[PPT] - On Minimal Corrections in ASP s Janota 1 Joao Marques-Silva 2 Mikol PowerPoint Presentation

SLIDE 1

On Minimal Corrections in ASP

Mikol´ aˇ s Janota1 Joao Marques-Silva2 RCRA 2017, Bari

1 INESC-ID/IST, University of Lisbon, Portugal 2 LaSIGE, Faculty of Science, University of Lisbon, Portugal Janota and Silva On Minimal Corrections in ASP 1 / 15

SLIDE 2

Context and History

Number of interesting problems about propositional formulae:

Janota and Silva On Minimal Corrections in ASP 2 / 15

SLIDE 3

Context and History

Number of interesting problems about propositional formulae:
Minimally Unsatisfiable Set (MUS), diagnostics, debugging,

SMT

Janota and Silva On Minimal Corrections in ASP 2 / 15

SLIDE 4

Context and History

Number of interesting problems about propositional formulae:
Minimally Unsatisfiable Set (MUS), diagnostics, debugging,

SMT

Minimal Correction Set (MCS), diagnostics, debugging

Janota and Silva On Minimal Corrections in ASP 2 / 15

SLIDE 5

Context and History

Number of interesting problems about propositional formulae:
Minimally Unsatisfiable Set (MUS), diagnostics, debugging,

SMT

Minimal Correction Set (MCS), diagnostics, debugging
Prime Implicant/Implicate, model checking

Janota and Silva On Minimal Corrections in ASP 2 / 15

SLIDE 6

Context and History

Number of interesting problems about propositional formulae:
Minimally Unsatisfiable Set (MUS), diagnostics, debugging,

SMT

Minimal Correction Set (MCS), diagnostics, debugging
Prime Implicant/Implicate, model checking
Minimal model, circumscription

Janota and Silva On Minimal Corrections in ASP 2 / 15

SLIDE 7

Context and History

Number of interesting problems about propositional formulae:
Minimally Unsatisfiable Set (MUS), diagnostics, debugging,

SMT

Minimal Correction Set (MCS), diagnostics, debugging
Prime Implicant/Implicate, model checking
Minimal model, circumscription
Backbone, fault-localization

Janota and Silva On Minimal Corrections in ASP 2 / 15

SLIDE 8

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Janota and Silva On Minimal Corrections in ASP 3 / 15

SLIDE 9

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ

Janota and Silva On Minimal Corrections in ASP 3 / 15

SLIDE 10

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ
φ ⊆ ψ ⇒
SAT(ψ) ⇒ SAT(φ)
Janota and Silva

On Minimal Corrections in ASP 3 / 15

SLIDE 11

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ
φ ⊆ ψ ⇒
SAT(ψ) ⇒ SAT(φ)
φ ⊆ ψ ⇒
UNSAT(φ) ⇒ UNSAT(ψ)
Janota and Silva

On Minimal Corrections in ASP 3 / 15

SLIDE 12

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ
φ ⊆ ψ ⇒
SAT(ψ) ⇒ SAT(φ)
φ ⊆ ψ ⇒
UNSAT(φ) ⇒ UNSAT(ψ)
MUS — subset minimum for the UNSAT predicate.

Janota and Silva On Minimal Corrections in ASP 3 / 15

SLIDE 13

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ
φ ⊆ ψ ⇒
SAT(ψ) ⇒ SAT(φ)
φ ⊆ ψ ⇒
UNSAT(φ) ⇒ UNSAT(ψ)
MUS — subset minimum for the UNSAT predicate.
MSS — subset maximum for the SAT predicate.

