Paraconsistent Modular Answer Set Programming Thomas Eiter joint - - PowerPoint PPT Presentation

Paraconsistent Modular ASP

Paraconsistent Modular Answer Set Programming

Thomas Eiter joint with M. Dao-Tran, M. Fink, T. Krennwallner

Institut für Informationsysteme, TU Wien 2nd Datalog 2.0 Workshop, Vienna, September 13, 2012

Austrian Science Fund (FWF) grant P20841 Vienna Science and Technology Fund (WWTF) grant ICT08-020. 1/58

Paraconsistent Modular ASP

• 1. Introduction

Needs for Knowledge Representation

Wanted: Interlink knowledge sources Cope with heterogeneity Modularization Handle inconsistency, incompleteness Dynamics, openness Issues: Frameworks, formal semantics Algorithms, implementations modular representation inconsistency management

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Paraconsistent Modular ASP

• 1. Introduction

1.1 Modularity in LP

Modularity in Logic Programming

Goal: Structured programming Gaifman and Shapiro [1989]: module architecture, interaction via input/output interfaces Different directions in general LP [Brogi et al., 1994]:

• Programming in the large:

compositional operators to build complex programs

• Programming in the small:

abstraction and scoping (e.g., macros, templates, . . . )

Issue: Modularity in Nonmonotonic LP

• rudimentary tools:

Stratified logic programs [Apt et al., 1988]. Splitting Sets [Lifschitz and Turner, 1994]

• more recent interest: module concepts [E_ et al., 1997b], [Janhunen

et al., 2007], [Dao-Tran et al., 2009a]

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• 1. Introduction

1.2 Paraconsistency

Paraconsistent Reasoning

Knowledge explosion: A, ¬A ⊢ B Belnap’s 4-valued logic: bilattice Paraconsistent Logic Programming: [Blair and Subrahmanian, 1989] Answer Set Programming: e.g. [Sakama and Inoue, 1995], [Alcântara et al. 2002,2004], [Arieli, 2002], [Odintsov and Pearce, 2005]

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• 1. Introduction

1.2 Paraconsistency

Roadmap

This talk: modularity and paraconsistency in nonmonotonic LP Focus: Answer Set Semantics, work at KBS

• Modular Answer Set Programming (MLP)
• Paraconsistent Answer Set Programming (ASP)
• Combination of Modular and Paraconsistent ASP
• Conclusion
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• 2. Modularity in NonMon LP

2.1 Replacement

Modular Replacement in NonMon LP

Problem: Replacement property (substitution) does not hold If P ≡ P ′ then Q ∪ P ≡ Q ∪ P ′ Counterexample: P = { a ← not b }, P ′ = { a }, Q = { b ← not a }. P and P ′ have the same model (answer set) {a} Q ∪ P ′ has the same model, but Q ∪ P in addition {b} Solution: Notion of strong equivalence [Lifschitz et al., 2001]: P ≡s P ′ iff Q ∪ P ≡ Q ∪ P ′ for all programs Q Refined notions (survey [Woltran, 2008]): uniform equivalence (restrict Q to sets of facts), modular equivalence [Oikarinen and Janhunen, 2006], projection equivalence [Oetsch et al., 2007], . . . Still not fully explored; computational complexity may be high

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Modular Answer Set Programming

Aim: [E_ et al., 1997b], [Dao-Tran et al., 2009a]

Provide module (“procedure”) concept as in ordinary programming realize libraries, code reuse establish call by value mechanism allow the use of recursive calls

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Program Modules

Conventional programming: Definition:

proc p(x, y: int): int begin ... a := ... return a end p;

Use: x := p(y, z); Nonmonotonic LP: Definition: Module m = (P[q1, q2], R), where

• P is a module name
• q1, q2 are predicate names
• R is a set of rules

Use: p(X) ← P[r, s].a

Modular Logic Programs

A modular (nonmon.) logic program (MLP) P = (m1, . . . , mn), consists

• f modules mi = (Pi[

qi], Ri) where at least one mi has void qi (=main). Rule bodies may contain module atoms P[p1, . . . , pk].o(t1, . . . , tl), where p1, . . . , pk are predicate names and o(t1, . . . , tl) is an ordinary atom.

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Semantics

[E_ et al., 1997b]: view modules as generalized quantifiers:

access a module P2 from module P1 by module atoms of the form P2[ p].q(X), using the extension of p as input from P1.

For an MLP P as above, answer sets were defined using a generalization of the GL-reduct [Gelfond and Lifschitz, 1991].

Drawbacks

Calls of modules must be acyclic (similar in other approaches). Lack of groundedness (minimality). Handling multiple models of called modules difficult.

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Example: Odd/Even

Canonical example of mutual recursion in LP

even(0) ← . even(f(X)) ← odd(X).

• dd(f(X)) ← even(X).

