ASP Solving for Expanding Universes Martin Gebser Tomi Janhunen - - PowerPoint PPT Presentation

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ASP Solving for Expanding Universes Martin Gebser Tomi Janhunen - - PowerPoint PPT Presentation

Feb 09, 2023 136 likes •575 views

ASP Solving for Expanding Universes Martin Gebser Tomi Janhunen Holger Jost Roland Kaminski Torsten Schaub Aalto University INRIA Rennes University of Potsdam G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 1 / 13

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ASP Solving for Expanding Universes

Martin Gebser Tomi Janhunen Holger Jost Roland Kaminski Torsten Schaub

Aalto University INRIA Rennes University of Potsdam

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 1 / 13

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Outline

1 Motivation 2 Expanding Logic Programs 3 Conclusions

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 2 / 13

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Motivation

Outline

1 Motivation 2 Expanding Logic Programs 3 Conclusions

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 3 / 13

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Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(X) ← bird(X), ∼penguin(X) bird(tweety) ← penguin(tweety) ← | = {bird(tweety), ¬penguin(tweety), flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(X) ← bird(X), ∼penguin(X) bird(tweety) ← penguin(tweety) ← | = {bird(tweety), ¬penguin(tweety), flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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SLIDE 6

Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(X) ← bird(X), ∼penguin(X) bird(tweety) ← penguin(tweety) ← | = {bird(tweety), ¬penguin(tweety), flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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SLIDE 7

Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(X) ← bird(X), ∼penguin(X) bird(tweety) ← penguin(tweety) ← | = {bird(tweety), ¬penguin(tweety), flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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SLIDE 8

Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(tweety) ↔ bird(tweety) ∧ ¬penguin(tweety) bird(tweety) ↔ ⊤ penguin(tweety) ↔ ⊥ | = {bird(tweety), ¬penguin(tweety), flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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SLIDE 9

Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(X) ← bird(X), ∼penguin(X) bird(tweety) ← penguin(tweety) ← | = {bird(tweety), ¬penguin(tweety), flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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SLIDE 10

Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(X) ← bird(X), ∼penguin(X) bird(tweety) ← penguin(tweety) ← | = {bird(tweety), ¬penguin(tweety), flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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SLIDE 11

Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(tweety) ↔ bird(tweety) ∧ ¬penguin(tweety) bird(tweety) ↔ ⊤ penguin(tweety) ↔ ⊤ | = {bird(tweety), penguin(tweety), ¬flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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SLIDE 12

Motivation

Multi-shot Solving

Input Ground Solve Output Traditional ASP systems were devised for one-shot solving Modern ASP systems allow for multi-shot solving in a reactive way New properties or objects must be integrated dynamically Due to non-monotonicity, new information can invalidate conclusions flies(tweety) ↔ bird(tweety) ∧ ¬penguin(tweety) bird(tweety) ↔ ⊤ penguin(tweety) ↔ ⊤ | = {bird(tweety), penguin(tweety), ¬flies(tweety)} General approach to integrate new information into reasoning process?

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 4 / 13

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SLIDE 13

Motivation

Basic Idea

View arrival of new objects as addition of new constants

➥ Successively expanding Herbrand universe

New constants induce new ground instances of rules

➥ Disjoint partition and modular composition of ground program

✘ New ground instances defining older atoms invalidate completion! Contribution ✔ Translation approach guaranteeing modularity at level of completion

1 New ground instances of rules define new expansion atoms 2 Expansion atoms are interconnected to accumulate derivations 3 Accumulated derivations are propagated to original ground atoms

a1 a2 a3 a4 a5 r1

1 · · · r1 n1

r2

1 · · · r2 n2

r3

1 · · · r3 n3

r4

1 · · · r4 n4

a

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 5 / 13

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SLIDE 14

Motivation

Basic Idea

View arrival of new objects as addition of new constants

➥ Successively expanding Herbrand universe

New constants induce new ground instances of rules

➥ Disjoint partition and modular composition of ground program

✘ New ground instances defining older atoms invalidate completion! Contribution ✔ Translation approach guaranteeing modularity at level of completion

1 New ground instances of rules define new expansion atoms 2 Expansion atoms are interconnected to accumulate derivations 3 Accumulated derivations are propagated to original ground atoms

a1 a2 a3 a4 a5 r1

1 · · · r1 n1

r2

1 · · · r2 n2

r3

1 · · · r3 n3

r4

1 · · · r4 n4

a

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 5 / 13

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SLIDE 15

Motivation

Basic Idea

View arrival of new objects as addition of new constants

➥ Successively expanding Herbrand universe

New constants induce new ground instances of rules

➥ Disjoint partition and modular composition of ground program

✘ New ground instances defining older atoms invalidate completion! Contribution ✔ Translation approach guaranteeing modularity at level of completion