Janota and Silva On Minimal Corrections in ASP 3 / 15

SLIDE 14

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ
φ ⊆ ψ ⇒
SAT(ψ) ⇒ SAT(φ)
φ ⊆ ψ ⇒
UNSAT(φ) ⇒ UNSAT(ψ)
MUS — subset minimum for the UNSAT predicate.
MSS — subset maximum for the SAT predicate.
L1, L2 sets of literals:

Janota and Silva On Minimal Corrections in ASP 3 / 15

SLIDE 15

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ
φ ⊆ ψ ⇒
SAT(ψ) ⇒ SAT(φ)
φ ⊆ ψ ⇒
UNSAT(φ) ⇒ UNSAT(ψ)
MUS — subset minimum for the UNSAT predicate.
MSS — subset maximum for the SAT predicate.
L1, L2 sets of literals:
L1 ⊆ L2 ⇒
L1 |

= ϕ ⇒ L2 | = ϕ

Janota and Silva

On Minimal Corrections in ASP 3 / 15

SLIDE 16

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ
φ ⊆ ψ ⇒
SAT(ψ) ⇒ SAT(φ)
φ ⊆ ψ ⇒
UNSAT(φ) ⇒ UNSAT(ψ)
MUS — subset minimum for the UNSAT predicate.
MSS — subset maximum for the SAT predicate.
L1, L2 sets of literals:
L1 ⊆ L2 ⇒
L1 |

= ϕ ⇒ L2 | = ϕ

prime implicant — subset minimum for the · |

= ϕ predicate.

Janota and Silva On Minimal Corrections in ASP 3 / 15

SLIDE 17

Monotone Predicates

These problems are instances of monotone predicates.

[Marques-Silva et al., 2013]

Example for sets of clauses φ, ψ
φ ⊆ ψ ⇒
SAT(ψ) ⇒ SAT(φ)
φ ⊆ ψ ⇒
UNSAT(φ) ⇒ UNSAT(ψ)
MUS — subset minimum for the UNSAT predicate.
MSS — subset maximum for the SAT predicate.
L1, L2 sets of literals:
L1 ⊆ L2 ⇒
L1 |

= ϕ ⇒ L2 | = ϕ

prime implicant — subset minimum for the · |

= ϕ predicate.

Literal a backbone if φ |

= l L1 ⊆ L2 ⇒

l∈L2 φ |

= l ⇒

l∈L1 φ |

= l

Janota and Silva

On Minimal Corrections in ASP 3 / 15

SLIDE 18

What about ASP?

Unlike propositional logic, ASP is not monotone.

Janota and Silva On Minimal Corrections in ASP 4 / 15

SLIDE 19

What about ASP?

Unlike propositional logic, ASP is not monotone.
How to define minimality?

Janota and Silva On Minimal Corrections in ASP 4 / 15

SLIDE 20

What about ASP?

Unlike propositional logic, ASP is not monotone.
How to define minimality?
Can the algorithms from propositional logic be adapted?

(or at least some)

Janota and Silva On Minimal Corrections in ASP 4 / 15

SLIDE 21

What about ASP?

Unlike propositional logic, ASP is not monotone.
How to define minimality?
Can the algorithms from propositional logic be adapted?

(or at least some) Example ← not move(a). % program move(a) ← stone(b), not stone(c). % program stone(c) ← . % fact (input)

Janota and Silva On Minimal Corrections in ASP 4 / 15

SLIDE 22

What about ASP?

Unlike propositional logic, ASP is not monotone.
How to define minimality?
Can the algorithms from propositional logic be adapted?

(or at least some) Example ← not move(a). % program move(a) ← stone(b), not stone(c). % program stone(c) ← . % fact (input) Possible fix: add stone(b), remove stone(c)

Janota and Silva On Minimal Corrections in ASP 4 / 15

SLIDE 23

Maximal Consistent Set in ASP

Definition

Janota and Silva On Minimal Corrections in ASP 5 / 15

SLIDE 24

Maximal Consistent Set in ASP

Definition

Let P be a consistent ASP program and S be a set of atoms.

