Analysis Graphs

The algorithm:

• Maintains local and global graphs with call/success pairs for the predicates and their dependencies .

• Deals incrementally with additions, deletions .

• Localizes as much as possible fixpoint (re)computation inside modules to minimize context swaps.

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Theorem 4 (Correctness of IncAnalyze starting from a partial analysis). Let P be a program, Q _α a set of ab- stract queries, and A 0 any analysis graph. Let A = IncAnalyze(P, Q α , ∅, A 0 ). A is correct for P and γ(Q α ) if for all concrete queries q ∈ γ(Q _α ) all nodes n from which there is a path in the concrete execution q n in JP KQ, that are abstracted in the analysis A 0 are included in Q _α , i.e.:

∀Q, n.Q ∈ γ(Q α ) ∧ q n ∈ JP KQ,

∀n α ∈ A 0 .n ∈ γ(n α )⇒n α ∈ Q α .

Theorem 6 (Precision of IncAnalyze). Let P, P ⁰ be pro- grams, such that P differs from P ⁰ by ∆, let Q _α a set of ab- stract queries, and A 0 = IncAnalyze(P ⁰ , Q _α , ∅, ∅) an analysis graph. The following hold:

• If A = IncAnalyze(P, Q α , ∅, ∅), then A is the least program analysis graph for P and γ(Q α ), and

• IncAnalyze(P, Q α , ∆, A 0 ) =

IncAnalyze(P, Q α , ∅, ∅).

Lemma 1 (Correctness of IncAnalyze modulo imported predicates). Let M be a module of program P , E a set of ab- stract queries. Let L 0 be an analysis graph such that ∀hA, λ ^c i ∈ L 0 .mod(A) ∈ imports(M ). The analysis result

L = IncAnalyze(M, E, ∅, L 0 ) is correct for M and γ(E) assuming L 0 .

Lemma 2 (Precision of IncAnalyze modulo imported pred- icates). Let M be a module of program P , E a set of abstract queries. Let L 0 be an analysis graph such that ∀hA, λ ^c i ∈ L 0 .mod(A) ∈ imports(M ) if L 0 contains the least fixed point as defined in Theorem 6. The analysis result

L = IncAnalyze(M, E, ∅, L 0 )

is the least program analysis graph for M and γ(E) assuming

Lemma 3 (Correctness updating L modulo G). Let M be a module of program P and E a set of entries. Let G be a previous state of the global analysis graph, if L M is correct for M and γ(E) assuming G. If G changes to G ⁰ the analysis result

L M ⁰ = LocIncAnalyze(M, E, G ⁰ , L M , ∅) is correct for M and γ(E) assuming G.

Theorem 10 (Correctness of ModIncAnalyze from scratch).

Let P be a modular program, and Q _α a set of abstract queries.

Then, if:

{G, { L M i }} = ModIncAnalyze(P, Q α , ∅, ∅) G is correct for P and γ(Q α ).

Lemma 4 (Precision updating L modulo G). Let M be a module contained in program P , E a set of entries. Let G be a previous state of the global analysis graph, if L M = LocIncAnalyze(M, E, G, ∅, ∅). If G changes to G ⁰ the analy- sis result:

LocIncAnalyze(M, E, G ⁰ , L M , ∅) = LocIncAnalyze(M, E, G ⁰ , ∅, ∅) is the same as analyzing from scratch, i.e., the lfp of M , E.

Theorem 11 (Precision of ModIncAnalyze from scratch).

Let P be a modular program and Q _α a set of abstract queries.

The analysis result

A = ModIncAnalyze(P, Q α , ∅, ∅) = ModAnalyze(P, Q α ) such that A = {G, {L M i }}, then G = G ⁰ .

Theorem 12 (Precision of ModIncAnalyze). Let P, P ⁰ be modular programs that differ by ∆, Q _α a set of queries, and A = ModIncAnalyze(P, Q α , ∅, (∅, ∅)), then

ModIncAnalyze(P ⁰ , Q , ∅, (∅, ∅)) = ModIncAnalyze(P ⁰ , Q , A , ∆).