1 New ground instances of rules define new expansion atoms 2 Expansion atoms are interconnected to accumulate derivations 3 Accumulated derivations are propagated to original ground atoms

a1 a2 a3 a4 a5 r1

1 · · · r1 n1

r2

1 · · · r2 n2

r3

1 · · · r3 n3

r4

1 · · · r4 n4

a

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 5 / 13

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SLIDE 16

Motivation

Basic Idea

View arrival of new objects as addition of new constants

➥ Successively expanding Herbrand universe

New constants induce new ground instances of rules

➥ Disjoint partition and modular composition of ground program

✘ New ground instances defining older atoms invalidate completion! Contribution ✔ Translation approach guaranteeing modularity at level of completion

1 New ground instances of rules define new expansion atoms 2 Expansion atoms are interconnected to accumulate derivations 3 Accumulated derivations are propagated to original ground atoms

a1 a2 a3 a4 a5 r1

1 · · · r1 n1

r2

1 · · · r2 n2

r3

1 · · · r3 n3

r4

1 · · · r4 n4

a

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 5 / 13

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SLIDE 17

Motivation

Basic Idea

View arrival of new objects as addition of new constants

➥ Successively expanding Herbrand universe

New constants induce new ground instances of rules

➥ Disjoint partition and modular composition of ground program

✘ New ground instances defining older atoms invalidate completion! Contribution ✔ Translation approach guaranteeing modularity at level of completion

1 New ground instances of rules define new expansion atoms 2 Expansion atoms are interconnected to accumulate derivations 3 Accumulated derivations are propagated to original ground atoms

a1 a2 a3 a4 a5 r1

1 · · · r1 n1

r2

1 · · · r2 n2

r3

1 · · · r3 n3

r4

1 · · · r4 n4

a

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 5 / 13

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SLIDE 18

Motivation

Basic Idea

View arrival of new objects as addition of new constants

➥ Successively expanding Herbrand universe

New constants induce new ground instances of rules

➥ Disjoint partition and modular composition of ground program

✘ New ground instances defining older atoms invalidate completion! Contribution ✔ Translation approach guaranteeing modularity at level of completion

1 New ground instances of rules define new expansion atoms 2 Expansion atoms are interconnected to accumulate derivations 3 Accumulated derivations are propagated to original ground atoms

a1 a2 a3 a4 a5 r1

1 · · · r1 n1

r2

1 · · · r2 n2

r3

1 · · · r3 n3

r4

1 · · · r4 n4

a

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 5 / 13

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SLIDE 19

Motivation

Basic Idea

View arrival of new objects as addition of new constants

➥ Successively expanding Herbrand universe

New constants induce new ground instances of rules

➥ Disjoint partition and modular composition of ground program

✘ New ground instances defining older atoms invalidate completion! Contribution ✔ Translation approach guaranteeing modularity at level of completion

1 New ground instances of rules define new expansion atoms 2 Expansion atoms are interconnected to accumulate derivations 3 Accumulated derivations are propagated to original ground atoms

a1 a2 a3 a4 a5 r1

1 · · · r1 n1

r2

1 · · · r2 n2

r3

1 · · · r3 n3

r4

1 · · · r4 n4

a

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 5 / 13

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Expanding Logic Programs

Outline

1 Motivation 2 Expanding Logic Programs 3 Conclusions

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 6 / 13

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Expanding Logic Programs

Translation Approach

Given a set R of rules, defining intensional predicates PI, let: Φ(R) = {pk(X1, . . . , Xn) ← B | (p(X1, . . . , Xn) ← B) ∈ R}, Π(PI) = {p(X1, . . . , Xn) ← pk(X1, . . . , Xn) | p/n ∈ PI}, ∆(PI) = {pk(X1, . . . , Xn) ← pk+1(X1, . . . , Xn) | p/n ∈ PI}. Example R =

  • k(C) ← cs(C), st(S), in(S, C)

ko(C) ← cs(C), ∼ok(C)

  • Φ(R) =
  • kk(C) ← cs(C), st(S), in(S, C)

kok(C) ← cs(C), ∼ok(C)

  • Π(PI) =
  • k(C) ← okk(C)

ko(C) ← kok(C)