Datalog version: Given unary relation q, decide whether |q| is even. Not expressible in plain datalog, nor in monadic ASP, but with an

• rdering (successor) on q using recursion

Use choice in a recursive module: P[q]:

q′(X) ← q(X), rem(Y ), X = Y . rem(X) ← q(X), not q′(X).

• dd ← rem(X), P[q′].even.

even ← not odd.

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Example: Order

Similar: generate an ordering (successor) Pord[q/1]: define relations succ, first, last

q′(X) ∨ q′(Y ) ← q(X), q(Y ), X = Y. rem(X) ← q(X), not q′(X). first(X) ← rem(X). succ(X, Y ) ← first(X), Pord[q′].first(Y ). succ(X, Y ) ← Pord[q′].succ(X, Y ). n_last(X) ← succ(X, Y ). last(X) ← q(X), not n_last(X).

No guess on binary predicates Answer sets correspond to orderings

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Example: Positive Loop

Consider P = (m1, m2), where m1 = (P1[], R1), with R1 = { a ← P2[ ].b } m2 = (P2[], R2),with R2 = { b ← P1[ ].a } Thus, has a loop via positive mutual calls. Desired semantics: an “empty” model

• Neither a in P1 nor b in P2 should be true.

Proposals like DLP functions [Janhunen et al., 2007] exclude positive recursion. Yet useful e.g. for modular embedding of nonmonotonic dl-programs into datalog (cf. [E_ et al., 2012])

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Syntax of MLPs

Datalog language (function-free first order) Module names P, each with a fixed associated list q = q1, . . . , qk (k ≥ 0) of predicates qi (the formal input parameters). Atoms:

• Ordinary atoms: o(t1, . . . , tℓ)
• Module atoms P[p1, . . . , pk].o(t1, . . . , tℓ)

p1, . . . , pk are predicates matching q1, . . . , qk (call input parameters)

Rules: α1 ∨ · · · ∨ αk ← β1, . . . , βj, not βj+1, . . . , not βm, k ≥ 1 where the αi are ordinary atoms, the βj are atoms Module: m = (P[ q], R), where R is finite set of rules

P[ q] R

Modular Nonmonotonic Logic Program: P = (m1, . . . , mn)

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Odd/Even: Mutual Recursion

P = (m1, m2, m3), where m1 = (P1, R1), m2 = (P2[q2], R2), m3 = (P3[q3], R3). R1 = {q(a). q(b).

• k ← P2[q].even2.}

R2 =            q′

2(X)∨q′ 2(Y ) ← q2(X), q2(Y ),

X = Y. rem2 ← q2(X), not q′

2(X).

even2 ← not rem2. even2 ← rem2, P3[q′

2].odd.

           R3 =        q′

3(X)∨q′ 3(Y ) ← q3(X), q3(Y ),

X = Y. rem3 ← q3(X), not q′

3(X).

• dd

← rem3, P2[q′

3].even2.

       main(): n := |q| ; if even(n) then return ok ; even(n): n′ := n − 1 ; if n′ < 0 then return true ; if n′ = 0 then return false ; if odd(n′) then return true ; else return false ;

• dd(n):

n′ := n − 1 ; if even(n′) then return true ; else return false ;

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Semantics of MLPs

Defined by grounding as usual, over the constants (Herbrand universe) of an MLP P. For a rule, set of rules, module, MLP X let gr(X) denote the grounding of X. In the sequel, we assume grounded modules / MLPs The Herbrand base of P, HBP, consists of all ground atoms An interpretation of a (ground) rule set R is a subset M ⊆ HBP containing only ordinary atoms. To interpret a (ground) module m = (P[ q, R]), we need to consider its instantiations

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Module Instantiation

Schema Instantiation

P [q1,q2] R ∅ P [q1,q2] R {q1(a)} P [q1,q2] R {q2(a)} P [q1,q2] R {q1(a) ,q2(a) } P [q1,q2] R

Value call: P[S], with input S ⊆ HBP|

q

VC(P) = set of all value calls P[S] in P IP(P[S]) = R ∪ S is an instantiation of m = (P[ q], R) with S ⊆ HBP|

q

I(P) = (IP(Pi[S]) | Pi[S] ∈ VC(P)) is the instantiation of P

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Interpretations and Models

Notation: Mi/S stands for MPi[S] M

Mi/S

S

Pi[ qi] Pk[ p].o( c) r( a) Mi/S r( a) Mk/T

• (

c) T = (Mi/S)|

qk

• p

Instantiation: I(P) = (IP(Pi[S]) | Pi[S] ∈ VC(P)) Interpretation: M = (Mi/S | Pi[S] ∈ VC(P)) M, Pi[S] | = r( a) iff r( a) ∈ Mi/S M, Pi[S] | = Pk[ p].o( c) iff o( c) ∈ Mk/T

