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Extending answer set programming using generalized possibilistic logic Steven Schockaert (joint work with Didier Dubois and Henri Prade) School of Computer Science & Informatics Cardi ff University, Cardi ff , UK schockaerts1@cardi ff .ac.uk

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Extending answer set programming using generalized possibilistic logic

School of Computer Science & Informatics Cardiff University, Cardiff, UK schockaerts1@cardiff.ac.uk http://users.cs.cf.ac.uk/S.Schockaert

Steven Schockaert (joint work with Didier Dubois and Henri Prade)

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  • 1. Possibilistic logic
  • 2. Generalized possibilistic logic
  • 3. Answer set programming
  • 4. Extending answer set programming
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Possibilistic logic: syntax

(p ∧ (¬q → r), 0.7)

propositional formula certainty degree, taken from

Λ = {0, 1 k , 2 k , ..., 1}

Knowledge bases in possibilistic logic are sets of weighted formulas

  • f the form:
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Possibilistic logic: syntax

(p ∧ (¬q → r), 0.7)

reflects my degree of surprise if I found out that the formula were false

Knowledge bases in possibilistic logic are sets of weighted formulas

  • f the form:

K = {(loc(IJCAI2015, Argentina), 1), (loc(IJCAI2015, Argentina) → loc(IJCAI2015, SouthAmerica), 1), (loc(IJCAI2016, NewYork) → loc(IJCAI2016, NorthAmerica), 1), (loc(IJCAI2015, SouthAmerica) → ¬loc(IJCAI2016, SouthAmerica), 0.75), (loc(IJCAI2015, SouthAmerica) → ¬loc(IJCAI2016, NorthAmerica), 0.5)}

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Possibilistic logic: syntax

Certainty degrees are interpreted qualitatively, i.e. possibilistic logic theories can be seen as stratified classical logic theories Using numbers makes it easier to describe the semantics and to formulate inference rules

→ ¬ loc(IJCAI2015, Argentina) loc(IJCAI2015, Argentina) → loc(IJCAI2015, SouthAmerica) loc(IJCAI2016, NewYork) → loc(IJCAI2016, NorthAmerica) loc(IJCAI2015, SouthAmerica) → ¬loc(IJCAI2016, SouthAmerica) loc(IJCAI2015, SouthAmerica) → ¬loc(IJCAI2016, NorthAmerica)

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Possibilistic logic: semantics

At the semantic level, the role of models is taken up by possibility distributions, which assign to every propositional interpretation (or possible world) a degree of possibility, e.g.

π(ω1) = 0 π(ω2) = 0.4 π(ω3) = 0.7 π(ω4) = 1

Intuitively, a model corresponds to the epistemic state of an agent, encoded as the weighted set of worlds it considers possible

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{ | | } − ¬ Π(α) = Π({ω | ω | = α}) = max{π(ω) | ω | = α}

Possibilistic logic: semantics

If π represents the epistemic state of an agent, then that agent considers a formula α possible to the degree that some model of α is considered possible

possibility measure

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Possibilistic logic: semantics

If π represents the epistemic state of an agent, then that agent considers a formula α possible to the degree that some model of α is considered possible

{ | | } − ¬ Π(α) = Π({ω | ω | = α}) = max{π(ω) | ω | = α}

The agent considers the formula necessary to the degree that all counter-models of α are impossible

N(α) = N({ω | ω | = α}) = min{1 π(ω) | ω 6| = α} = 1 Π(¬α)

necessity measure

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Possibilistic logic: semantics

A possibility distribution π satisfies a formula (α,λ) iff N(α) ≥ λ, with N the necessity measure induced by π A possibility distribution π is called a model of a set of formulas K iff π satisfies all formulas in K

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Possibilistic logic: semantics

A possibility distribution π satisfies a formula (α,λ) iff N(α) ≥ λ, with N the necessity measure induced by π K = {(a, 0.8), (a ! b, 1)}

π({a, b}) = 1 π({a}) = 0 π({b}) = 0.2 π({}) = 0.2

A possibility distribution π is called a model of a set of formulas K iff π satisfies all formulas in K

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Possibilistic logic: semantics

A possibility distribution π satisfies a formula (α,λ) iff N(α) ≥ λ, with N the necessity measure induced by π K = {(a, 0.8), (a ! b, 1)}

