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Strong Negation in Well-Founded and Partial Stable Semantics for Logic Programs Pedro Cabalar 1 Sergei Odintsov 2 David Pearce 3 1 University of Corunna (Spain) 2 Sobolev Institute of Mathematics (Novosibirsk, Russia) 3 Universidad Rey Juan Carlos

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SLIDE 1

Strong Negation in Well-Founded and Partial Stable Semantics for Logic Programs

Pedro Cabalar1 Sergei Odintsov2 David Pearce3

1University of Corunna (Spain) 2Sobolev Institute of Mathematics (Novosibirsk, Russia) 3Universidad Rey Juan Carlos (Spain)

IBERAMIA 2006, Ribeirão Preto

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 1 / 35

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SLIDE 2

Outline

1

Introduction Overview of Logic Programming semantics Logical foundations Partial Equilibrium Logic

2

Contributions Routley semantics and strong negation HT 2 with strong negation Partial Equilibrium Logic with strong negation

3

Conclusions

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 2 / 35

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SLIDE 3

Outline

1

Introduction Overview of Logic Programming semantics Logical foundations Partial Equilibrium Logic

2

Contributions Routley semantics and strong negation HT 2 with strong negation Partial Equilibrium Logic with strong negation

3

Conclusions

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 3 / 35

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SLIDE 4

Logic programming: semantics for default negation

Semantics for default negation Stable Models [Gelfond & Lifschitz 88] Partial Stable Models [Przymusinski 91] Well-Founded semantics (WFS) [van Gelder, Ross & Schlipf 91]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 4 / 35

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SLIDE 5

Logic programming: semantics for default negation

Semantics for default negation Stable Models [Gelfond & Lifschitz 88] Partial Stable Models [Przymusinski 91] Well-Founded semantics (WFS) [van Gelder, Ross & Schlipf 91]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 4 / 35

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SLIDE 6

Logic programming: semantics for default negation

Semantics for default negation Stable Models [Gelfond & Lifschitz 88] Partial Stable Models [Przymusinski 91] Well-Founded semantics (WFS) [van Gelder, Ross & Schlipf 91]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 4 / 35

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SLIDE 7

Logic programming: semantics for default negation

LP definitions rely on: syntax transformations (reduct) + fixpoint constructions Stable models [Gelfond & Lifschitz 88] M stable model iff classical minimal model of ΠM Example: We guess some M say M = {q, r} to interprete ¬α’s p ← r ∧ ¬q q ← r ∧ ¬p r ← ¬p

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 5 / 35

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SLIDE 8

Logic programming: semantics for default negation

LP definitions rely on: syntax transformations (reduct) + fixpoint constructions Stable models [Gelfond & Lifschitz 88] M stable model iff classical minimal model of ΠM Example: We guess some M say M = {q, r} to interprete ¬α’s p ← r ∧ ¬q q ← r ∧ ¬p r ← ¬p

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 5 / 35

slide-9
SLIDE 9

Logic programming: semantics for default negation

LP definitions rely on: syntax transformations (reduct) + fixpoint constructions Stable models [Gelfond & Lifschitz 88] M stable model iff classical minimal model of ΠM Example: We guess some M say M = {q, r} to interprete ¬α’s p ← r ∧ ¬q q ← r ∧ ¬p r ← ¬p

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 5 / 35

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SLIDE 10

Logic programming: semantics for default negation

LP definitions rely on: syntax transformations (reduct) + fixpoint constructions Stable models [Gelfond & Lifschitz 88] M stable model iff classical minimal model of ΠM Example: We guess some M say M = {q, r} to interprete ¬α’s p ← r ∧ ⊥ q ← r ∧ ⊤ r ← ⊤

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 5 / 35

slide-11
SLIDE 11

Logic programming: semantics for default negation

LP definitions rely on: syntax transformations (reduct) + fixpoint constructions Stable models [Gelfond & Lifschitz 88] M stable model iff classical minimal model of ΠM Example: We guess some M say M = {q, r} to interprete ¬α’s q ← r r

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 5 / 35

slide-12
SLIDE 12

Logic programming: semantics for default negation

LP definitions rely on: syntax transformations (reduct) + fixpoint constructions Stable models [Gelfond & Lifschitz 88] M stable model iff classical minimal model of ΠM Example: We guess some M say M = {q, r} to interprete ¬α’s q ← r r Minimal model {q, r} = M stable model !

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 5 / 35

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SLIDE 13

Logic programming: semantics for default negation

Partial stable models [Przymusinski 91] M partial stable model iff 3-valued minimal-truth model of ΠM Again similar idea: reduct + fixpoint condition Note that interpretations are now 3-valued. Well-founded model = partial stable model with minimal info. (defined atoms) Example: p ← ¬p has no stable model. It has one partial stable model leaving p undefined.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 6 / 35

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SLIDE 14

Logic programming: semantics for default negation

Partial stable models [Przymusinski 91] M partial stable model iff 3-valued minimal-truth model of ΠM Again similar idea: reduct + fixpoint condition Note that interpretations are now 3-valued. Well-founded model = partial stable model with minimal info. (defined atoms) Example: p ← ¬p has no stable model. It has one partial stable model leaving p undefined.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 6 / 35

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SLIDE 15

Logic programming: semantics for default negation

Partial stable models [Przymusinski 91] M partial stable model iff 3-valued minimal-truth model of ΠM Again similar idea: reduct + fixpoint condition Note that interpretations are now 3-valued. Well-founded model = partial stable model with minimal info. (defined atoms) Example: p ← ¬p has no stable model. It has one partial stable model leaving p undefined.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 6 / 35

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SLIDE 16

Logic programming: semantics for default negation

Partial stable models [Przymusinski 91] M partial stable model iff 3-valued minimal-truth model of ΠM Again similar idea: reduct + fixpoint condition Note that interpretations are now 3-valued. Well-founded model = partial stable model with minimal info. (defined atoms) Example: p ← ¬p has no stable model. It has one partial stable model leaving p undefined.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 6 / 35

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SLIDE 17

Logic programming: semantics for default negation

Partial stable models [Przymusinski 91] M partial stable model iff 3-valued minimal-truth model of ΠM Again similar idea: reduct + fixpoint condition Note that interpretations are now 3-valued. Well-founded model = partial stable model with minimal info. (defined atoms) Example: p ← ¬p has no stable model. It has one partial stable model leaving p undefined.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 6 / 35