Informally, T assigns the formal parameters qk the value of p in Mi/S

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Answer Sets (Stable Models)

For rules r and set R of rules, define M, P[S] | = r and M, P[S] | = R as usual Analgously, M | = r iff M, P[S] | = r, resp. M, P[S] | = R iff M | = R, for all P[S] ∈ V C For answer sets, we require stability of M at all P[S] using the FLP reduct [Faber et al., 2004]. For any P[S] ∈ VC(P), the FLP reduct of P wrt. M is fP(P[S])M = {H ← B ∈ Igr(P)(P[S]) | M, P[S] | = B}

Definition

M is an answer set (stable model) of P iff M is a ⊆-minimal model of fPM, where ⊆ is componentwise. (N ⊆ M iff NP[S] ⊆ MP[S] for all P[S] ∈ VC)

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Example

Loop (reconsidered): P1 : a ← P2[ ].b P2 : b ← P1[ ].a Answer set: M1 = (∅, ∅)

• fP(P1[])M1 = {}
• fP(P2[])M1 = {}
• Clearly, M1 is a minimal model of fPM1

Non-answer set: M2 = ({a}, {b})

• fPM2 = P
• M1 |

= P

• As M1 ⊂ M2, M2 is not an answer set of P.
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Odd/Even (cont’d)

P = (m1, m2, m3), where m1 = (P1, R1), m2 = (P2[q2], R2), m3 = (P3[q3], R3). R1 = {q(a). q(b).

• k ← P2[q].even2.}

R2 =            q′

2(X)∨q′ 2(Y ) ← q2(X), q2(Y ),

X = Y. rem2 ← q2(X), not q′

2(X).

even2 ← not rem2. even2 ← rem2, P3[q′

2].odd3.

           R3 =        q′

3(X)∨q′ 3(Y ) ← q3(X), q3(Y ),

X = Y. rem3 ← q3(X), not q′

3(X).

• dd3

← rem3, P2[q′

3].even2.

       M M1/∅ : {ok, q(a), q(b)} M2/{q2(a), q2(b)} : even2, rem2, q2(a), q2(b), q′

2(b)

• M2/∅ : {even2}

. . . M3/{q3(b)} : {odd3, rem3, q3(b)} . . .

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Semantic Properties

Horn MLPs P (rules α1 ← β1, . . . , βj):

• P satisfies model intersection M ∩ N, where

(M ∩ N)P [S] =

S′⊇S(MP [S′] ∩ NP [S′]).

• P has a unique answer set (minimal model), LM(P)
• LM(P) = lfp(TP) for the natural TP operator.

The answer sets of a not-free MLP P are its minimal models. For a generalized notion of stratification Σ = Σ1, . . . , Σl [Apt et al., 1988], the iterative model semantics (computable via operators T Σi

P )

coincides with the answer set semantics. Generalizes the concept of DLP-function of Janhunen et al. [2007]

• Allow for positive recursion through modules
• Value call mechanism (i.e., not only empty inputs P[ ].o)
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Computational Complexity

Horn MLP P Computing lfp(P) Deciding α ∈ lfp(P) propositional, empty inputs polynomial time P-c propositional exponential time EXP-c non-ground double exponential time 2EXP-c MLP P Answer set existence / brave inference propositional, normal, empty inputs NP-c propositional, empty inputs Σp

2-c

propositional, normal NEXP-c propositional NEXPNP-c non-ground, normal 2NEXP-c non-ground 2NEXPNP-c

In general, one exponential higher than for ordinary programs Intuitively, input p1, . . . , pk encodes k·nm bits (m = max. arity of pi) Also exponentially higher data complexity No increase for constant bound on input arities (m)

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Implementation

Top-Down evaluation algorithm TD-MLP [Dao-Tran et al., 2009b]

• for ic-stratified MLPs (e.g., the odd/even example)
• works in exponential space
• exploits Splitting Sets extended to MLPs
• implementation (+ optimizations): [Wijaya, 2011]
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• 3. Paraconsistent ASP

Sources of Inconsistency

Contradiction

☞ Example P = {p; −p ← q q}.

Ako paraconsistency may be supported in implementations of ASP:

• −p(

X) is viewed as atom with predicate name “−p”

• the constraint false ← p(

X), “−p”( X), not false is added ☞ disable this constraint

Unfoundedness / Instability: Due to nonmonotonicity, programs may be incoherent, i.e., no answer set due to cyclic dependencies of an atom from its default negation.

☞ Example P = { p ← not p }.

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• 3. Paraconsistent ASP

Sources of Inconsistency (cont’d)

Focus: genuine problem of unfoundedness ☞paracoherence contradiction (“classical negation”) can be “added on top” (cf. ordinary ASP)

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• 3. Paraconsistent ASP

Desiderata

Ideally, a paraconsistent semantics for ASP satisfies the following properties:

• 1. Classical Coherence: If a program has a classical model,

then it has a model.