Fundamental results

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Theorem 4 (Correctness of IncAnalyze starting from a partial analysis). Let P be a program, Q _α a set of ab- stract queries, and A 0 any analysis graph. Let A = IncAnalyze(P, Q α , ∅, A 0 ). A is correct for P and γ(Q α ) if for all concrete queries q ∈ γ(Q _α ) all nodes n from which there is a path in the concrete execution q n in JP KQ, that are abstracted in the analysis A 0 are included in Q _α , i.e.:

∀Q, n.Q ∈ γ(Q α ) ∧ q n ∈ JP KQ,

∀n α ∈ A 0 .n ∈ γ(n α )⇒n α ∈ Q α .

Theorem 6 (Precision of IncAnalyze). Let P, P ⁰ be pro- grams, such that P differs from P ⁰ by ∆, let Q _α a set of ab- stract queries, and A 0 = IncAnalyze(P ⁰ , Q _α , ∅, ∅) an analysis graph. The following hold:

• If A = IncAnalyze(P, Q α , ∅, ∅), then A is the least program analysis graph for P and γ(Q α ), and

• IncAnalyze(P, Q α , ∆, A 0 ) =

IncAnalyze(P, Q α , ∅, ∅).

Lemma 1 (Correctness of IncAnalyze modulo imported predicates). Let M be a module of program P , E a set of ab- stract queries. Let L 0 be an analysis graph such that ∀hA, λ ^c i ∈ L 0 .mod(A) ∈ imports(M ). The analysis result

L = IncAnalyze(M, E, ∅, L 0 ) is correct for M and γ(E) assuming L 0 .

Lemma 2 (Precision of IncAnalyze modulo imported pred- icates). Let M be a module of program P , E a set of abstract queries. Let L 0 be an analysis graph such that ∀hA, λ ^c i ∈ L 0 .mod(A) ∈ imports(M ) if L 0 contains the least fixed point as defined in Theorem 6. The analysis result

L = IncAnalyze(M, E, ∅, L 0 )

is the least program analysis graph for M and γ(E) assuming L 0 .

Lemma 3 (Correctness updating L modulo G). Let M be a module of program P and E a set of entries. Let G be a previous state of the global analysis graph, if L M is correct for M and γ(E) assuming G. If G changes to G ⁰ the analysis result

L M ⁰ = LocIncAnalyze(M, E, G ⁰ , L M , ∅) is correct for M and γ(E) assuming G.

Theorem 10 (Correctness of ModIncAnalyze from scratch).

Let P be a modular program, and Q _α a set of abstract queries.

Then, if:

{G, { L M i }} = ModIncAnalyze(P, Q α , ∅, ∅) G is correct for P and γ(Q α ).

Lemma 4 (Precision updating L modulo G). Let M be a module contained in program P , E a set of entries. Let G be a previous state of the global analysis graph, if L M = LocIncAnalyze(M, E, G, ∅, ∅). If G changes to G ⁰ the analy- sis result:

LocIncAnalyze(M, E, G ⁰ , L M , ∅) = LocIncAnalyze(M, E, G ⁰ , ∅, ∅) is the same as analyzing from scratch, i.e., the lfp of M , E.

Theorem 11 (Precision of ModIncAnalyze from scratch).

Let P be a modular program and Q _α a set of abstract queries.

The analysis result

A = ModIncAnalyze(P, Q α , ∅, ∅) = ModAnalyze(P, Q α ) such that A = {G, {L M i }}, then G = G ⁰ .

Theorem 12 (Precision of ModIncAnalyze). Let P, P ⁰ be modular programs that differ by ∆, Q _α a set of queries, and A = ModIncAnalyze(P, Q α , ∅, (∅, ∅)), then

ModIncAnalyze(P ⁰ , Q _α , ∅, (∅, ∅)) = ModIncAnalyze(P ⁰ , Q _α , A , ∆).

Fundamental results

Contributions

The results from our incremental, modular analysis are:

• Correct over-approximations of the AND tree semantics.