π({a, b}) = 1 π({a}) = 0 π({b}) = 0.2 π({}) = 0.2

N(a) = 1 Π(¬a) = 1 max(π({b}), π({})) = 0.8 N(a ! b) = 1 Π(a ^ ¬b) = 1 max(π({a})) = 1

A possibility distribution π is called a model of a set of formulas K iff π satisfies all formulas in K

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Possibilistic logic: semantics

⌃ω . π(ω) = 1

π(ω0) = 1 and ⌃ω ⇧= ω0 . π(ω) = 0

Completely uninformative: every world remains possible Maximally informative: exactly one world is considered possible π1 is less specific than π2 if

⌃ω . π1(ω) ⇥ π2(ω)

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Possibilistic logic: semantics

⌃ω . π(ω) = 1

π(ω0) = 1 and ⌃ω ⇧= ω0 . π(ω) = 0

Completely uninformative: every world remains possible Maximally informative: exactly one world is considered possible π1 is less specific than π2 if

⌃ω . π1(ω) ⇥ π2(ω)

Every consistent set of formulas K in possibilistic logic has a unique least specific model πK

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Possibilistic logic: inference

inference rules

if (α, λ) 2 K then K ` (α, λ) ( if α ⌘ β and K ` (α, λ) then K ` (β, λ) ( if λ1 λ2 and K ` (α, λ1) then K ` (α, λ2) ( if K ` (α _ β, λ1) and K ` (¬α _ γ, λ2) then K ` (β _ γ, min(λ1, λ2))

possibilistic logic is based

  • n a weakest link idea
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Possibilistic logic: inference

inference rules soundness and completeness

The following statements are equivalent:

if (α, λ) 2 K then K ` (α, λ) ( if α ⌘ β and K ` (α, λ) then K ` (β, λ) ( if λ1 λ2 and K ` (α, λ1) then K ` (α, λ2) ( if K ` (α _ β, λ1) and K ` (¬α _ γ, λ2) then K ` (β _ γ, min(λ1, λ2))

  • 1. K ` (α, λ) can be derived from (1)–(4).
  • 2. Every model π of K is a model of (α, λ).
  • 3. The least specific model πK of K is a model of (α, λ).
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Possibilistic logic: applications

Possibilistic logic is closely related to AGM belief revision and the rational closure of default rules

(loc(IJCAI2016, NewYork), 1)

+

K = {(loc(IJCAI2015, Argentina), 1), (loc(IJCAI2015, Argentina) → loc(IJCAI2015, SouthAmerica), 1), (loc(IJCAI2016, NewYork) → loc(IJCAI2016, NorthAmerica), 1), (loc(IJCAI2015, SouthAmerica) → ¬loc(IJCAI2016, SouthAmerica), 0.75), (loc(IJCAI2015, SouthAmerica) → ¬loc(IJCAI2016, NorthAmerica), 0.5)}

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Possibilistic logic: applications

Possibilistic logic is closely related to AGM belief revision and the rational closure of default rules

K = {(loc(IJCAI2015, Argentina), 1), (loc(IJCAI2016, NewYork), 1), (loc(IJCAI2015, Argentina) ! loc(IJCAI2015, SouthAmerica), 1), (loc(IJCAI2016, NewYork) ! loc(IJCAI2016, NorthAmerica), 1), (loc(IJCAI2015, SouthAmerica) ! ¬loc(IJCAI2016, SouthAmerica), 0.75), (loc(IJCAI2015, SouthAmerica) ! ¬loc(IJCAI2016, NorthAmerica), 0.5)}

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Possibilistic logic: applications

Possibilistic logic is closely related to AGM belief revision and the rational closure of default rules

K = {(loc(IJCAI2015, Argentina), 1), (loc(IJCAI2016, NewYork), 1), (loc(IJCAI2015, Argentina) ! loc(IJCAI2015, SouthAmerica), 1), (loc(IJCAI2016, NewYork) ! loc(IJCAI2016, NorthAmerica), 1), (loc(IJCAI2015, SouthAmerica) ! ¬loc(IJCAI2016, SouthAmerica), 0.75), (loc(IJCAI2015, SouthAmerica) ! ¬loc(IJCAI2016, NorthAmerica), 0.5)}