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SLIDE 18

A second negation

Default negation ¬p means no evidence on p What if we want to represent p is false (∼p)? Semantics for default negation Second negation Stable Models Answer sets [Gelfond & Lifschitz 88] [Gelfond & Lifschitz 91] Partial Stable Models with classical negation [Przymusinski 91] [Przymusinski 91] with strong negation [Alferes & Pereira 92] with explicit negation (WFSX) [Alferes & Pereira 92] Well-Founded semantics (WFS) In all cases, WF model [van Gelder, Ross & Schlipf 91] is the minimal info. part. s. model

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 7 / 35

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SLIDE 19

A second negation

Default negation ¬p means no evidence on p What if we want to represent p is false (∼p)? Semantics for default negation Second negation Stable Models Answer sets [Gelfond & Lifschitz 88] [Gelfond & Lifschitz 91] Partial Stable Models with classical negation [Przymusinski 91] [Przymusinski 91] with strong negation [Alferes & Pereira 92] with explicit negation (WFSX) [Alferes & Pereira 92] Well-Founded semantics (WFS) In all cases, WF model [van Gelder, Ross & Schlipf 91] is the minimal info. part. s. model

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 7 / 35

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SLIDE 20

A second negation

Default negation ¬p means no evidence on p What if we want to represent p is false (∼p)? Semantics for default negation Second negation Stable Models Answer sets [Gelfond & Lifschitz 88] [Gelfond & Lifschitz 91] Partial Stable Models with classical negation [Przymusinski 91] [Przymusinski 91] with strong negation [Alferes & Pereira 92] with explicit negation (WFSX) [Alferes & Pereira 92] Well-Founded semantics (WFS) In all cases, WF model [van Gelder, Ross & Schlipf 91] is the minimal info. part. s. model

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 7 / 35

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SLIDE 21

A second negation

Default negation ¬p means no evidence on p What if we want to represent p is false (∼p)? Semantics for default negation Second negation Stable Models Answer sets [Gelfond & Lifschitz 88] [Gelfond & Lifschitz 91] Partial Stable Models with classical negation [Przymusinski 91] [Przymusinski 91] with strong negation [Alferes & Pereira 92] with explicit negation (WFSX) [Alferes & Pereira 92] Well-Founded semantics (WFS) In all cases, WF model [van Gelder, Ross & Schlipf 91] is the minimal info. part. s. model

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 7 / 35

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SLIDE 22

A second negation

Default negation ¬p means no evidence on p What if we want to represent p is false (∼p)? Semantics for default negation Second negation Stable Models Answer sets [Gelfond & Lifschitz 88] [Gelfond & Lifschitz 91] Partial Stable Models with classical negation [Przymusinski 91] [Przymusinski 91] with strong negation [Alferes & Pereira 92] with explicit negation (WFSX) [Alferes & Pereira 92] Well-Founded semantics (WFS) In all cases, WF model [van Gelder, Ross & Schlipf 91] is the minimal info. part. s. model

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 7 / 35

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SLIDE 23

Outline

1

Introduction Overview of Logic Programming semantics Logical foundations Partial Equilibrium Logic

2

Contributions Routley semantics and strong negation HT 2 with strong negation Partial Equilibrium Logic with strong negation

3

Conclusions

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 8 / 35

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SLIDE 24

Fixing logical foundations of LP

Reduct: not exactly a semantic definition. Syntax is restricted: no arbitrary formulas. Our goal: look for a logical style definition. Get minimal models inside some (monotonic) logic. Advantages:

◮ Logically equivalent programs ⇒ same minimal models. ◮ Full logical interpretation of connectives. ◮ “Import” logical stuff (inference, tableaux, model checking, . . . ) P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 9 / 35

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SLIDE 25

Fixing logical foundations of LP

Reduct: not exactly a semantic definition. Syntax is restricted: no arbitrary formulas. Our goal: look for a logical style definition. Get minimal models inside some (monotonic) logic. Advantages:

◮ Logically equivalent programs ⇒ same minimal models. ◮ Full logical interpretation of connectives. ◮ “Import” logical stuff (inference, tableaux, model checking, . . . ) P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 9 / 35

slide-26
SLIDE 26

Fixing logical foundations of LP

Reduct: not exactly a semantic definition. Syntax is restricted: no arbitrary formulas. Our goal: look for a logical style definition. Get minimal models inside some (monotonic) logic. Advantages:

◮ Logically equivalent programs ⇒ same minimal models. ◮ Full logical interpretation of connectives. ◮ “Import” logical stuff (inference, tableaux, model checking, . . . ) P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 9 / 35

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SLIDE 27

Known logical foundations

Stable Models Partial Stable Models Monotonic Here-and-There HT 2 (HT) [Heyting 30] [Cabalar 01] Nonmonotonic Equilibrium Logic Partial Equil. Logic (PEL) (min. models) [Pearce 96] [Cabalar,Odintsov&Pearce 06] What about the second negation? Answer sets Partial Stable Models Monotonic N5 = HT + strong neg. HT 3 [Nelson 45] No axioms. [Vorob’ev 52] We study PEL+strong neg. Nonmonotonic Equilibrium Logic WFSXp (min. models) [Pearce 96] [Alcântara,Damásio&Pereira]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 10 / 35

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SLIDE 28

Known logical foundations

Stable Models Partial Stable Models Monotonic Here-and-There HT 2 (HT) [Heyting 30] [Cabalar 01] Nonmonotonic Equilibrium Logic Partial Equil. Logic (PEL) (min. models) [Pearce 96] [Cabalar,Odintsov&Pearce 06] What about the second negation? Answer sets Partial Stable Models Monotonic N5 = HT + strong neg. HT 3 [Nelson 45] No axioms. [Vorob’ev 52] We study PEL+strong neg. Nonmonotonic Equilibrium Logic WFSXp (min. models) [Pearce 96] [Alcântara,Damásio&Pereira]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 10 / 35

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SLIDE 29

Known logical foundations

Stable Models Partial Stable Models Monotonic Here-and-There HT 2 (HT) [Heyting 30] [Cabalar 01] Nonmonotonic Equilibrium Logic Partial Equil. Logic (PEL) (min. models) [Pearce 96] [Cabalar,Odintsov&Pearce 06] What about the second negation? Answer sets Partial Stable Models Monotonic N5 = HT + strong neg. HT 3 [Nelson 45] No axioms. [Vorob’ev 52] We study PEL+strong neg. Nonmonotonic Equilibrium Logic WFSXp (min. models) [Pearce 96] [Alcântara,Damásio&Pereira]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 10 / 35

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SLIDE 30

Known logical foundations

Stable Models Partial Stable Models Monotonic Here-and-There HT 2 (HT) [Heyting 30] [Cabalar 01] Nonmonotonic Equilibrium Logic Partial Equil. Logic (PEL) (min. models) [Pearce 96] [Cabalar,Odintsov&Pearce 06] What about the second negation? Answer sets Partial Stable Models Monotonic N5 = HT + strong neg. HT 3 [Nelson 45] No axioms. [Vorob’ev 52] We study PEL+strong neg. Nonmonotonic Equilibrium Logic WFSXp (min. models) [Pearce 96] [Alcântara,Damásio&Pereira]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 10 / 35