  • ∆(PI) =
  • kk(C) ← okk+1(C)

kok(C) ← kok+1(C)

  • G, Janhunen, Jost, Kaminski, Schaub

ASP Solving for Expanding Universes 7 / 13

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SLIDE 22

Expanding Logic Programs

Translation Approach

Given a set R of rules, defining intensional predicates PI, let: Φ(R) = {pk(X1, . . . , Xn) ← B | (p(X1, . . . , Xn) ← B) ∈ R}, Π(PI) = {p(X1, . . . , Xn) ← pk(X1, . . . , Xn) | p/n ∈ PI}, ∆(PI) = {pk(X1, . . . , Xn) ← pk+1(X1, . . . , Xn) | p/n ∈ PI}. Example R =

  • k(C) ← cs(C), st(S), in(S, C)

ko(C) ← cs(C), ∼ok(C)

  • Φ(R) =
  • kk(C) ← cs(C), st(S), in(S, C)

kok(C) ← cs(C), ∼ok(C)

  • Π(PI) =
  • k(C) ← okk(C)

ko(C) ← kok(C)

  • ∆(PI) =
  • kk(C) ← okk+1(C)

kok(C) ← kok+1(C)

  • G, Janhunen, Jost, Kaminski, Schaub

ASP Solving for Expanding Universes 7 / 13

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SLIDE 23

Expanding Logic Programs

Translation Approach

Given a set R of rules, defining intensional predicates PI, let: Φ(R) = {pk(X1, . . . , Xn) ← B | (p(X1, . . . , Xn) ← B) ∈ R}, Π(PI) = {p(X1, . . . , Xn) ← pk(X1, . . . , Xn) | p/n ∈ PI}, ∆(PI) = {pk(X1, . . . , Xn) ← pk+1(X1, . . . , Xn) | p/n ∈ PI}. Example R =

  • k(C) ← cs(C), st(S), in(S, C)

ko(C) ← cs(C), ∼ok(C)

  • Φ(R) =
  • kk(C) ← cs(C), st(S), in(S, C)

kok(C) ← cs(C), ∼ok(C)

  • Π(PI) =
  • k(C) ← okk(C)

ko(C) ← kok(C)

  • ∆(PI) =
  • kk(C) ← okk+1(C)

kok(C) ← kok+1(C)

  • G, Janhunen, Jost, Kaminski, Schaub

ASP Solving for Expanding Universes 7 / 13

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SLIDE 24

Expanding Logic Programs

Translation Approach

Given a set R of rules, defining intensional predicates PI, let: Φ(R) = {pk(X1, . . . , Xn) ← B | (p(X1, . . . , Xn) ← B) ∈ R}, Π(PI) = {p(X1, . . . , Xn) ← pk(X1, . . . , Xn) | p/n ∈ PI}, ∆(PI) = {pk(X1, . . . , Xn) ← pk+1(X1, . . . , Xn) | p/n ∈ PI}. Example R =

  • k(C) ← cs(C), st(S), in(S, C)

ko(C) ← cs(C), ∼ok(C)

  • Φ(R) =
  • kk(C) ← cs(C), st(S), in(S, C)

kok(C) ← cs(C), ∼ok(C)

  • Π(PI) =
  • k(C) ← okk(C)

ko(C) ← kok(C)

  • ∆(PI) =
  • kk(C) ← okk+1(C)

kok(C) ← kok+1(C)

  • G, Janhunen, Jost, Kaminski, Schaub

ASP Solving for Expanding Universes 7 / 13

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SLIDE 25

Expanding Logic Programs

Expansible Instantiation

Given a set R of rules and a constant stream (c1, . . . , ci, . . . , cj, . . . ), the expansible instantiation of R for j ≥ 0 is j

i=0 Ri, where:

Ri = {(r[i])σ | r ∈ Φ(R) ∪ Π(PI), σ is new ground substitution for i} ∪ {(r[i])σ | r ∈ ∆(PI), σ is ground substitution for i}. Constant stream (c1, s1, . . .)

  • kk(C) ← cs(C), st(S), in(S, C)

kok(C) ← cs(C), ∼ok(C) [Φ(R)]

  • k(C) ← okk(C)

ko(C) ← kok(C) [Π(PI)]

  • kk(C) ← okk+1(C)

kok(C) ← kok+1(C) [∆(PI)]

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 8 / 13

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SLIDE 26

Expanding Logic Programs

Expansible Instantiation

Given a set R of rules and a constant stream (c1, . . . , ci, . . . , cj, . . . ), the expansible instantiation of R for j ≥ 0 is j

i=0 Ri, where:

Ri = {(r[i])σ | r ∈ Φ(R) ∪ Π(PI), σ is new ground substitution for i} ∪ {(r[i])σ | r ∈ ∆(PI), σ is ground substitution for i}. Constant stream (c1, s1, . . .)