• 2. Answer Set Coverage: Every (consistent) answer set of a program

corresponds to some model.

• 3. Answer Set Incidence (aka Congruence): If a (consistent) answer

set exists for a program, then every model corresponds to some answer set. Widely-known semantics satisfy only part of these requirements:

• 3-valued stable models [Przymusinski 91], L-stable models [Eiter et
• al. 95], revised stable models [Pereira & Pinto 95],
• regular models [You & Yuan 94], pstable models [Osorio et al. 08].

Semi-stable models [Inoue & Sakama 95] satisfy all three properties.

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3.1 Semi-Stable Models

Semi-Stable Models

Semi-stable models [Sakama and Inoue, 1995]: an approach to paraconsistent answer set semantics. Consider an extended propositional language Lκ = L ∪ {Ka | a ∈ L}. Interpretations Iκ are subsets of Lκ assigning truth values

• false ⊥,
• believed (assumed) true bt, and
• true t,

where ⊥ bt t. The truth value assigned by Iκ to a variable a is defined by Iκ(a) =      t if a ∈ Iκ, bt if Ka ∈ Iκ and a ∈ Iκ, ⊥ otherwise. Definition via epistemic transformation (syntactic transformation)

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3.1 Semi-Stable Models

Semi-Stable Models (cont’d)

Definition (P κ [Sakama and Inoue, 1995])

The epistemic transformation of a program P is the positive program P κ

• btained from P by replacing each rule r of the form

a1 ∨ · · · ∨ al ←b1, . . . , bm, not bm+1, . . . , not bn, (1) in P, such that B−(r) = ∅, with: λr,1 ∨ . . . ∨ λr,l ∨ Kbm+1 ∨ . . . ∨ Kbn ← b1, . . . , bm, ai ← λr,i, ← λr,i, bj, λr,i ← ai, λr,k, for 1 ≤ i ≤ l, m < j ≤ n, and 1 ≤ k ≤ l. Note that P κ is positive for every program P.

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3.1 Semi-Stable Models

Semi-Stable Models (cont’d)

Semi-stable models are maximal canonical answer sets of P κ: Given an interpretation Iκ over L′ ⊇ Lκ, let gap(Iκ) = {Ka | Ka ∈ Iκ and a ∈ Iκ}. An interpretation Iκ ∈ S is maximal canonical in S iff no interpretation Jκ ∈ S such that gap(Jκ) ⊂ gap(Iκ) exists. Let mc(S) denote maximal canonical interpretations in S and let SST (P) be the semi-stable models of a program P. Then, we can paraphrase the definition of semi-stable models:

Definition

Let P be a program over L. Then, SST (P) = {Iκ ∩ Lκ | Iκ ∈ mc(AS(P κ))}.

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3.1 Semi-Stable Models

Examples

☞ Example 1: incoherent program The program P = {a ← not a} is incoherent, with signle semi- stable model {Ka}. ☞ Example 2: incoherent program, II The program P = {b ← a; a ← not a} has the single semi-stable model {Ka}. ☞ Example 3: many to one coverage The program P = {a; b; c; d ← not a, not b; d ← not b, not c}. is coherent and has the single answer set {a, b, c}. As SST (P) = {{a, b, c, Kb},{a, b, c, Ka, Kc}}, the semi-stable models do not not uniquely cover answer sets.

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3.1 Semi-Stable Models

Anomalies

The examples exhibit anomalies wrt. desirable rationality properties. knowledge generalization (or necessitation, resp. modal axiom N): Property N: a ∈ Iκ implies Ka ∈ Iκ, for all a ∈ L. This property does not hold as witnessed by Example 3. distribution axiom (modal axiom K): a1 ∨ · · · ∨ al ← b1, . . . , bm, not bm+1, . . . , not bn. Property K: If Iκ | = Kb1 ∧ . . . ∧ Kbm and Iκ | = Kbm+1 ∨ . . . ∨ Kbn (that is Iκ | = K not bm+1, . . . , Iκ | = K not bn) then Iκ | = Ka1 ∨ . . . ∨ Kal. Note that this does not hold for rule b ← a in Example 2.

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3.2 Semi-Equilibrium Models

Alternative Paracoherent Semantics

Alternative: semi-equilibrium models [E_ et al., 2010] Aim: Enforce Properties N and K. Have a nice model-theoretic definition resp. characterization of the semantics, rather than an transformational definition. Approach: use Equilibrium Logic [Pearce, 2006]

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3.2 Semi-Equilibrium Models

Equilibrium Logic

Basis for semantic charachterization of answer sets: the logic of here-and-there (HT) (Heyting, 1930), i.e., 2-world intuitionistic logic. We restrict our attention to program-like theories T (sets of formulas) of the form b1 ∧ . . . ∧ bm ∧ ¬bm+1 ∧ . . . ∧ ¬bn → a1 ∨ · · · ∨ al, (2) Semantically characterized by Kripke models of two worlds, h(ere) ≤ t(here).