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SLIDE 31

Known logical foundations

Stable Models Partial Stable Models Monotonic Here-and-There HT 2 (HT) [Heyting 30] [Cabalar 01] Nonmonotonic Equilibrium Logic Partial Equil. Logic (PEL) (min. models) [Pearce 96] [Cabalar,Odintsov&Pearce 06] What about the second negation? Answer sets Partial Stable Models Monotonic N5 = HT + strong neg. HT 3 [Nelson 45] No axioms. [Vorob’ev 52] We study PEL+strong neg. Nonmonotonic Equilibrium Logic WFSXp (min. models) [Pearce 96] [Alcântara,Damásio&Pereira]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 10 / 35

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SLIDE 32

Known logical foundations

Stable Models Partial Stable Models Monotonic Here-and-There HT 2 (HT) [Heyting 30] [Cabalar 01] Nonmonotonic Equilibrium Logic Partial Equil. Logic (PEL) (min. models) [Pearce 96] [Cabalar,Odintsov&Pearce 06] What about the second negation? Answer sets Partial Stable Models Monotonic N5 = HT + strong neg. HT 3 [Nelson 45] No axioms. [Vorob’ev 52] We study PEL+strong neg. Nonmonotonic Equilibrium Logic WFSXp (min. models) [Pearce 96] [Alcântara,Damásio&Pereira]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 10 / 35

slide-33
SLIDE 33

Known logical foundations

Stable Models Partial Stable Models Monotonic Here-and-There HT 2 (HT) [Heyting 30] [Cabalar 01] Nonmonotonic Equilibrium Logic Partial Equil. Logic (PEL) (min. models) [Pearce 96] [Cabalar,Odintsov&Pearce 06] What about the second negation? Answer sets Partial Stable Models Monotonic N5 = HT + strong neg. HT 3 [Nelson 45] No axioms. [Vorob’ev 52] We study PEL+strong neg. Nonmonotonic Equilibrium Logic WFSXp (min. models) [Pearce 96] [Alcântara,Damásio&Pereira]

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 10 / 35

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SLIDE 34

Stable models and Equilibrium Logic

(Monotonic) intermediate logic of here-and-there (HT) Intuitionistic ⊆ HT ⊆ Classical h

  • t
  • Pearce’s Equilibrium Logic: minimal HT models

Intuition: t world is fixed (plays the role of “reduct”), h world is minimized Interesting results:

◮ Equilibrium models = stable models [Pearce 97] ◮ HT captures strong equivalence [Lifschitz, Pearce & Valverde 01]

(we’ll see later. . . )

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 11 / 35

slide-35
SLIDE 35

Stable models and Equilibrium Logic

(Monotonic) intermediate logic of here-and-there (HT) Intuitionistic ⊆ HT ⊆ Classical h

  • t
  • Pearce’s Equilibrium Logic: minimal HT models

Intuition: t world is fixed (plays the role of “reduct”), h world is minimized Interesting results:

◮ Equilibrium models = stable models [Pearce 97] ◮ HT captures strong equivalence [Lifschitz, Pearce & Valverde 01]

(we’ll see later. . . )

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 11 / 35

slide-36
SLIDE 36

Stable models and Equilibrium Logic

(Monotonic) intermediate logic of here-and-there (HT) Intuitionistic ⊆ HT ⊆ Classical h

  • t
  • Pearce’s Equilibrium Logic: minimal HT models

Intuition: t world is fixed (plays the role of “reduct”), h world is minimized Interesting results:

◮ Equilibrium models = stable models [Pearce 97] ◮ HT captures strong equivalence [Lifschitz, Pearce & Valverde 01]

(we’ll see later. . . )

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 11 / 35

slide-37
SLIDE 37

Outline

1

Introduction Overview of Logic Programming semantics Logical foundations Partial Equilibrium Logic

2

Contributions Routley semantics and strong negation HT 2 with strong negation Partial Equilibrium Logic with strong negation

3

Conclusions

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 12 / 35

slide-38
SLIDE 38

Logical foundation of WFS: recently solved

[Cabalar,Odintsov & Pearce KR’06] Partial Equilibrium Logic

1

takes minimal models on monotonic logic HT 2

2

HT 2 classified inside [Došen 86] framework N combined with [Routley & Routley 72].

3

Main idea: each world h t founded ⊆ non-unfounded

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 13 / 35

slide-39
SLIDE 39

Logical foundation of WFS: recently solved

[Cabalar,Odintsov & Pearce KR’06] Partial Equilibrium Logic

1

takes minimal models on monotonic logic HT 2

2

HT 2 classified inside [Došen 86] framework N combined with [Routley & Routley 72].

3

Main idea: each world h t founded ⊆ non-unfounded

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 13 / 35

slide-40
SLIDE 40

Logical foundation of WFS: recently solved

[Cabalar,Odintsov & Pearce KR’06] Partial Equilibrium Logic

1

takes minimal models on monotonic logic HT 2

2

HT 2 classified inside [Došen 86] framework N combined with [Routley & Routley 72].

3

Main idea: each world h t founded ⊆ non-unfounded

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 13 / 35

slide-41
SLIDE 41

Logical foundation of WFS: recently solved

[Cabalar,Odintsov & Pearce KR’06] Partial Equilibrium Logic

1

takes minimal models on monotonic logic HT 2

2

HT 2 classified inside [Došen 86] framework N combined with [Routley & Routley 72].

3

Main idea: each world h t founded ⊆ non-unfounded

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 13 / 35

slide-42
SLIDE 42

Logical foundation of WFS: recently solved

[Cabalar,Odintsov & Pearce KR’06] Partial Equilibrium Logic

1

takes minimal models on monotonic logic HT 2

2

HT 2 classified inside [Došen 86] framework N combined with [Routley & Routley 72].

3

Main idea: each world h h′ has now a primed version t t′ founded ⊆ non-unfounded

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 13 / 35

slide-43
SLIDE 43

Logical foundation of WFS: recently solved

[Cabalar,Odintsov & Pearce KR’06] Partial Equilibrium Logic

1

takes minimal models on monotonic logic HT 2

2

HT 2 classified inside [Došen 86] framework N combined with [Routley & Routley 72].