  • kk(C) ← cs(C), st(S), in(S, C)

kok(C) ← cs(C), ∼ok(C) [Φ(R)]

  • k(C) ← okk(C)

ko(C) ← kok(C) [Π(PI)]

  • kk(C) ← okk+1(C)

kok(C) ← kok+1(C) [∆(PI)]

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 8 / 13

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SLIDE 27

Expanding Logic Programs

Expansible Instantiation

Given a set R of rules and a constant stream (c1, . . . , ci, . . . , cj, . . . ), the expansible instantiation of R for j ≥ 0 is j

i=0 Ri, where:

Ri = {(r[i])σ | r ∈ Φ(R) ∪ Π(PI), σ is new ground substitution for i} ∪ {(r[i])σ | r ∈ ∆(PI), σ is ground substitution for i}. Constant stream (c1, s1, . . .)

  • kk(C) ← cs(C), st(S), in(S, C)

kok(C) ← cs(C), ∼ok(C) [Φ(R)]

  • k(C) ← okk(C)

ko(C) ← kok(C) [Π(PI)]

  • kk(C) ← okk+1(C)

kok(C) ← kok+1(C) [∆(PI)]

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 8 / 13

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SLIDE 28

Expanding Logic Programs

Expansible Instantiation

Given a set R of rules and a constant stream (c1, . . . , ci, . . . , cj, . . . ), the expansible instantiation of R for j ≥ 0 is j

i=0 Ri, where:

Ri = {(r[i])σ | r ∈ Φ(R) ∪ Π(PI), σ is new ground substitution for i} ∪ {(r[i])σ | r ∈ ∆(PI), σ is ground substitution for i}. Constant stream (c1, s1, . . .) R1 =         

  • k1(c1) ← cs(c1), st(c1), in(c1, c1)

ko1(c1) ← cs(c1), ∼ok(c1) [Φ(R)]

  • k(c1) ← ok1(c1)

ko(c1) ← ko1(c1) [Π(PI)]

  • k1(c1) ← ok2(c1)

ko1(c1) ← ko2(c1) [∆(PI)]         

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 8 / 13

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SLIDE 29

Expanding Logic Programs

Expansible Instantiation

Given a set R of rules and a constant stream (c1, . . . , ci, . . . , cj, . . . ), the expansible instantiation of R for j ≥ 0 is j

i=0 Ri, where:

Ri = {(r[i])σ | r ∈ Φ(R) ∪ Π(PI), σ is new ground substitution for i} ∪ {(r[i])σ | r ∈ ∆(PI), σ is ground substitution for i}. Constant stream (c1, s1, . . .) R2 =                       

  • k2(c1) ← cs(c1), st(s1), in(s1, c1)
  • k2(s1) ← cs(s1), st(c1), in(c1, s1)
  • k2(s1) ← cs(s1), st(s1), in(s1, s1)

ko2(s1) ← cs(s1), ∼ok(s1) [Φ(R)]

  • k(s1) ← ok2(s1)

ko(s1) ← ko2(s1) [Π(PI)]

  • k2(c1) ← ok3(c1)

ko2(c1) ← ko3(c1) [∆(PI)]

  • k2(s1) ← ok3(s1)

ko2(s1) ← ko3(s1)                       

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 8 / 13

slide-30
SLIDE 30

Expanding Logic Programs

Expansible Instantiation

Given a set R of rules and a constant stream (c1, . . . , ci, . . . , cj, . . . ), the expansible instantiation of R for j ≥ 0 is j

i=0 Ri, where:

Ri = {(r[i])σ | r ∈ Φ(R) ∪ Π(PI), σ is new ground substitution for i} ∪ {(r[i])σ | r ∈ ∆(PI), σ is ground substitution for i}. Constant stream (c1, s1, . . .) R2 =                       

  • k2(c1) ← cs(c1), st(s1), in(s1, c1)
  • k2(s1) ← cs(s1), st(c1), in(c1, s1)
  • k2(s1) ← cs(s1), st(s1), in(s1, s1)

ko2(s1) ← cs(s1), ∼ok(s1) [Φ(R)]