➥ Propositional HT-interpretations are pairs (X, Y ) of sets of atoms,

such that X ⊆ Y .

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3.2 Semi-Equilibrium Models

Satisfaction

Satisfaction at the world w of an HT model M = (X, Y ): Notation: (X, Y ) (X, Y )|h = X, (X, Y )|t = Y

• 1. M, w |

= a if a ∈ M|w, for any atom a,

• 2. M, w |

= ⊥,

• 3. M, w |

= φ ∧ ψ if M, w | = φ and M, w | = ψ,

• 4. M, w |

= φ ∨ ψ if M, w | = φ or M, w | = ψ,

• 5. M, w |

= φ → ψ if for every w′ ≥ w: M|w′ | = φ or M|w′ | = ψ,

where ¬φ is φ → ⊥ M | = φ iff M, w | = φ for all w ∈ {h, t} As M, w | = φ is monotonic in w, M | = φ iff M, h | = φ Thus M | = ¬φ iff M|t | = φ.

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Satisfaction (cont’d)

Alternatively define M | = φ, M = (X, Y ), straight:

• 1. (X, Y ) |

= a if a ∈ X, for any atom a,

• 2. (X, Y ) |

= ⊥,

• 3. (X, Y ) |

= φ ∧ ψ if (X, Y ) | = φ and (X, Y ) | = ψ,

• 4. (X, Y ) |

= φ ∨ ψ if (X, Y ) | = φ or (X, Y ) | = ψ,

• 5. (X, Y ) |

= φ → ψ if (i) (X, Y ) | = φ or (X, Y ) | = ψ, and (ii) Y | = φ → ψ.

where ¬φ is φ → ⊥; thus (X, Y ) | = ¬φ iff Y | = φ. An HT-interpretation (X, Y ) satisfies a theory T, iff it satisfies all formulas Axiomatic proof system: e.g. [Lifschitz et al., 2007]

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• 3. Paraconsistent ASP

3.2 Semi-Equilibrium Models

Equilibrium Models

Definition

An HT-interpretation (Y, Y ) is an equilibrium model of a theory T, iff (Y, Y ) | = T and for all X ⊂ Y , (X, Y ) | = T. (h-minimality) ☞ Examples P = {a ← not b; a ← not b} has the eq. models (a, a) and (b, b) P = {a ← not a} has no equilibrium model

Theorem ([Pearce, 2006])

M is an answer set of P iff (M, M) is an equilibrium model of P. The use of HT/Equilibrium Logic has many attractive features: easy generalization to richer syntax (strong negation, nesting etc). lifts to non-ground programs (under suitable provisos) HT captures strong equivalence: P ≡s Q iff | =HT P ↔ Q

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3.2 Semi-Equilibrium Models

Semi-Equilibrium Models

In a HT-model (X, Y ), we may view Y as believed (assumed) atoms and X as a possible model of the rules under this belief The atoms in Y \ X of an HT-model (X, Y ) of P are “unfounded” In an equilibrium model X = Y , i.e., all atoms are founded By allowing (minimal) gaps Y \ X, coherence is ensured.

Definition

Let P be a program over L. An semi-equilibrium (seq) model of P is any HT-model (X, Y ) of P such that (i) (X′, Y ) | = P, for all X′ ⊂ X, and (h-minimality) (ii) there is no HT-model (X′, Y ′) of P that satisfies (i) and Y ′ \ X′ ⊂ Y \ X. (gap-minimality) SEQ(P) denotes the set of semi-equilibrium models of P.

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3.2 Semi-Equilibrium Models

Desiterata, Revisited

Semi-equilibrium models satisfy the desiderata 1-3:

• 1. Classical Coherence: If a program has a classical model,

then it has a semi-equilibrium model (as some HT-model (X, Y ) exists).

• 2. Answer Set Coverage: Every answer set X of a program P

corresponds to a semi-equilibrum model (X, X) of P (zero gap)

• 3. Answer Set Incidence: If an answer set X exists for a program,

then every seq-model (X, Y ) of P corresponds to some answer set of P (Y , and X = Y ) Furthermore: generalization from programs to theories is straightforward.

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3.2 Semi-Equilibrium Models

Examples

☞ Example: coherent (cont’d) Reconsider P = {a; b; c; d ← not a, not b; d ← not b, not c}. Its unique semi-equilibrium model is (abc, abc), corresponding to the single answer set {a, b, c}. As a consequence of Property K, semi-equilibrium semantics differs from semi-stable semantics not only with respect to believed consequences. ☞ Example Consider P = {a ← b; b ← not b; c ← not a}.