3

Main idea: each world h h′ has now a primed version t t′ with the intended meaning founded ⊆ non-unfounded

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 13 / 35

slide-44
SLIDE 44

Logical foundation of WFS: recently solved

[Cabalar,Odintsov & Pearce KR’06] Partial Equilibrium Logic

1

takes minimal models on monotonic logic HT 2

2

HT 2 classified inside [Došen 86] framework N combined with [Routley & Routley 72].

3

Main idea: each world h h′ has now a primed version t t′ with the intended meaning founded ⊆ non-unfounded

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 13 / 35

slide-45
SLIDE 45

Semantics: HT 2 Frames

h

  • t
  • h′
  • t′
  • ≤ Accessibility relation like any intermediate logic

(w | = p and w ≤ w′) implies w′ | = p ≤ used for implication: w | = ϕ → ψ when ∀w′ ≥ w, w′ | = ϕ implies w′ | = ψ But negation ¬φ is no longer defined as φ → ⊥ h

  • t
  • h′
  • t′

∗ star function (from Routley semantics) satisfies: v ≤ w iff w∗ ≤ v∗ w | = ¬ϕ when w∗ | = ϕ

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 14 / 35

slide-46
SLIDE 46

Semantics: HT 2 Frames

h

  • t
  • h′
  • t′
  • ≤ Accessibility relation like any intermediate logic

(w | = p and w ≤ w′) implies w′ | = p ≤ used for implication: w | = ϕ → ψ when ∀w′ ≥ w, w′ | = ϕ implies w′ | = ψ But negation ¬φ is no longer defined as φ → ⊥ h

  • t
  • h′
  • t′

∗ star function (from Routley semantics) satisfies: v ≤ w iff w∗ ≤ v∗ w | = ¬ϕ when w∗ | = ϕ

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 14 / 35

slide-47
SLIDE 47

Semantics: HT 2 Frames

h

  • t
  • h′
  • t′
  • ≤ Accessibility relation like any intermediate logic

(w | = p and w ≤ w′) implies w′ | = p ≤ used for implication: w | = ϕ → ψ when ∀w′ ≥ w, w′ | = ϕ implies w′ | = ψ But negation ¬φ is no longer defined as φ → ⊥ h

  • t
  • h′
  • t′

∗ star function (from Routley semantics) satisfies: v ≤ w iff w∗ ≤ v∗ w | = ¬ϕ when w∗ | = ϕ

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 14 / 35

slide-48
SLIDE 48

Semantics: HT 2 Frames

h

  • t
  • h′
  • t′
  • ≤ Accessibility relation like any intermediate logic

(w | = p and w ≤ w′) implies w′ | = p ≤ used for implication: w | = ϕ → ψ when ∀w′ ≥ w, w′ | = ϕ implies w′ | = ψ But negation ¬φ is no longer defined as φ → ⊥ h

  • t
  • h′
  • t′

∗ star function (from Routley semantics) satisfies: v ≤ w iff w∗ ≤ v∗ w | = ¬ϕ when w∗ | = ϕ

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 14 / 35

slide-49
SLIDE 49

Semantics: HT 2 Frames

h

  • t
  • h′
  • t′
  • ≤ Accessibility relation like any intermediate logic

(w | = p and w ≤ w′) implies w′ | = p ≤ used for implication: w | = ϕ → ψ when ∀w′ ≥ w, w′ | = ϕ implies w′ | = ψ But negation ¬φ is no longer defined as φ → ⊥ h

  • t
  • h′
  • t′

∗ star function (from Routley semantics) satisfies: v ≤ w iff w∗ ≤ v∗ w | = ¬ϕ when w∗ | = ϕ

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 14 / 35

slide-50
SLIDE 50

Partial equilibrium models

Let H, H′, T, T ′ denote sets of atoms verified at h, h′, t, t′. A model can be seen as a pair H, T of 3-valued interp. where H = (H, H′) and T = (T, T ′). Define an ordering among models, H1, T1 H2, T2 if:

(i) T1 = T2 (this is fixed) (ii) H1 less truth than H2 (H1 ⊆ H2 and H′

1 ⊆ H′ 2).

H, T is said to be total if H = T.

Definition (Partial equilibrium model)

A model M of theory Π is a partial equilibrium (PE) model of Π if it is total and -minimal.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 15 / 35

slide-51
SLIDE 51

Partial equilibrium models

Let H, H′, T, T ′ denote sets of atoms verified at h, h′, t, t′. A model can be seen as a pair H, T of 3-valued interp. where H = (H, H′) and T = (T, T ′). Define an ordering among models, H1, T1 H2, T2 if:

(i) T1 = T2 (this is fixed) (ii) H1 less truth than H2 (H1 ⊆ H2 and H′

1 ⊆ H′ 2).

H, T is said to be total if H = T.

Definition (Partial equilibrium model)

A model M of theory Π is a partial equilibrium (PE) model of Π if it is total and -minimal.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 15 / 35

slide-52
SLIDE 52

Partial equilibrium models

Let H, H′, T, T ′ denote sets of atoms verified at h, h′, t, t′. A model can be seen as a pair H, T of 3-valued interp. where H = (H, H′) and T = (T, T ′). Define an ordering among models, H1, T1 H2, T2 if:

(i) T1 = T2 (this is fixed) (ii) H1 less truth than H2 (H1 ⊆ H2 and H′

1 ⊆ H′ 2).

H, T is said to be total if H = T.

Definition (Partial equilibrium model)

A model M of theory Π is a partial equilibrium (PE) model of Π if it is total and -minimal.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 15 / 35

slide-53
SLIDE 53

Partial equilibrium models

Let H, H′, T, T ′ denote sets of atoms verified at h, h′, t, t′. A model can be seen as a pair H, T of 3-valued interp. where H = (H, H′) and T = (T, T ′). Define an ordering among models, H1, T1 H2, T2 if:

(i) T1 = T2 (this is fixed) (ii) H1 less truth than H2 (H1 ⊆ H2 and H′

1 ⊆ H′ 2).

H, T is said to be total if H = T.

Definition (Partial equilibrium model)

A model M of theory Π is a partial equilibrium (PE) model of Π if it is total and -minimal.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 15 / 35

slide-54
SLIDE 54

Partial equilibrium models

Let H, H′, T, T ′ denote sets of atoms verified at h, h′, t, t′. A model can be seen as a pair H, T of 3-valued interp. where H = (H, H′) and T = (T, T ′). Define an ordering among models, H1, T1 H2, T2 if:

(i) T1 = T2 (this is fixed) (ii) H1 less truth than H2 (H1 ⊆ H2 and H′

1 ⊆ H′ 2).

H, T is said to be total if H = T.