• The most accurate (lfp) if no widening is performed.

Additionally:

• Extended traditional algorithm with widening (not formalized before).

• Split correctness and precision of incremental analysis.

• New results reanalyzing starting from a partial analysis .

• Formalized results of an existing modular algorithm (non incremental).

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

0 5 10 15 20 25

0 2 4 6 8 10 12 14 16

Time (ms)

# of clauses

Adding - aiakl

mon mon_inc mod_incmod

0 5 10 15 20 25

0 2 4 6 8 10 12 14 16

Time (ms)

# of clauses

Deleting - aiakl

mon mon_inc_td mon_inc_sccmod mod_inc_td mod_inc_scc

0 20 40 60 80 100 120 140 160 180 200

0 50 100 150 200 250

Time (ms)

# of clauses

Adding - ann

0 50 100 150 200 250 300

0 50 100 150 200 250

Time (ms)

# of clauses

Deleting - ann

mon mon_inc_td mon_inc_scc mod_inc_tdmod mod_inc_scc

0 5 10 15 20 25 30 35 40 45

0 10 20 30 40 50

Time (ms)

# of clauses

Adding - bid

10 0 20 30 40 50 60 70 80 90

0 10 20 30 40 50

Time (ms)

# of clauses

Deleting - bid

0 5 10 15 20 25 30 35 40 45 50

0 20 40 60 80 100 120 140 160

Time (ms)

# of clauses

Adding - boyer

mon_incmon mod_incmod

0 10 20 30 40 50 60 70

0 20 40 60 80 100 120 140 160

Time (ms)

# of clauses

Deleting - boyer

mon_inc_tdmon mon_inc_scc mod mod_inc_td mod_inc_scc

0 20 40 60 80 100 120 140

0 10 20 30 40 50 60 70 80

Time (ms)

# of clauses

Adding - cleandirs

0 100 200 300 400 500 600

0 10 20 30 40 50 60 70 80

Time (ms)

# of clauses

Deleting - cleandirs

0 20 40 60 80 100 120

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Adding - peephole

0 20 40 60 80 100 120 140 160

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Deleting - peephole

0 2 4 6 10 8 12 14

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Adding - progeom

mon mon_inc mod mod_inc

50 0 100 150 200 250 300 350

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Deleting - progeom

mon mon_inc_td mon_inc_scc mod mod_inc_td mod_inc_scc

0 50 100 150 200 250

0 10 20 30 40 50 60 70 80 90 100

Time (ms)

# of clauses

Adding - read

20 0 40 60 100 80 120 140 160 180 200

0 10 20 30 40 50 60 70 80 90 100

Time (ms)

# of clauses

Deleting - read

mon_inc_tdmon mon_inc_scc mod_inc_tdmod mod_inc_scc

0 2 4 6 10 8 12 14

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Adding - qsort

0 5 10 15 20 25 30 35

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Deleting - qsort

10 20 30 40 50 60 70

Time (ms)

Adding - rdtok

10 20 30 40 50 60 70 80

Time (ms)

Deleting - rdtok

5 10 15 20 25 30 35 40

Time (ms)

Adding - warplan

mon-incmon

mod mod-inc

10 20 30 40 50 60 70 80

Time (ms)

Deleting - warplan

mon-incmon mon-scc mod mod-inc mod-scc

100 200 300 400 500 600 700 800

Time (ms)

Adding - witt

200 400 600 800 1000 1200

Time (ms)

Deleting - witt

Experimental evaluation

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

0 5 10 15 20 25

0 2 4 6 8 10 12 14 16

Time (ms)

# of clauses

Adding - aiakl

0 5 10 15 20 25

0 2 4 6 8 10 12 14 16

Time (ms)

# of clauses

Deleting - aiakl

0 20 40 60 80 100 120 140 160 180 200

0 50 100 150 200 250

Time (ms)

# of clauses

Adding - ann

0 50 100 150 200 250 300

0 50 100 150 200 250

Time (ms)

# of clauses

Deleting - ann

0 5 10 15 20 25 30 35 40 45

0 10 20 30 40 50

Time (ms)