• SST (P) = {{c, Kb}}
• SEQ(P) = {(∅, ab)}

Note that c is not even believed to be true.

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3.2 Semi-Equilibrium Models

SEQ Transformation

A transformation to standard ASP is possible by an extension of the epistemic program transformation P κ.

Definition (P HT)

The epistemic HT-transformation of program P over L, P HT , is P κ plus all rules: Ka ←a, Ka1 ∨ . . . ∨ Kal ∨ Kbm+1 ∨ . . . ∨ Kbn ←Kb1, . . . , Kbm, for a ∈ L, respectively for every rule r ∈ P of the form (1): a1 ∨ · · · ∨ al ← b1, . . . , bm, not bm+1, . . . , not bn. The maximal canonical answer sets of P HT coincide with SEQ(P).

Theorem

Let P be a program over L, and let Iκ be an interpretation over Lκ. Then, Iκ ∈ SEQ(P) iff Iκ ∈ {M ∩ Lκ | M ∈ mc(AS(P HT ))}.

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3.2 Semi-Equilibrium Models

Computational Complexity of SEQ

Computational complexity of major reasoning tasks (propositional case): Problem normal P disjunctive P SEQ(P) = ∅ ? NP-c NP-c (X, Y ) ∈ SEQ(P) ? co-NP-c Πp

2-c

P | =v

b a

Σp

2-c

Σp

3-c

P | =v

c a

Πp

2-c

Πp

3-c

| =v

b / |

=v

c: a has value v ∈ {⊥, bt, t} in some / every seq-model of P.

• ne level up in the polyonmial hierarchy (global gap minimization)

seq-model existence is trivial if no constraints (empty heads) are present

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• 4. Paraconsistent Modular ASP

Paraconsistent Modular ASP

Issue: Combine modular + paraconsistent ASP Can this be done easily? Challenges, problems? Evaluation / Efficiency ?

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• 4. Paraconsistent Modular ASP

HT Logic

Approach: formulate HT-Logic on MLPs, do gap minimization Interpretations: M = (MP[S] | P[S] ∈ V C) ❀ M = (MP[S] | P[S] ∈ V C) where MP[S] = (XP[S], YP[S]) is an HT-interpretation. Satisfaction: M, P[S] | = P ❀ M, P[S], w | = P, w ∈ {h, t} Key Issue: M, P[S], w | = Pj[ p].o ? handled in intuitionistic manner Notation: h- resp. t-projection M|h = (XP[S] | P[S] ∈ VC) and M|t = (YP[S] | P[S] ∈ VC) M, P[S], t | = Pj[ p].o iff M|t , P[S], | = Pj[ p].o M, P[S], h | = Pj[ p].o iff M|h , P[S], | = Pj[ p].o and M, P[S], t | = Pj[ p].o Thus M, P[S] | = Pj[ p].o iff M|w , P[S] | = Pj[ p].o for w ∈ {h, t}

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4.1 Equilibrium MLP Models

Equilibrium MLP Models

Define equilibrium models as usual

Definition

An MLP HT-interpretation M = (M, M) is an equilibrium model of an MLP theory T, iff M | = T and for every M′ ⊂ M, it holds that (M′, M) | = T. (h-minimality) Correspondence result:

Theorem

M is an answer set of P iff M = (M, M) is an equilibrium model of P. Thus, we have an EQ-Logic view of MLPs Generalizes to richer MLP language Next: formulate semi-equilibrium models

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4.2 Semi-Equilibrium MLP Models

Semi-Equilibrium MLP Models

Notation: for MLP HT-model M = (X, Y), let gap(M) = (YP[S] − XP[S] | P[S] ∈ V C)

Definition

Let P be a program over L. An semi-equilibrium (seq) model of an MLP P is any HT-model M = (X, Y) of P such that (i) for every P[S] ∈ V C and formal input atom q, q ∈ S iff q ∈ YP[S] (input congruence) (ii) (X, Y) | = P, for all X′ ⊂ X, and (h-minimality) (iii) there is no HT-model M′ = (X′, Y′) of P that satisfies (i), (ii) and gap(M′) ⊂ gap(M). (gap-minimality) SEQ(P) denotes the set of semi-equilibrium models of P.