Definition (Partial equilibrium model)

A model M of theory Π is a partial equilibrium (PE) model of Π if it is total and -minimal.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 15 / 35

slide-55
SLIDE 55

Partial equilibrium models

Let H, H′, T, T ′ denote sets of atoms verified at h, h′, t, t′. A model can be seen as a pair H, T of 3-valued interp. where H = (H, H′) and T = (T, T ′). Define an ordering among models, H1, T1 H2, T2 if:

(i) T1 = T2 (this is fixed) (ii) H1 less truth than H2 (H1 ⊆ H2 and H′

1 ⊆ H′ 2).

H, T is said to be total if H = T.

Definition (Partial equilibrium model)

A model M of theory Π is a partial equilibrium (PE) model of Π if it is total and -minimal.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 15 / 35

slide-56
SLIDE 56

Some properties of PEL

Theorem (Corresp. to Partial Stable Models)

For a normal or disjunctive logic program Π, T, T is a partial equilibrium model of Π iff T is a partial stable model of Π.

Definition (strong equivalence)

Two theories Π1, Π2 are said to be strongly equivalent if for any set of formulas Γ, Π1 ∪ Γ and Π2 ∪ Γ have the same partial stable models.

Theorem (from KR’06 paper)

Π1, Π2 are PEL strongly equivalent iff they are equivalent in HT 2. The same holds for Well-Founded (WF) model(s), understood as those partial stable models with minimal information.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 16 / 35

slide-57
SLIDE 57

Some properties of PEL

Theorem (Corresp. to Partial Stable Models)

For a normal or disjunctive logic program Π, T, T is a partial equilibrium model of Π iff T is a partial stable model of Π.

Definition (strong equivalence)

Two theories Π1, Π2 are said to be strongly equivalent if for any set of formulas Γ, Π1 ∪ Γ and Π2 ∪ Γ have the same partial stable models.

Theorem (from KR’06 paper)

Π1, Π2 are PEL strongly equivalent iff they are equivalent in HT 2. The same holds for Well-Founded (WF) model(s), understood as those partial stable models with minimal information.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 16 / 35

slide-58
SLIDE 58

Some properties of PEL

Theorem (Corresp. to Partial Stable Models)

For a normal or disjunctive logic program Π, T, T is a partial equilibrium model of Π iff T is a partial stable model of Π.

Definition (strong equivalence)

Two theories Π1, Π2 are said to be strongly equivalent if for any set of formulas Γ, Π1 ∪ Γ and Π2 ∪ Γ have the same partial stable models.

Theorem (from KR’06 paper)

Π1, Π2 are PEL strongly equivalent iff they are equivalent in HT 2. The same holds for Well-Founded (WF) model(s), understood as those partial stable models with minimal information.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 16 / 35

slide-59
SLIDE 59

Some properties of PEL

Theorem (Corresp. to Partial Stable Models)

For a normal or disjunctive logic program Π, T, T is a partial equilibrium model of Π iff T is a partial stable model of Π.

Definition (strong equivalence)

Two theories Π1, Π2 are said to be strongly equivalent if for any set of formulas Γ, Π1 ∪ Γ and Π2 ∪ Γ have the same partial stable models.

Theorem (from KR’06 paper)

Π1, Π2 are PEL strongly equivalent iff they are equivalent in HT 2. The same holds for Well-Founded (WF) model(s), understood as those partial stable models with minimal information.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 16 / 35

slide-60
SLIDE 60

Outline

1

Introduction Overview of Logic Programming semantics Logical foundations Partial Equilibrium Logic

2

Contributions Routley semantics and strong negation HT 2 with strong negation Partial Equilibrium Logic with strong negation

3

Conclusions

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 17 / 35

slide-61
SLIDE 61

N∗∼

HT 2 special case of N∗ family = intuitionistic Kripke frames with a weaker negation [Routley & Routley 72]. We define next N∗∼, adding strong negation ∼, as follows. Syntax: atoms, ∧, ∨, →, ¬ (weak negation) and ∼ (strong negation) Inference rules: modus ponens, plus (RC) α → β ¬β → ¬α

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 18 / 35

slide-62
SLIDE 62

N∗∼

HT 2 special case of N∗ family = intuitionistic Kripke frames with a weaker negation [Routley & Routley 72]. We define next N∗∼, adding strong negation ∼, as follows. Syntax: atoms, ∧, ∨, →, ¬ (weak negation) and ∼ (strong negation) Inference rules: modus ponens, plus (RC) α → β ¬β → ¬α

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 18 / 35

slide-63
SLIDE 63

N∗∼

HT 2 special case of N∗ family = intuitionistic Kripke frames with a weaker negation [Routley & Routley 72]. We define next N∗∼, adding strong negation ∼, as follows. Syntax: atoms, ∧, ∨, →, ¬ (weak negation) and ∼ (strong negation) Inference rules: modus ponens, plus (RC) α → β ¬β → ¬α

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 18 / 35

slide-64
SLIDE 64

N∗∼ axioms

1

the axiom schemes of positive logic,

2

weak negation axioms:

  • W1. ¬α ∧ ¬β → ¬(α ∨ β)
  • W2. ¬(α ∧ β) → ¬α ∨ ¬β
  • W3. ¬(α → α) → β

Until now, N∗

3

and for N∗∼, we add the schemata for strong negation from [Vorob’ev 52]:

  • N1. ∼ (α → β) ↔ α ∧ ∼β
  • N2. ∼(α ∧ β) ↔ ∼α∨ ∼ β
  • N3. ∼(α ∨ β) ↔ ∼α ∧ ∼β
  • N4. ∼ ∼α ↔ α
  • N5. ∼¬α ↔ α

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 19 / 35

slide-65
SLIDE 65

N∗∼ models

Definition (N∗∼ frame)

is a triple W, ≤, ∗, where:

1

W is a set of worlds

2

≤ a partial order on W

3

∗ : W − → W such that x ≤ y iff y∗ ≤ x∗.

Definition (N∗∼ model)

is an N∗∼ frame W, ≤, ∗, V +, V − plus two valuations V +, V − : At × W − → {0, 1} such that: V +(−)(p, u) = 1 & u ≤ w ⇒ V +(−)(p, w) = 1

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 20 / 35

slide-66
SLIDE 66

N∗∼ models

Definition (N∗∼ frame)

is a triple W, ≤, ∗, where:

1

W is a set of worlds

2

≤ a partial order on W

3

∗ : W − → W such that x ≤ y iff y∗ ≤ x∗.

Definition (N∗∼ model)

is an N∗∼ frame W, ≤, ∗, V +, V − plus two valuations V +, V − : At × W − → {0, 1} such that: V +(−)(p, u) = 1 & u ≤ w ⇒ V +(−)(p, w) = 1

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 20 / 35

slide-67
SLIDE 67

N∗∼ models

Definition (N∗∼ frame)

is a triple W, ≤, ∗, where:

1

W is a set of worlds

2

≤ a partial order on W

3

∗ : W − → W such that x ≤ y iff y∗ ≤ x∗.