Janota and Silva On Minimal Corrections in ASP 5 / 15

SLIDE 25

Maximal Consistent Set in ASP

Definition

Let P be a consistent ASP program and S be a set of atoms.
A set L ⊆ S is a maximal consistent subset of S w.r.t. P

Janota and Silva On Minimal Corrections in ASP 5 / 15

SLIDE 26

Maximal Consistent Set in ASP

Definition

Let P be a consistent ASP program and S be a set of atoms.
A set L ⊆ S is a maximal consistent subset of S w.r.t. P
if the program P ∪ {s. | s ∈ L} is consistent

Janota and Silva On Minimal Corrections in ASP 5 / 15

SLIDE 27

Maximal Consistent Set in ASP

Definition

Let P be a consistent ASP program and S be a set of atoms.
A set L ⊆ S is a maximal consistent subset of S w.r.t. P
if the program P ∪ {s. | s ∈ L} is consistent
and for any L′, such that L L′ ⊆ S, the program

P ∪

s. | s ∈ L′

is inconsistent.

Observe: In monotone case L is maximally consistent iff L ∪{s} is inconsistent for any s ∈ S \ L. Does not hold in non-monotone.

Janota and Silva On Minimal Corrections in ASP 5 / 15

SLIDE 28

Choice Rules

Notation

Consider set of atoms: S = {s1, . . . , sk}.

Janota and Silva On Minimal Corrections in ASP 6 / 15

SLIDE 29

Choice Rules

Notation

Consider set of atoms: S = {s1, . . . , sk}.
Let choice(S) denote the choice rule 0 ≤ {s1, . . . , sk}

Janota and Silva On Minimal Corrections in ASP 6 / 15

SLIDE 30

Choice Rules

Notation

Consider set of atoms: S = {s1, . . . , sk}.
Let choice(S) denote the choice rule 0 ≤ {s1, . . . , sk}
Let atleast1({s1, . . . , sk}) denote the choice rule

1 ≤ {s1, . . . , sk}.

Janota and Silva On Minimal Corrections in ASP 6 / 15

SLIDE 31

Choice Rules

Notation

Consider set of atoms: S = {s1, . . . , sk}.
Let choice(S) denote the choice rule 0 ≤ {s1, . . . , sk}
Let atleast1({s1, . . . , sk}) denote the choice rule

1 ≤ {s1, . . . , sk}. Idea

Janota and Silva On Minimal Corrections in ASP 6 / 15

SLIDE 32

Choice Rules

Notation

Consider set of atoms: S = {s1, . . . , sk}.
Let choice(S) denote the choice rule 0 ≤ {s1, . . . , sk}
Let atleast1({s1, . . . , sk}) denote the choice rule

1 ≤ {s1, . . . , sk}. Idea

Define P′ = P ∪ {s. | s ∈ L} ∪{choice(S \ L).}.

Janota and Silva On Minimal Corrections in ASP 6 / 15

SLIDE 33

Choice Rules

Notation

Consider set of atoms: S = {s1, . . . , sk}.
Let choice(S) denote the choice rule 0 ≤ {s1, . . . , sk}
Let atleast1({s1, . . . , sk}) denote the choice rule

1 ≤ {s1, . . . , sk}. Idea

Define P′ = P ∪ {s. | s ∈ L} ∪{choice(S \ L).}.
There exists a consistent set L′ s.t. L ⊆ L′ ⊆ S iff P′ has an

answer set µ such that L′ = S ∩µ.

Janota and Silva On Minimal Corrections in ASP 6 / 15

SLIDE 34

Algorithm: At-least-1

1 L ← ∅

// consistency lower bound

2 while true do 3

P′ ← P ∪ {s. | s ∈ L}

4

P′ ← P′ ∪ {atleast1(S L).}

5

(res, µ) ← solve(P′)

6

if ¬res then return L

7

L ← L ∪(µ ∩ S)

Janota and Silva On Minimal Corrections in ASP 7 / 15

SLIDE 35

Algorithm: Unit addition

1 L ← ∅

// consistency lower bound

2 while S = ∅ do 3

sf ← pick an arbitrary element from S

4

S ← S {sf }

5

L ← L ∪{sf }

6

P′ ← P′ ∪ {s. | s ∈ L}

7

P′ ← P ∪{choice(S).}

8

(res, µ) ← solve(P′)