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4.2 Semi-Equilibrium MLP Models

SEQ MLP Transformation

Modify and adapt P HT above for P:

Definition (PHT)

For each rule r a1 ∨ · · · ∨ al ← b1, . . . , bm, not bm+1, . . . , not bn, in P, set up (1) a1 ∨ · · · ∨ al ∨ Kbm+1 ∨ . . . ∨ Kbn ← b1, . . . , bm, (2) Ka1 ∨ · · · ∨ Kal ∨ . . . ∨ Kbm+1 ∨ . . . ∨ Kbn ← Kb1, . . . , Kbm, (3)

• Ka ← a,
• rdinary a

(4) ← Ka, not Ka, module atom a (5) ← Kq, not q, formal input atom q where

b Ka = Kp a = p KoP.[

p] a = P.[

p].o Ka = Kp a = p P.[Kp1, . . . , Kpk].Ko a = P.[p1, . . . , pk].o

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4.2 Semi-Equilibrium MLP Models

SEQ MLP Transformation (cont’d)

The answer sets of PHT provide the basis for

Lemma

The equilibrium models of PHT , projected to the original language L, coincide with the HT-models of P that fulfill (i) and (ii). Perform gap minimization on top. Let M ∩ L be the projection of M to L, and Let mc(S) = {M ∈ S | ∀M′ ∈ S : gap(M′ ∩ L) ⊂ gap(M ∩ L)} The maximal canonical answer sets of PHT yield SEQ(P).

Theorem

For any MLP program P over L, SEQ(P) = {ML | M ∈ mc(EQ(PHT )).

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4.3 Computational Complexity

Computational Complexity

Gap minimization adds for ordinary answer set programs a guess for refutation (recognition) For MLPs, an anlog increase is plausible Expected results:

MLP P pseq-model existence brave inference propositional, normal, empty inputs NP-c Σp

2-c

propositional, empty inputs NP-c Σp

3-c

propositional, normal NEXP-c NEXPNP-c propositional NEXP-c NEXPΣp

2 -c

non-ground, normal 2NEXP-c 2NEXPNP-c non-ground 2NEXP-c 2NEXPΣp

2 -c

pseq-model existence is trivial, if no constraints (empty heads) are present

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4.4 Classical Negation

Classical Negation

Classical (strong) negation ∼φ can be added using paraconsistent Nelson logic (N9), a 9-valued logic with explicit strong negation. Can be semantically described by Routley models: (X, Y, X∗, Y ∗) [Odintsov and Pearce, 2005]:

• (X, Y ) is an HT interpretation.
• (X∗, Y ∗) is an HT interpretation to evaluate ∼φ.
• Intuition: truth of ∼φ at w is equivalent to falsity at w∗ (and

w∗∗ = w).

A quantified version of N9 (with static domain), QN9, is available (cf. [Fink, 2012]). For foundedness, (i) minimize X, X∗ in parallel (keeping the there-worlds Y , Y ∗ fixed) On top, (ii) minimize gaps Y \ X, Y ∗ \ X∗ in parallel

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4.4 Classical Negation

Bilattice NINE

The nine truth values form a bilatice wrt. truth order (≤t) and knowledge

• rder (≤k).

Where bt = (0, 1), t = (1, 1), f = (1, 0), ⊥ = (0, 0)

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4.4 Classical Negation

Properties

Paraconsistent semi-equilibrium (pseq) models (defined using (i) and (ii)) for MLPs has desirable properties:

• 1. Classical Coherence:

If a program has a classical model, MLP P has some pseq-model if P has some model

• 2. Answer Set Coverage:

If M is an equilibrium model (answer set) of MLP P, then ((X, X, X, X) | X = MP[S], P[S] ∈ V C) is a pseq-model of P

• 3. Answer Set Incidence:

If an MLP P has some equilibrium model (answer set) M, then the pseq models of P correspond to the answer sets of P. No increase in complexity

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4.5 Structured Rule Sets

Structured Rule Sets

Gap minimization is too weak! P = c ← b, not c. b ← not a.

• Two semi equilibrium models: (b, bc) and (∅, a)

(b, bc) is more appealing than (∅, a):

• a can not be derived for sure (no rule with a in the head),
• P1 = {b ← not a} is a “lower” part feeding into the “upper” part

P2 = {c ← b, not c}

• The lower program part is coherent
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4.5 Structured Rule Sets

Splitting Sets

Splitting sets [Lifschitz and Turner, 1994] capture implicit program modularity A set U of atoms is a splitting set of (propositional) program P, if for every rule r ∈ P: ] head(r) ∩ U = ∅ implies Ats(r) ⊆ U. Split program into P = bU(P) ∪ tU(P) ☞ Example (cont’d) U = {a, b}, bU(P) = P1, tU(P) = P2. Splitting sets allow for a compositional evaluation (bottom up) Datalog: P2 ⊲ P1 E_ et al. [1994,1997a] They generalize (local) stratification Easy to recognize Dilemma: not unique

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4.5 Structured Rule Sets

Explicit Modularity

Reformulate P as MLP P = (P1, P2): P1 = { b ← not a }, P2 = { c ← P1[ ].b, not c } Semi-equilibrium models: ((b, b), (∅, c)), ((∅, a), (∅, ∅)) Preferred: ((b, b), (∅, c)). In structured programming, the semantics of basic modules (at least) should be invariant and compositional.