Definition (N∗∼ model)

is an N∗∼ frame W, ≤, ∗, V +, V − plus two valuations V +, V − : At × W − → {0, 1} such that: V +(−)(p, u) = 1 & u ≤ w ⇒ V +(−)(p, w) = 1

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 20 / 35

slide-68
SLIDE 68

N∗∼ valuation

V +, V − are extended to arbitrary formulas as follows: V +(ϕ ∧ ψ, w) = 1 iff V +(ϕ, w) = V +(ψ, w) = 1 V +(ϕ ∨ ψ, w) = 1 iff V +(ϕ, w) = 1 or V +(ψ, w) = 1 V +(ϕ → ψ, w) = 1 iff for every w′ such that w ≤ w′, V +(ϕ, w′) = 1 ⇒ V +(ψ, w′) = 1 V +(¬ϕ, w) = 1 iff V +(ϕ, w∗) = 0 V +(∼ ϕ, w) = 1 iff V −(ϕ, w) = 1 V −(ϕ ∧ ψ, w) = 1 iff V −(ϕ, w) = 1 or V −(ψ, w) = 1 V −(ϕ ∨ ψ, w) = 1 iff V −(ϕ, w) = V −(ψ, w) = 1 V −(ϕ → ψ, w) = 1 iff V +(ϕ, w) = 1 and V −(ψ, w) = 1 V −(¬ϕ, w) = 1 iff V +(ϕ, w) = 1 V −(∼ ϕ, w) = 1 iff V +(ϕ, w) = 1

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 21 / 35

slide-69
SLIDE 69

N∗∼ properties

Axiom (W3) allows defining an intuitionistic negation ⊥ := ¬(p0 → p0) and − α := α → ⊥

Proposition

The ∨, ∧, →, −-fragment of N∗∼ coincides with intuitionistic logic.

Proposition

N∗∼ is a conservative extension of N∗ and of Nelson’s paraconsistent logic N−.

Proposition

For each formula φ there exists some N∗∼-equivalent formula ψ in negation normal form (∼ only applied to atoms).

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 22 / 35

slide-70
SLIDE 70

N∗∼ properties

Axiom (W3) allows defining an intuitionistic negation ⊥ := ¬(p0 → p0) and − α := α → ⊥

Proposition

The ∨, ∧, →, −-fragment of N∗∼ coincides with intuitionistic logic.

Proposition

N∗∼ is a conservative extension of N∗ and of Nelson’s paraconsistent logic N−.

Proposition

For each formula φ there exists some N∗∼-equivalent formula ψ in negation normal form (∼ only applied to atoms).

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 22 / 35

slide-71
SLIDE 71

N∗∼ properties

Axiom (W3) allows defining an intuitionistic negation ⊥ := ¬(p0 → p0) and − α := α → ⊥

Proposition

The ∨, ∧, →, −-fragment of N∗∼ coincides with intuitionistic logic.

Proposition

N∗∼ is a conservative extension of N∗ and of Nelson’s paraconsistent logic N−.

Proposition

For each formula φ there exists some N∗∼-equivalent formula ψ in negation normal form (∼ only applied to atoms).

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 22 / 35

slide-72
SLIDE 72

N∗∼ properties

Axiom (W3) allows defining an intuitionistic negation ⊥ := ¬(p0 → p0) and − α := α → ⊥

Proposition

The ∨, ∧, →, −-fragment of N∗∼ coincides with intuitionistic logic.

Proposition

N∗∼ is a conservative extension of N∗ and of Nelson’s paraconsistent logic N−.

Proposition

For each formula φ there exists some N∗∼-equivalent formula ψ in negation normal form (∼ only applied to atoms).

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 22 / 35

slide-73
SLIDE 73

N∗∼ properties

Theorem (Vorob’ev reduction)

For each formula φ, let φ′ be the result of:

1

  • btaining its negation normal form and

2

replacing each ∼ p by a new atom p′. Then: N∗∼ ⊢ φ iff N∗∼ ⊢ φ′.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 23 / 35

slide-74
SLIDE 74

Outline

1

Introduction Overview of Logic Programming semantics Logical foundations Partial Equilibrium Logic

2

Contributions Routley semantics and strong negation HT 2 with strong negation Partial Equilibrium Logic with strong negation

3

Conclusions

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 24 / 35

slide-75
SLIDE 75

HT 2 with strong negation

HT 2 = N∗ + Ax where Ax are more axioms for weak negation Nothing new is required: HT 2∼ = N∗∼ + Ax The (common) set of axioms Ax is the following: W4. −α ∨ − − α

  • W5. −α ∨ (α → (β ∨ (β → (γ ∨ −γ))))

W6. 2

i=0((αi → j=i αj) → j=i αj) → 2 i=0 αi

W7. α → ¬¬α W8. α ∧ ¬α → ¬β ∨ ¬¬β W9. ¬α ∧ ¬(α → β) → ¬¬α W10. ¬¬α ∨ ¬¬β ∨ ¬(α → β) ∨ ¬¬(α → β) W11. ¬¬α ∧ ¬¬β → (α → β) ∨ (β → α) plus the rule (EC) α∨(β∧¬β)

α

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 25 / 35

slide-76
SLIDE 76

HT 2 with strong negation

HT 2 = N∗ + Ax where Ax are more axioms for weak negation Nothing new is required: HT 2∼ = N∗∼ + Ax The (common) set of axioms Ax is the following: W4. −α ∨ − − α

  • W5. −α ∨ (α → (β ∨ (β → (γ ∨ −γ))))

W6. 2

i=0((αi → j=i αj) → j=i αj) → 2 i=0 αi

W7. α → ¬¬α W8. α ∧ ¬α → ¬β ∨ ¬¬β W9. ¬α ∧ ¬(α → β) → ¬¬α W10. ¬¬α ∨ ¬¬β ∨ ¬(α → β) ∨ ¬¬(α → β) W11. ¬¬α ∧ ¬¬β → (α → β) ∨ (β → α) plus the rule (EC) α∨(β∧¬β)

α

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 25 / 35

slide-77
SLIDE 77

HT 2 with strong negation

HT 2 = N∗ + Ax where Ax are more axioms for weak negation Nothing new is required: HT 2∼ = N∗∼ + Ax The (common) set of axioms Ax is the following: W4. −α ∨ − − α

  • W5. −α ∨ (α → (β ∨ (β → (γ ∨ −γ))))

W6. 2

i=0((αi → j=i αj) → j=i αj) → 2 i=0 αi

W7. α → ¬¬α W8. α ∧ ¬α → ¬β ∨ ¬¬β W9. ¬α ∧ ¬(α → β) → ¬¬α W10. ¬¬α ∨ ¬¬β ∨ ¬(α → β) ∨ ¬¬(α → β) W11. ¬¬α ∧ ¬¬β → (α → β) ∨ (β → α) plus the rule (EC) α∨(β∧¬β)

α

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 25 / 35

slide-78
SLIDE 78

HT 2 with strong negation

HT 2∼ = HT 2 + {N1, . . . , N5}. HT 2∼ frames coincide with HT 2 ones seen before: h

  • t
  • h′
  • t′
  • h
  • t
  • h′
  • t′

relation ≤ ∗ function Note: we allow paraconsistency: p and ∼ p can be both founded.