# of clauses

Adding - bid

10 0 20 30 40 50 60 70 80 90

0 10 20 30 40 50

Time (ms)

# of clauses

Deleting - bid

0 5 10 15 20 25 30 35 40 45 50

0 20 40 60 80 100 120 140 160

Time (ms)

# of clauses

Adding - boyer

0 10 20 30 40 50 60 70

0 20 40 60 80 100 120 140 160

Time (ms)

# of clauses

Deleting - boyer

0 20 40 60 80 100 120 140

0 10 20 30 40 50 60 70 80

Time (ms)

# of clauses

Adding - cleandirs

0 100 200 300 400 500 600

0 10 20 30 40 50 60 70 80

Time (ms)

# of clauses

Deleting - cleandirs

0 20 40 60 80 100 120

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Adding - peephole

0 20 40 60 80 100 120 140 160

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Deleting - peephole

0 2 4 6 10 8 12 14

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Adding - progeom

50 0 100 150 200 250 300 350

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Deleting - progeom

0 50 100 150 200 250

0 10 20 30 40 50 60 70 80 90 100

Time (ms)

# of clauses

Adding - read

20 0 40 60 100 80 120 140 160 180 200

0 10 20 30 40 50 60 70 80 90 100

Time (ms)

# of clauses

Deleting - read

0 2 4 6 10 8 12 14

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Adding - qsort

0 5 10 15 20 25 30 35

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Deleting - qsort

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60

Time (ms)

# of clauses

Adding - rdtok

0 10 20 30 40 50 60 70 80

0 10 20 30 40 50 60

Time (ms)

# of clauses

Deleting - rdtok

0 5 10 15 20 25 30 35 40

0 20 40 60 80 100 120

Time (ms)

# of clauses

Adding - warplan

mon-incmon

mod mod-inc

0 10 20 30 40 50 60 70 80

0 20 40 60 80 100 120

Time (ms)

# of clauses

Deleting - warplan

0 100 200 300 400 500 600 700 800

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Adding - witt

0 200 400 600 800 1000 1200

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Deleting - witt

Experimental evaluation

Addition experiment (time in ms) – def domain

0 5 10 15 20 25 30 35 40

0 20 40 60 80 100 120

Time (ms)

# of clauses

Adding - warplan

mon mon-inc mod mod-inc

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(20)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

0 5 10 15 20 25

0 2 4 6 8 10 12 14 16

Time (ms)

# of clauses

Adding - aiakl

0 5 10 15 20 25

0 2 4 6 8 10 12 14 16

Time (ms)

# of clauses

Deleting - aiakl

0 20 40 60 80 100 120 140 160 180 200

0 50 100 150 200 250

Time (ms)

# of clauses

Adding - ann

0 50 100 150 200 250 300

0 50 100 150 200 250

Time (ms)

# of clauses

Deleting - ann

0 5 10 15 20 25 30 35 40 45

0 10 20 30 40 50

Time (ms)

# of clauses

Adding - bid

10 0 20 30 40 50 60 70 80 90

0 10 20 30 40 50

Time (ms)

# of clauses

Deleting - bid

0 5 10 15 20 25 30 35 40 45 50

0 20 40 60 80 100 120 140 160

Time (ms)

# of clauses

Adding - boyer

0 10 20 30 40 50 60 70

0 20 40 60 80 100 120 140 160

Time (ms)

# of clauses

Deleting - boyer

0 20 40 60 80 100 120 140

0 10 20 30 40 50 60 70 80

Time (ms)

# of clauses

Adding - cleandirs

0 100 200 300 400 500 600

0 10 20 30 40 50 60 70 80

Time (ms)

# of clauses

Deleting - cleandirs

0 20 40 60 80 100 120

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Adding - peephole

0 20 40 60 80 100 120 140 160

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Deleting - peephole

0 2 4 6 10 8 12 14

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Adding - progeom

50 0 100 150 200 250 300 350

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Deleting - progeom

0 50 100 150 200 250

0 10 20 30 40 50 60 70 80 90 100

Time (ms)