  • k(s1) ← ok2(s1)

ko(s1) ← ko2(s1) [Π(PI)]

  • k2(c1) ← ok3(c1)

ko2(c1) ← ko3(c1) [∆(PI)]

  • k2(s1) ← ok3(s1)

ko2(s1) ← ko3(s1)                       

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 8 / 13

slide-31
SLIDE 31

Expanding Logic Programs

Expansible Instantiation

Given a set R of rules and a constant stream (c1, . . . , ci, . . . , cj, . . . ), the expansible instantiation of R for j ≥ 0 is j

i=0 Ri, where:

Ri = {(r[i])σ | r ∈ Φ(R) ∪ Π(PI), σ is new ground substitution for i} ∪ {(r[i])σ | r ∈ ∆(PI), σ is ground substitution for i}. Constant stream (c1, s1, . . .) R3 =             

  • k3(c1) ← . . .

. . . [Φ(R)] . . . [Π(PI)] . . . [∆(PI)]             

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 8 / 13

slide-32
SLIDE 32

Expanding Logic Programs

Modularity Properties

✔ Expansion atoms guarantee disjointness of constraints at ground level

1 Rules 2 Completion 3 Loop formulas

✔ Union of local constraints for Ri (1 ≤ i ≤ j) matches those of j

i=0 Ri

Expansible instantiation can be produced in successive parts that are:

1 Sound 2 Complete 3 Cumulative

Non-monotone semantics is broken down into monotone constraints

➥ Reasoning process can integrate successive parts in multi-shot solving

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 9 / 13

slide-33
SLIDE 33

Expanding Logic Programs

Modularity Properties

✔ Expansion atoms guarantee disjointness of constraints at ground level

1 Rules 2 Completion 3 Loop formulas

✔ Union of local constraints for Ri (1 ≤ i ≤ j) matches those of j

i=0 Ri

Expansible instantiation can be produced in successive parts that are:

1 Sound 2 Complete 3 Cumulative

Non-monotone semantics is broken down into monotone constraints

➥ Reasoning process can integrate successive parts in multi-shot solving

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 9 / 13

slide-34
SLIDE 34

Expanding Logic Programs

Modularity Properties

✔ Expansion atoms guarantee disjointness of constraints at ground level

1 Rules 2 Completion 3 Loop formulas

✔ Union of local constraints for Ri (1 ≤ i ≤ j) matches those of j

i=0 Ri

Expansible instantiation can be produced in successive parts that are:

1 Sound 2 Complete 3 Cumulative

Non-monotone semantics is broken down into monotone constraints

➥ Reasoning process can integrate successive parts in multi-shot solving

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Expanding Logic Programs

Modularity Properties

✔ Expansion atoms guarantee disjointness of constraints at ground level

1 Rules 2 Completion 3 Loop formulas

✔ Union of local constraints for Ri (1 ≤ i ≤ j) matches those of j

i=0 Ri

Expansible instantiation can be produced in successive parts that are:

1 Sound 2 Complete 3 Cumulative

Non-monotone semantics is broken down into monotone constraints

➥ Reasoning process can integrate successive parts in multi-shot solving

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 9 / 13

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Expanding Logic Programs

Original Semantics

Associate interpretation I for rules R with extended interpretation I ∗, augmenting I with expansion atoms ai (based on a predicate pi) such that I | = B for a ground instance a ← B, where i is a stream position in between the maximum of constants in a and those in a or B Constant stream (c1, s1, . . . ) revisited I = {cs(c1), st(s1), in(s1, c1), ok(c1)} I ∗ = I ∪ {ok1(c1), ok2(c1)}

1 If I is a stable (or supported) model of R, given constants

{c1, . . . , cj} and facts over extensional predicates, then I ∗ is a stable (or supported) model of the expansible instantiation of R for j ≥ 0

2 If I ′ is a stable (or supported) model of the expansible instantiation

  • f R for j ≥ 0, given facts over extensional predicates, then I ′ = I ∗ for

a stable (or supported) model I of R relative to constants {c1, . . . , cj}

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 10 / 13

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Expanding Logic Programs

Original Semantics

Associate interpretation I for rules R with extended interpretation I ∗, augmenting I with expansion atoms ai (based on a predicate pi) such that I | = B for a ground instance a ← B, where i is a stream position in between the maximum of constants in a and those in a or B Constant stream (c1, s1, . . . ) revisited I = {cs(c1), st(s1), in(s1, c1), ok(c1)} I ∗ = I ∪ {ok1(c1), ok2(c1)}