9

if ¬res then L ← L {sf }

10

else L ← L ∪(µ ∩ S)

11 return L

Janota and Silva On Minimal Corrections in ASP 8 / 15

SLIDE 36

Algorithm: Progression

1 L ← ∅

// consistency lower bound

2 K ← 1

// chunk size

3 while S = ∅ do 4

C ← pick min(| S |, K) arbitrary elements from S

5

S ← S C

6

L ← L ∪ C

7

P′ ← P′ ∪ {s. | s ∈ L}

8

P′ ← P ∪{choice(S).}

9

(res, µ) ← solve(P′)

10

if ¬res then // re-analyze chunk more finely

11

L ← L C

12

if K > 1 then S ← S ∪ C

13

K = 1 // reset chunk size

14

else

15

K ← 2K // double chunk size

16

L ← L ∪(µ ∩ S)

17 return L Janota and Silva On Minimal Corrections in ASP 9 / 15

SLIDE 37

Minimal Correction Sets in ASP

Definition

Janota and Silva On Minimal Corrections in ASP 10 / 15

SLIDE 38

Minimal Correction Sets in ASP

Definition

P be an inconsistent logic program

Janota and Silva On Minimal Corrections in ASP 10 / 15

SLIDE 39

Minimal Correction Sets in ASP

Definition

P be an inconsistent logic program
A and R be sets of rules

Janota and Silva On Minimal Corrections in ASP 10 / 15

SLIDE 40

Minimal Correction Sets in ASP

Definition

P be an inconsistent logic program
A and R be sets of rules
An (A, R)-correction of P is a pair (Mr, Ma) s.t.

Janota and Silva On Minimal Corrections in ASP 10 / 15

SLIDE 41

Minimal Correction Sets in ASP

Definition

P be an inconsistent logic program
A and R be sets of rules
An (A, R)-correction of P is a pair (Mr, Ma) s.t.
Mr ⊆ R and Ma ⊆ A and the program

Janota and Silva On Minimal Corrections in ASP 10 / 15

SLIDE 42

Minimal Correction Sets in ASP

Definition

P be an inconsistent logic program
A and R be sets of rules
An (A, R)-correction of P is a pair (Mr, Ma) s.t.
Mr ⊆ R and Ma ⊆ A and the program
(P \ Mr) ∪ Ma is consistent.

Janota and Silva On Minimal Corrections in ASP 10 / 15

SLIDE 43

Minimal Correction Sets in ASP

Definition

P be an inconsistent logic program
A and R be sets of rules
An (A, R)-correction of P is a pair (Mr, Ma) s.t.
Mr ⊆ R and Ma ⊆ A and the program
(P \ Mr) ∪ Ma is consistent.
An (A, R)-correction (Mr, Ma) is minimal if

for any (A, R)-correction (M′

r, M′ a) such that M′ r ⊆ Mr and

M′

a ⊆ Ma, it holds that Ma = M′ a and Mr = M′ r.

Janota and Silva On Minimal Corrections in ASP 10 / 15

SLIDE 44

Minimal Correction Sets and Maximal Consistency

To calculate (A, R)-correction via Maximal Consistency:

Introduce fresh atoms sr

r for each r ∈ R.

Janota and Silva On Minimal Corrections in ASP 11 / 15

SLIDE 45

Minimal Correction Sets and Maximal Consistency

To calculate (A, R)-correction via Maximal Consistency:

Introduce fresh atoms sr

r for each r ∈ R.

Introduce fresh atoms sa

r for each r ∈ A.

Janota and Silva On Minimal Corrections in ASP 11 / 15

SLIDE 46

Minimal Correction Sets and Maximal Consistency

To calculate (A, R)-correction via Maximal Consistency:

Introduce fresh atoms sr

r for each r ∈ R.

Introduce fresh atoms sa

r for each r ∈ A.