# of clauses

Adding - read

20 0 40 60 100 80 120 140 160 180 200

0 10 20 30 40 50 60 70 80 90 100

Time (ms)

# of clauses

Deleting - read

0 2 4 6 10 8 12 14

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Adding - qsort

0 5 10 15 20 25 30 35

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Deleting - qsort

10 20 30 40 50 60 70

Time (ms)

Adding - rdtok

10 20 30 40 50 60 70 80

Time (ms)

Deleting - rdtok

5 10 15 20 25 30 35 40

Time (ms)

Adding - warplan

mon-incmon

mod mod-inc

10 20 30 40 50 60 70 80

Time (ms)

Deleting - warplan

100 200 300 400 500 600 700 800

Time (ms)

Adding - witt

200 400 600 800 1000 1200

Time (ms)

Deleting - witt

Experimental evaluation

Accumulated normalized time (def) – clause addition

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

hanoi aiakl qsort progeom bid rdtok cleand read warplan boyer peephole witt ann mproj clinks

The order inside each set of bars is: |mon|mon_inc|mod|mod_inc|.

19

(21)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

0 5 10 15 20 25

0 2 4 6 8 10 12 14 16

Time (ms)

# of clauses

Adding - aiakl

0 5 10 15 20 25

0 2 4 6 8 10 12 14 16

Time (ms)

# of clauses

Deleting - aiakl

0 20 40 60 80 100 120 140 160 180 200

0 50 100 150 200 250

Time (ms)

# of clauses

Adding - ann

0 50 100 150 200 250 300

0 50 100 150 200 250

Time (ms)

# of clauses

Deleting - ann

0 5 10 15 20 25 30 35 40 45

0 10 20 30 40 50

Time (ms)

# of clauses

Adding - bid

10 0 20 30 40 50 60 70 80 90

0 10 20 30 40 50

Time (ms)

# of clauses

Deleting - bid

0 5 10 15 20 25 30 35 40 45 50

0 20 40 60 80 100 120 140 160

Time (ms)

# of clauses

Adding - boyer

0 10 20 30 40 50 60 70

0 20 40 60 80 100 120 140 160

Time (ms)

# of clauses

Deleting - boyer

0 20 40 60 80 100 120 140

0 10 20 30 40 50 60 70 80

Time (ms)

# of clauses

Adding - cleandirs

0 100 200 300 400 500 600

0 10 20 30 40 50 60 70 80

Time (ms)

# of clauses

Deleting - cleandirs

0 20 40 60 80 100 120

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Adding - peephole

0 20 40 60 80 100 120 140 160

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Deleting - peephole

0 2 4 6 10 8 12 14

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Adding - progeom

50 0 100 150 200 250 300 350

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Deleting - progeom

0 50 100 150 200 250

0 10 20 30 40 50 60 70 80 90 100

Time (ms)

# of clauses

Adding - read

20 0 40 60 100 80 120 140 160 180 200

0 10 20 30 40 50 60 70 80 90 100

Time (ms)

# of clauses

Deleting - read

0 2 4 6 10 8 12 14

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Adding - qsort

0 5 10 15 20 25 30 35

0 2 4 6 8 10 12 14 16 18

Time (ms)

# of clauses

Deleting - qsort

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60

Time (ms)

# of clauses

Adding - rdtok

0 10 20 30 40 50 60 70 80

0 10 20 30 40 50 60

Time (ms)

# of clauses

Deleting - rdtok

0 5 10 15 20 25 30 35 40

0 20 40 60 80 100 120

Time (ms)

# of clauses

Adding - warplan

mon-incmon

mod mod-inc

0 10 20 30 40 50 60 70 80

0 20 40 60 80 100 120

Time (ms)

# of clauses

Deleting - warplan

0 100 200 300 400 500 600 700 800

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Adding - witt

0 200 400 600 800 1000 1200

0 20 40 60 80 100 120 140 160 180

Time (ms)

# of clauses

Incremental and Modular Context-Sensitive Analysis