1 If I is a stable (or supported) model of R, given constants

{c1, . . . , cj} and facts over extensional predicates, then I ∗ is a stable (or supported) model of the expansible instantiation of R for j ≥ 0

2 If I ′ is a stable (or supported) model of the expansible instantiation

  • f R for j ≥ 0, given facts over extensional predicates, then I ′ = I ∗ for

a stable (or supported) model I of R relative to constants {c1, . . . , cj}

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 10 / 13

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Expanding Logic Programs

Original Semantics

Associate interpretation I for rules R with extended interpretation I ∗, augmenting I with expansion atoms ai (based on a predicate pi) such that I | = B for a ground instance a ← B, where i is a stream position in between the maximum of constants in a and those in a or B Constant stream (c1, s1, . . . ) revisited I = {cs(c1), st(s1), in(s1, c1), ok(c1)} I ∗ = I ∪ {ok1(c1), ok2(c1)}

1 If I is a stable (or supported) model of R, given constants

{c1, . . . , cj} and facts over extensional predicates, then I ∗ is a stable (or supported) model of the expansible instantiation of R for j ≥ 0

2 If I ′ is a stable (or supported) model of the expansible instantiation

  • f R for j ≥ 0, given facts over extensional predicates, then I ′ = I ∗ for

a stable (or supported) model I of R relative to constants {c1, . . . , cj}

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 10 / 13

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SLIDE 39

Expanding Logic Programs

Original Semantics

Associate interpretation I for rules R with extended interpretation I ∗, augmenting I with expansion atoms ai (based on a predicate pi) such that I | = B for a ground instance a ← B, where i is a stream position in between the maximum of constants in a and those in a or B Constant stream (c1, s1, . . . ) revisited I = {cs(c1), st(s1), in(s1, c1), ok(c1)} I ∗ = I ∪ {ok1(c1), ok2(c1)}

1 If I is a stable (or supported) model of R, given constants

{c1, . . . , cj} and facts over extensional predicates, then I ∗ is a stable (or supported) model of the expansible instantiation of R for j ≥ 0

2 If I ′ is a stable (or supported) model of the expansible instantiation

  • f R for j ≥ 0, given facts over extensional predicates, then I ′ = I ∗ for

a stable (or supported) model I of R relative to constants {c1, . . . , cj}

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 10 / 13

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Conclusions

Outline

1 Motivation 2 Expanding Logic Programs 3 Conclusions

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 11 / 13

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Conclusions

Partner Units

Problem domain and instances from ASP Competition 2014 Expansible instantiation encoded via predicates providing substitutions clingo 4 control adding objects to be assigned or resources on demand Single-shot solving Multi-shot solving Instance #S ∅S #U ∅U #S ∅S #U ∅U 026 40 0.10 10 34.69 40 0.04 10 3.00 091 40 0.10 10 3.71 40 0.04 10 8.42 100 40 0.09 10 57.05 40 0.04 10 2.13 127 40 0.10 10 4.99 40 0.04 10 9.38 175 40 0.12 10 48.44 40 0.04 10 4.86 188 40 0.11 10 54.67 40 0.03 10 2.69 Multi-shot solving can significantly reduce #conflicts and runtime

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 12 / 13

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Conclusions

Partner Units

Problem domain and instances from ASP Competition 2014 Expansible instantiation encoded via predicates providing substitutions clingo 4 control adding objects to be assigned or resources on demand Single-shot solving Multi-shot solving Instance #S ∅S #U ∅U #S ∅S #U ∅U 026 40 0.10 10 34.69 40 0.04 10 3.00 091 40 0.10 10 3.71 40 0.04 10 8.42 100 40 0.09 10 57.05 40 0.04 10 2.13 127 40 0.10 10 4.99 40 0.04 10 9.38 175 40 0.12 10 48.44 40 0.04 10 4.86 188 40 0.11 10 54.67 40 0.03 10 2.69 Multi-shot solving can significantly reduce #conflicts and runtime

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 12 / 13

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Conclusions

Discussion

Expansible instantiation induces monotone constraints, enabling successive integration into reasoning process in multi-shot solving Translation approach provides scheme for introducing expansion atoms through which later additions take care of non-monotonicity New substitutions or ground rules, respectively, must be distinguished to guarantee modular composition of ground program parts Future work includes automatic support for introducing expansion atoms by need in multi-shot solving with ASP systems like clingo 4

G, Janhunen, Jost, Kaminski, Schaub ASP Solving for Expanding Universes 13 / 13