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  • 1. Possibilistic logic
  • 2. Generalized possibilistic logic
  • 3. Answer set programming
  • 4. Extending answer set programming
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Generalized possibilistic logic

In possibilistic logic, a formula (α,λ) corresponds to the constraint N(α) ≥ λ at the semantic level, and a knowledge base corresponds to a conjunction of such constraints Π(α) ≥ λ

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Generalized possibilistic logic

In possibilistic logic, a formula (α,λ) corresponds to the constraint N(α) ≥ λ at the semantic level, and a knowledge base corresponds to a conjunction of such constraints In generalized possibilistic logic (GPL), arbitrary propositional combinations of such formulas are allowed To emphasize the view of possibilistic logic as a modal logic, we use the following notations

) means that whereas Nλ(α) with suggests that

alternative notation for (α,λ)

ailable beliefs while Πλ(α) with (Π( ) ⌅

abbreviation for ¬N1−λ+ 1

k (¬α)

corresponds to the constraint Π(α) ≥ λ

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Generalized possibilistic logic

Syntax Note: every propositional atom is encapsulated by a modality Note: no nesting of modalities

  • N0.4(a ∧ ¬b) ∨ Π0.3(a → (b ∨ c))

⇥ → N0.7(b)

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Generalized possibilistic logic

Syntax Note: every propositional atom is encapsulated by a modality Note: no nesting of modalities Semantics

  • N0.4(a ∧ ¬b) ∨ Π0.3(a → (b ∨ c))

⇥ → N0.7(b)

  • ⇥
  • N(a ∧ ¬b) < 0.4 ∧ Π(a → (b ∨ c)) < 0.3

⇥ ∨ N(b) ≥ 0.7

Note: models are possibility distributions, which are interpreted as epistemic states Axiomatization Weighted version of the axiomatization of the modal logic KD without introspection (KR 2012)

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possibilistic logic GPL

formulas:

express lower bounds on the necessity of a propositional formula

models:

express propositional combinations of lower bounds on the necessity of a propositional formula

weighted epistemic states

minimally specific models:

unique 0, 1 or more weighted epistemic states

useful for reasoning about the consequences

  • f one’s own beliefs

useful for reasoning about the revealed beliefs

  • f another agent
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Inference relations

GPL naturally leads to one monotonic and two non-monotonic inference relations (all of which are equivalent in normal possibilistic logic):

ten K | =b Φ en K | =c Φ

iff Φ is true in at least one minimally specific model of K iff Φ is true in all minimally specific models of K

Brave reasoning Cautious reasoning

ten K | =b Φ

iff Φ is true in all models of K

Standard reasoning

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Computational complexity

| = | =b | =c GPL coNP ΣP

2

ΠP

2

GPL∆ ΘP

2

ΣP

2

ΠP

2

GPL∆

R

ΠP

3

ΣP

4

ΠP

4

Same complexity as inference in (disjunctive) answer set programming and many non-monotonic reasoning frameworks Extension of the language of GPL with a modality that corresponds to the guaranteed possibility measure Extension based on a refinement of the guaranteed possibility measure

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  • 1. Possibilistic logic
  • 2. Generalized possibilistic logic
  • 3. Answer set programming
  • 4. Extending answer set programming
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Answer set programming

a1 ∨ ¬a2 ∨ a3 ← b1 ∧ b2 ∧ not c1 ∧ not c2

A (disjunctive) answer set program (ASP) consists of rules of the form: Intuition: if we already know that b1 and b2 are true, then we should

accept that a1 is true, that a2 is false, or that a3 is true, unless we know that c1 or c2 is true.

head body

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Answer set programming

A (disjunctive) answer set program (ASP) consists of rules of the form: Intuition: if we already know that b1 and b2 are true, then we should

accept that a1 is true, that a2 is false, or that a3 is true, unless we know that c1 or c2 is true.