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4.5 Structured Rule Sets

Structural Gap Minimization

Recipe: refine gap minimization For explicit modularity, preference respecting modules

• minimize gap on bottom modules

For implicit modularity, optional preference of parts

• minimize gap on bottom parts

Uniqueness ? ☞ use strongly connected components (SCCs) of a (syntactic) dependency graph

• module call dependency graph
• atom dependency graph

The SCCs of the ordinary dependency graph induce canonical splitting sets.

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4.5 Structured Rule Sets

SSCs

atom dependency graph vs module call dependency graph More about dependency graphs: [Dao-Tran et al., 2011]

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• 5. Related Work & Conclusion

Related Work

Semi-stable models [Sakama and Inoue, 1995] Paraconsistent Answer-Set Semantics (PAS) [Alcântara et al., 2004]

• logical axiomatization in [Odintsov and Pearce, 2005] using Routley

models

Consistency Restoring Rules (CR-Prolog) [Balduccini and Gelfond, 2003] Paraconsistent hybrid theories [Fink, 2012], hybrid theories [Fink and Pearce, 2010] First-order stable models [Ferraris et al., 2007], [Lee and Meng, 2012]

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• 5. Related Work & Conclusion

Conclusion

Modularity in NonMon LP: MLPs Paracoherence / -consistency: HT gap minimization

• lifts to MLPs
• shows versatility of HT / Equilibrium Logic

Issues

• Semantics: refine paracoherence

Preference on (gap-minimal) HT-models

modular composition implicit vs. explicit modularity

• Complexity: low cost paracoherent / -consistent fragments
• Implementation
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• Ann. Math. Artif. Intell., 36(4):381–417, 2002.

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• T. Eiter, G. Gottlob, and H. Mannila.

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Thomas Eiter, Georg Gottlob, and Helmuth Veith. Modular Logic Programming and Generalized Quantifiers. In LPNMR-1997, volume 1265 of LNCS, pages 290–309. Springer, 1997. Thomas Eiter, Michael Fink, and Joao Moura. Paracoherent answer set programming. In Proceedings 12th International Conference on Principles of Knowledge Representation and Reasoning (KR 2010), May 9-13, 2010, Toronto, Canada, 2010. Thomas Eiter, Thomas Krennwallner, Patrik Schneider, and Guohui Xiao. Uniform evaluation of nonmonotonic dl-programs. In T. Lukasiewicz and A. Sali, editors, Proceedings 7th International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2012), Kiel, Germany, March 5–9, 2012, number 7153 in LNCS, pages 1–22. Springer, 2012.

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Michael Fink. Paraconsistent hybrid theories. In Gerhard Brewka, Thomas Eiter, and Sheila A. McIlraith, editors, KR. AAAI Press, 2012. Haim Gaifman and Ehud Y. Shapiro. Fully abstract compositional semantics for logic programs. In POPL, pages 134–142. ACM Press, 1989.

• M. Gelfond and V. Lifschitz.

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Vladimir Lifschitz, David Pearce, and Agustín Valverde. Strongly equivalent logic programs. ACM Trans. Comput. Log., 2(4):526–541, 2001. Vladimir Lifschitz, David Pearce, and Agustín Valverde. A characterization of strong equivalence for logic programs with variables. In Chitta Baral, Gerhard Brewka, and John S. Schlipf, editors, LPNMR, volume 4483 of Lecture Notes in Computer Science, pages 188–200. Springer, 2007. Sergei P. Odintsov and David Pearce. Routley semantics for answer sets. In Chitta Baral, Gianluigi Greco, Nicola Leone, and Giorgio Terracina, editors, LPNMR, volume 3662 of Lecture Notes in Computer Science, pages 343–355. Springer, 2005.

References X

Johannes Oetsch, Hans Tompits, and Stefan Woltran. Facts do not cease to exist because they are ignored: Relativised uniform equivalence with answer-set projection. In AAAI, pages 458–464. AAAI Press, 2007. Emilia Oikarinen and Tomi Janhunen. Modular equivalence for normal logic programs. In Gerhard Brewka, Silvia Coradeschi, Anna Perini, and Paolo Traverso, editors, ECAI, volume 141 of Frontiers in Artificial Intelligence and Applications, pages 412–416. IOS Press, 2006. David Pearce. Equilibrium logic.

• Ann. Math. Artif. Intell., 47(1-2):3–41, 2006.

References XI

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• J. Log. Comput., 5(3):265–285, 1995.

Tri Kurniawan Wijaya. Top-down evaluation techniques for modular nonmonotonic logic programs. Master’s thesis, TU Wien, August 2011. Stefan Woltran. A common view on strong, uniform, and other notions of equivalence in answer-set programming. TPLP, 8(2):217–234, 2008.