Replace each rule r ∈ R with head(r) ← sr

r , body(r)

Janota and Silva On Minimal Corrections in ASP 11 / 15

SLIDE 47

Minimal Correction Sets and Maximal Consistency

To calculate (A, R)-correction via Maximal Consistency:

Introduce fresh atoms sr

r for each r ∈ R.

Introduce fresh atoms sa

r for each r ∈ A.

Replace each rule r ∈ R with head(r) ← sr

r , body(r)

Replace each rule r ∈ A with head(r) ← not sa

r , body(r).

Janota and Silva On Minimal Corrections in ASP 11 / 15

SLIDE 48

Minimal Correction Sets and Maximal Consistency

To calculate (A, R)-correction via Maximal Consistency:

Introduce fresh atoms sr

r for each r ∈ R.

Introduce fresh atoms sa

r for each r ∈ A.

Replace each rule r ∈ R with head(r) ← sr

r , body(r)

Replace each rule r ∈ A with head(r) ← not sa

r , body(r).

Maximal consistent subset of the fresh atoms gives a minimal

correction.

Janota and Silva On Minimal Corrections in ASP 11 / 15

SLIDE 49

Experimental Results

Family a p u x VBS knight [8,10] (95) 74 75 78 60 80 knight [8,4] (51) 7 13 13 7 14 patterns [16,10] (100) 100 100 100 100 100 patterns [20,15] (100) 100 100 100 100 100 solitaire [12] (18) 18 18 18 17 18 solitaire [14] (16) 12 9 11 4 13 graceful graphs [10,50] (100) 57 75 63 62 83 graceful graphs [30,20] (57) 56 57 57 55 57 total (537) 424 447 440 405 465

Janota and Silva On Minimal Corrections in ASP 12 / 15

SLIDE 50

Experimental Results (Cont.)

300 350 400 450 500 instances 200 400 600 800 1000 CPU time (s)

VBS p u a x

Janota and Silva On Minimal Corrections in ASP 13 / 15

SLIDE 51

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 52

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

What about ASP, which is not monotone?

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 53

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

What about ASP, which is not monotone?
Idea: Using a choice rule let the solver choose one of the

supersets.

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 54

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

What about ASP, which is not monotone?
Idea: Using a choice rule let the solver choose one of the

supersets.

3 different algorithms developed.

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 55

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

What about ASP, which is not monotone?
Idea: Using a choice rule let the solver choose one of the

supersets.

3 different algorithms developed.
Link between maximal consistency and corrections.

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 56

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

What about ASP, which is not monotone?
Idea: Using a choice rule let the solver choose one of the

supersets.

3 different algorithms developed.
Link between maximal consistency and corrections.
More experiments.

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 57

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

What about ASP, which is not monotone?
Idea: Using a choice rule let the solver choose one of the

supersets.

3 different algorithms developed.
Link between maximal consistency and corrections.
More experiments.
More algorithms?

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 58

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

What about ASP, which is not monotone?
Idea: Using a choice rule let the solver choose one of the

supersets.

3 different algorithms developed.
Link between maximal consistency and corrections.
More experiments.
More algorithms?
How to obtain the “addition set”?

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 59

Summary

Many recent results on algorithms for propositional logic

involving monotone predicates.

What about ASP, which is not monotone?
Idea: Using a choice rule let the solver choose one of the

supersets.

3 different algorithms developed.
Link between maximal consistency and corrections.
More experiments.
More algorithms?
How to obtain the “addition set”?
What are the good means for users to specify the addition

and removal sets?

Janota and Silva On Minimal Corrections in ASP 14 / 15

SLIDE 60

Thank You for Your Attention! Questions?

Janota and Silva On Minimal Corrections in ASP 15 / 15

SLIDE 61

Marques-Silva, J., Janota, M., and Belov, A. (2013). Minimal sets over monotone predicates in boolean formulae. In Computer Aided Verification - International Conference (CAV), pages 592–607.

Janota and Silva On Minimal Corrections in ASP 15 / 15