Usual way to define the semantics is in terms of the Gelfond-Lifschitz reduct: guess what can be derived from the program, use this guess to evaluate literals preceded by not, and verify that the guess was correct The set of literals that can be derived from the program is then called an answer set or stable model

a1 ∨ ¬a2 ∨ a3 ← b1 ∧ b2 ∧ not c1 ∧ not c2

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Answer set programming

1 the program w ← not c c ← not w

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Answer set programming

1 the program w ← not c c ← not w is it ? 2

guess

{w}

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Answer set programming

1 the program w ← not c c ← not w 3

reduct

w ← not c c ← not w is it ? 2

guess

{w}

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Answer set programming

1 the program w ← not c c ← not w 3

reduct

w ← not c c ← not w is it ? 2

guess

{w} Minimal model of the reduct is indeed {w}, which means that {w} is an answer set (or stable model)

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Relationship with GPL

We associate a GPL theory ϴP with an answer set program P as follows

l1 ... ln ⇤ p1 ⌥ ... ⌥ pm ⌥ not c1 ⌥ ... ⌥ not cr

ASP rule GPL formula

N1(p1) ^ ... ^ N1(pm) ^ Π1(¬c1) ^ ... ^ Π1(¬cr) ! N1(l1) _ ... _ N1(ln)

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Relationship with GPL

A set of literals M is an answer set of P iff the possibility distribution π defined as

π(ω) =

  • 1

if ω | = l for all l ∈ M

  • therwise

is a minimally specific model of ϴP Note that π is the least specific possibility distribution that satisfies N1(l) for each l in M

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Relationship with GPL: intuition

There are three ways of making the following GPL formula satisfied: Make N1(pi) false Make 𝚸1(¬ci) false, i.e. make N1/k(ci) true Make N1(li) true

N1(p1) ^ ... ^ N1(pm) ^ Π1(¬c1) ^ ... ^ Π1(¬cr) ! N1(l1) _ ... _ N1(ln)

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Relationship with GPL: intuition

There are three ways of making the following GPL formula satisfied: Make N1(pi) false Make 𝚸1(¬ci) false, i.e. make N1/k(ci) true Make N1(li) true

This will always be preferred when we are looking for minimally specific models

N1(p1) ^ ... ^ N1(pm) ^ Π1(¬c1) ^ ... ^ Π1(¬cr) ! N1(l1) _ ... _ N1(ln)

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Relationship with GPL: intuition

There are three ways of making the following GPL formula satisfied: Make N1(pi) false Make 𝚸1(¬ci) false, i.e. make N1/k(ci) true Make N1(li) true

This corresponds to the guess that ci will be derivable Boolean models are those for which the guess can be verified, i.e. those for which N1(ci) can be derived

N1(p1) ^ ... ^ N1(pm) ^ Π1(¬c1) ^ ... ^ Π1(¬cr) ! N1(l1) _ ... _ N1(ln)

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Relationship with GPL: intuition

There are three ways of making the following GPL formula satisfied: Make N1(pi) false Make 𝚸1(¬ci) false, i.e. make N1/k(ci) true Make N1(li) true

This intuitively corresponds to forward chaining In minimal models, this option will only be chosen if all of the pi literals can be derived and none of the ci literals.

N1(p1) ^ ... ^ N1(pm) ^ Π1(¬c1) ^ ... ^ Π1(¬cr) ! N1(l1) _ ... _ N1(ln)

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Relationship with GPL

P has an answer set iff:

ΘP | =b Φ

Let us define:

Φ ≡ ^

a∈At

N1(a) ∨ N1(¬a) ∨ (Π1(a) ∧ Π1(¬a))

The literal l is contained in at least one answer set of P iff:

ΘP | =b Φ ∧ N1(l)

The literal l is contained in all answer sets of P iff:

ΘP | =c Φ → N1(l)

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Restricting the set of certainty degrees

N1(α) = ¬Π1(¬α)

When only two certainty degrees, 0 and 1, are used, we have This would mean that the following two ASP rules are incorrectly translated to the same GPL formula

a ← not b b ← not a

The characterization of ASP requires that at least 3 certainty degrees are considered

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  • 1. Possibilistic logic
  • 2. Generalized possibilistic logic
  • 3. Answer set programming
  • 4. Extending answer set programming
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Disjunction inside N

strong disjunction weak disjunction

loc(IJCAI2016, NewYork) ∨ loc(IJCAI2016, SanFrancisco) ←

N1

  • loc(IJCAI2016, NewYork)
  • ∨ N1
  • loc(IJCAI2016, SanFrancisco)
  • N1
  • loc(IJCAI2016, NewYork) ∨ loc(IJCAI2016, SanFrancisco)
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Stable models of propositional theories