Proposition

Vorob’ev reduction also holds for HT 2∼.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 26 / 35

slide-79
SLIDE 79

HT 2 with strong negation

HT 2∼ = HT 2 + {N1, . . . , N5}. HT 2∼ frames coincide with HT 2 ones seen before: h

  • t
  • h′
  • t′
  • h
  • t
  • h′
  • t′

relation ≤ ∗ function Note: we allow paraconsistency: p and ∼ p can be both founded.

Proposition

Vorob’ev reduction also holds for HT 2∼.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 26 / 35

slide-80
SLIDE 80

HT 2 with strong negation

HT 2∼ = HT 2 + {N1, . . . , N5}. HT 2∼ frames coincide with HT 2 ones seen before: h

  • t
  • h′
  • t′
  • h
  • t
  • h′
  • t′

relation ≤ ∗ function Note: we allow paraconsistency: p and ∼ p can be both founded.

Proposition

Vorob’ev reduction also holds for HT 2∼.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 26 / 35

slide-81
SLIDE 81

HT 2 with strong negation

HT 2∼ = HT 2 + {N1, . . . , N5}. HT 2∼ frames coincide with HT 2 ones seen before: h

  • t
  • h′
  • t′
  • h
  • t
  • h′
  • t′

relation ≤ ∗ function Note: we allow paraconsistency: p and ∼ p can be both founded.

Proposition

Vorob’ev reduction also holds for HT 2∼.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 26 / 35

slide-82
SLIDE 82

HT 2 with strong negation

We extend HT 2 with a new truth constant u (undefinedness).

Definition (HT 2

u )

V(u, h) = V(u, t) = 0 and V(u, h′) = V(u, t′) = 1. That is, always undefined.

Theorem

HT 2

u = HT 2 + {u ↔ ¬u}

The same extension can be done on HT 2∼.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 27 / 35

slide-83
SLIDE 83

HT 2 with strong negation

We extend HT 2 with a new truth constant u (undefinedness).

Definition (HT 2

u )

V(u, h) = V(u, t) = 0 and V(u, h′) = V(u, t′) = 1. That is, always undefined.

Theorem

HT 2

u = HT 2 + {u ↔ ¬u}

The same extension can be done on HT 2∼.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 27 / 35

slide-84
SLIDE 84

HT 2 with strong negation

We extend HT 2 with a new truth constant u (undefinedness).

Definition (HT 2

u )

V(u, h) = V(u, t) = 0 and V(u, h′) = V(u, t′) = 1. That is, always undefined.

Theorem

HT 2

u = HT 2 + {u ↔ ¬u}

The same extension can be done on HT 2∼.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 27 / 35

slide-85
SLIDE 85

HT 2 with strong negation

Other useful logics: Semi-consistency: HT 2

sc := HT 2∼ u

+ {p∧ ∼ p → u yields the effect:

p, ∼ p can be both non-unfounded, but not both founded.

Coherence: HT 2

coh := HT 2∼ u

+ {p → ¬ ∼ p ∨ u, ∼ p → ¬p ∨ u} yields the effect:

p founded ⇒ ∼ p unfounded ∼ p founded ⇒ p unfounded

Proposition

HT 2

coh (coherence) is stronger than HT 2 sc (semi-consistency).

Vorob’ev reductions for these variants: just apply translation to axiom schemata too.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 28 / 35

slide-86
SLIDE 86

HT 2 with strong negation

Other useful logics: Semi-consistency: HT 2

sc := HT 2∼ u

+ {p∧ ∼ p → u yields the effect:

p, ∼ p can be both non-unfounded, but not both founded.

Coherence: HT 2

coh := HT 2∼ u

+ {p → ¬ ∼ p ∨ u, ∼ p → ¬p ∨ u} yields the effect:

p founded ⇒ ∼ p unfounded ∼ p founded ⇒ p unfounded

Proposition

HT 2

coh (coherence) is stronger than HT 2 sc (semi-consistency).

Vorob’ev reductions for these variants: just apply translation to axiom schemata too.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 28 / 35

slide-87
SLIDE 87

HT 2 with strong negation

Other useful logics: Semi-consistency: HT 2

sc := HT 2∼ u

+ {p∧ ∼ p → u yields the effect:

p, ∼ p can be both non-unfounded, but not both founded.

Coherence: HT 2

coh := HT 2∼ u

+ {p → ¬ ∼ p ∨ u, ∼ p → ¬p ∨ u} yields the effect:

p founded ⇒ ∼ p unfounded ∼ p founded ⇒ p unfounded

Proposition

HT 2

coh (coherence) is stronger than HT 2 sc (semi-consistency).

Vorob’ev reductions for these variants: just apply translation to axiom schemata too.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 28 / 35

slide-88
SLIDE 88

HT 2 with strong negation

Other useful logics: Semi-consistency: HT 2

sc := HT 2∼ u

+ {p∧ ∼ p → u yields the effect:

p, ∼ p can be both non-unfounded, but not both founded.

Coherence: HT 2

coh := HT 2∼ u

+ {p → ¬ ∼ p ∨ u, ∼ p → ¬p ∨ u} yields the effect:

p founded ⇒ ∼ p unfounded ∼ p founded ⇒ p unfounded

Proposition

HT 2

coh (coherence) is stronger than HT 2 sc (semi-consistency).

Vorob’ev reductions for these variants: just apply translation to axiom schemata too.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 28 / 35

slide-89
SLIDE 89

Outline

1

Introduction Overview of Logic Programming semantics Logical foundations Partial Equilibrium Logic

2

Contributions Routley semantics and strong negation HT 2 with strong negation Partial Equilibrium Logic with strong negation

3

Conclusions

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 29 / 35

slide-90
SLIDE 90

PEL with strong negation

H, H′, T, T ′ are now sets of literals (p, ∼ p). PEL definitions remain unchanged: PE model = total and -minimal. Well-founded model = PE model with minimal info. We can get PE models for any strong neg. version of HT 2: HT 2∼

u , HT 2 sc, HT 2 coh.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 30 / 35

slide-91
SLIDE 91

PEL with strong negation

H, H′, T, T ′ are now sets of literals (p, ∼ p). PEL definitions remain unchanged: PE model = total and -minimal. Well-founded model = PE model with minimal info. We can get PE models for any strong neg. version of HT 2: HT 2∼

u , HT 2 sc, HT 2 coh.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 30 / 35

slide-92
SLIDE 92

PEL with strong negation

Theorem (Strong equivalence)

Let Γ1, Γ2 be sets of N∗∼ formulas. Γ1, Γ2 are strongly equivalent (wrt each version of PEL models) iff Γ1, Γ2 equivalent in the corresp. monotonic logic HT 2∼

u , HT 2 sc, HT 2 coh.

Proposition

For all PEL variants with strong neg., complexity of reasoning tasks is the same class as that of ordinary PEL (in particular, decision problem is ΠP

2 -hard).

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 31 / 35

slide-93
SLIDE 93

PEL with strong negation

Theorem (Strong equivalence)

Let Γ1, Γ2 be sets of N∗∼ formulas. Γ1, Γ2 are strongly equivalent (wrt each version of PEL models) iff Γ1, Γ2 equivalent in the corresp. monotonic logic HT 2∼

u , HT 2 sc, HT 2 coh.

Proposition

For all PEL variants with strong neg., complexity of reasoning tasks is the same class as that of ordinary PEL (in particular, decision problem is ΠP

2 -hard).

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 31 / 35

slide-94
SLIDE 94

Correspondence theorems

An extended logic program Π is a set of rules r: Hd(r) ← B(r) where Hd(r) is a literal (p, ∼ p) and B(r) a conjunction of expressions like L or ¬L (L=literal).

Theorem

T, T is an HT 2

sc PE model of an extended program Π iff T is a

classical-negation [Przymusinski 91] part. stable model of Π.

Theorem

T, T is an HT 2

coh PE model of an extended program Π iff T is a

strong-negation [Alferes & Pereira 92] part. stable model of Π.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 32 / 35

slide-95
SLIDE 95

Correspondence theorems

An extended logic program Π is a set of rules r: Hd(r) ← B(r) where Hd(r) is a literal (p, ∼ p) and B(r) a conjunction of expressions like L or ¬L (L=literal).

Theorem

T, T is an HT 2

sc PE model of an extended program Π iff T is a

classical-negation [Przymusinski 91] part. stable model of Π.

Theorem

T, T is an HT 2

coh PE model of an extended program Π iff T is a

strong-negation [Alferes & Pereira 92] part. stable model of Π.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 32 / 35

slide-96
SLIDE 96

Correspondence theorems

An extended logic program Π is a set of rules r: Hd(r) ← B(r) where Hd(r) is a literal (p, ∼ p) and B(r) a conjunction of expressions like L or ¬L (L=literal).

Theorem

T, T is an HT 2

sc PE model of an extended program Π iff T is a

classical-negation [Przymusinski 91] part. stable model of Π.

Theorem

T, T is an HT 2

coh PE model of an extended program Π iff T is a

strong-negation [Alferes & Pereira 92] part. stable model of Π.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 32 / 35

slide-97
SLIDE 97

Correspondence theorems

Given an extended program Π we define Π′ by replacing each rule r by: Hd(r) ← B(r) ∧ u ∧ ¬ ∼ Hd(r) Hd(r) ∨ u ← B(r)

Theorem

A pair T = (T, T ′) is a WFSX part. stable model [Alferes & Pereira 92]

  • f an extended logic program Π

iff T, T is an HT 2

sc PE model of Π′.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 33 / 35

slide-98
SLIDE 98

Correspondence theorems

Given an extended program Π we define Π′ by replacing each rule r by: Hd(r) ← B(r) ∧ u ∧ ¬ ∼ Hd(r) Hd(r) ∨ u ← B(r)

Theorem

A pair T = (T, T ′) is a WFSX part. stable model [Alferes & Pereira 92]

  • f an extended logic program Π

iff T, T is an HT 2

sc PE model of Π′.

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 33 / 35

slide-99
SLIDE 99

Conclusions

PEL is a natural logical foundation for partial stable models. Strong negation added preserving complexity and strong equivalence results. We provided a Routley-style general family N∗∼ of strong neg. logics We explored 3 different options:

◮ HT 2∼

u

paraconsistency

◮ HT 2

sc semi-consistency

◮ HT 2

coh coherence ∼ L ⇒ ¬L

Coherence:

◮ not so natural when handling paraconsistency ◮ for capturing WFSX, HT 2

coh is too strong

◮ WFSX can be encoded into HT 2

sc

Future work: detailed comparison to frame-based characterisation

  • f WFSX [Alcântara,Damásio&Pereira].

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 34 / 35

slide-100
SLIDE 100

Conclusions

PEL is a natural logical foundation for partial stable models. Strong negation added preserving complexity and strong equivalence results. We provided a Routley-style general family N∗∼ of strong neg. logics We explored 3 different options:

◮ HT 2∼

u

paraconsistency

◮ HT 2

sc semi-consistency

◮ HT 2

coh coherence ∼ L ⇒ ¬L

Coherence:

◮ not so natural when handling paraconsistency ◮ for capturing WFSX, HT 2

coh is too strong

◮ WFSX can be encoded into HT 2

sc

Future work: detailed comparison to frame-based characterisation

  • f WFSX [Alcântara,Damásio&Pereira].

P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 34 / 35

slide-101
SLIDE 101

Further reading

P . Cabalar, S. Odintsov & D. Pearce. Logical Foundations of Well-Founded Semantics. In Proceedings KR 06. P . Cabalar, S. Odintsov, D. Pearce & A. Valverde. Analysing and Extending Well-Founded and Partial Stable Semantics using Partial Equilibrium Logic. In Proceedings of ICLP’06, (LNCS 4079).

◮ Strong equivalence, complexity results, properties of PEL inference,

disjunctive WFS, . . . P . Cabalar, S. Odintsov, D. Pearce & A. Valverde. On the logic and computation of Partial Equilibrium Models. In Proceedings of JELIA’06, (LNAI 4160).

◮ Tableaux proof system ◮ Splitting theorem for PEL P . Cabalar, A. Odintsov & D. Pearce ( University of Corunna (Spain), Sobolev Institute of Mathematics (Novosibirsk, Russia), Strong negation in WF and PS semantics . . . IBERAMIA 2006 35 / 35