((a ← b) ∧ (c ← not d)) ∨ (x ∨ y ← z)

  • N1(b) → N1(a)
  • ∧
  • Π1(¬d) → N1(c)
  • ∨
  • N1(z) → N1(x) ∨ N1(y)
  • More flexible than equilibrium logic (e.g. weak disjunction)

More intuitive semantics: all models respect the idea of minimal commitment

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Possibilistic ASP

1 : loc(IJCAI2015, Argentina) ← 0.6 : loc(IJCAI2016, NewYork) ← not unreliable(website) 1 : loc(IJCAI2015, SouthAmerica) ← loc(IJCAI2015, Argentina) 1 : loc(IJCAI2016, NorthAmerica) ← loc(IJCAI2016, NewYork) 0.75 : ¬loc(IJCAI2015, SouthAmerica) ← loc(IJCAI2015, SouthAmerica) 0.5 : ¬loc(IJCAI2015, NorthAmerica) ← loc(IJCAI2015, SouthAmerica) Certainty weights indicate our belief that the rule is correct Semantics: possibility distribution

  • ver classical answer sets

Certainty weights indicate our belief in the conclusion of the rule, if the body is satisfied Semantics: set of possibilistic answer sets

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Possibilistic ASP

1 : loc(IJCAI2015, Argentina) ← 0.6 : loc(IJCAI2016, NewYork) ← not unreliable(website) 1 : loc(IJCAI2015, SouthAmerica) ← loc(IJCAI2015, Argentina) 1 : loc(IJCAI2016, NorthAmerica) ← loc(IJCAI2016, NewYork) 0.75 : ¬loc(IJCAI2015, SouthAmerica) ← loc(IJCAI2015, SouthAmerica) 0.5 : ¬loc(IJCAI2015, NorthAmerica) ← loc(IJCAI2015, SouthAmerica) Certainty weights indicate our belief that the rule is correct Semantics: possibility distribution

  • ver classical answer sets

Certainty weights indicate our belief in the conclusion of the rule, if the body is satisfied Semantics: set of possibilistic answer sets

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Possibilistic ASP

Intuition: certainty(head) ≥ min(l/k, certainty(body))

l k : a1 ∨ ... ∨ an ← b1 ∧ ... ∧ bm ∧ not c1 ∧ ... ∧ not cr

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Possibilistic ASP

l k : a1 ∨ ... ∨ an ← b1 ∧ ... ∧ bm ∧ not c1 ∧ ... ∧ not cr

l

^

i=1

  • N i

k (b1) ∧ ... ∧ N i k (bm) ∧ Π i k (¬c1) ∧ ... ∧ Π i k (¬cr) → N i k (a1) ∨ ... ∨ N i k (an)

  • assume that l is even

Intuition: for each i≤l, if the body can be derived with certainty i/k then we should also believe the head with certainty i/k

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Conclusions

Possibilistic logic offers a convenient way to encode the epistemic state of an agent ➡ natural characterization of (AGM) belief revision and default reasoning Generalized possibilistic logic (GPL) offers a convenient way to encode

  • ur knowledge about the possible epistemic states of another agent

➡ natural characterization of NMR frameworks based on multiple extensions,

such as answer set programming (ASP)

The GPL characterization of ASP suggests several natural generalizations

➡ Weak disjunction ➡ Stable models of arbitrary propositional theories ➡ Modelling uncertainty in answer set programming

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References and acknowledgments

Didier Dubois, Henri Prade, Steven Schockaert. Stable models in generalized possibilistic logic, Proc. KR, 2012 Didier Dubois, Henri Prade, Steven Schockaert. Reasoning about uncertainty and explicit ignorance in generalized possibilistic logic, Proc. ECAI, 2014 Kim Bauters, Steven Schockaert, Martine De Cock, Dirk Vermeir. Semantics for possibilistic answer set programs: uncertain rules versus rules with uncertain conclusions, International Journal of Approximate Reasoning, 2014 Kim Bauters, Steven Schockaert, Martine De Cock, Dirk Vermeir. Possibilistic answer set programming revisited, Proc. UAI, 2010

Generalized possibilistic logic Possibilistic answer set programming