Proof-search and countermodel generation for non-normal modal logics › gncs › wp-content › uploads › sites › … · Non-Normal Modal Logics Bi-neighbourhood semantics Sequent

Non-Normal Modal LogicsBi-neighbourhood semantics

Sequent calculi LSE∗PRONOM

Hypersequent Calculi HE?

HYPNOConclusions

Universitá degli Studi di Torino

Proof-search and countermodel generation fornon-normal modal logics

Gian Luca POZZATO

Progetti GNCS 2019, Hotel Belvedere, Montecatini Terme

13 febbraio 2020

Gian Luca POZZATO 13 febbraio 2020 Proof theory for NNMLs 1 / 46




HYPNOConclusions


Table of Contents1. Non-Normal Modal Logics2. Bi-neighbourhood semantics

Basic System EExtensions

3. Sequent calculi LSE∗

4. PRONOMFinding a derivationCounter-model construction

5. Hypersequent Calculi HE?

6. HYPNOFinding a derivationCounter-model construction

7. ConclusionsGian Luca POZZATO 13 febbraio 2020 Proof theory for NNMLs 2 / 46




HYPNOConclusions












HYPNOConclusions


Non-normal modal logics

Generalization of ordinary modal logics that do not satisfy someaxioms or rules of normal modal logic K

Why NNMLAvoid the commitment to some logic principles that are theoremsof K but are not acceptable under possible interpretations of themodality:• Deontic• Epistemic• · · ·





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• Epistemic reasoning• 2A read as “the agent knows/believes A”• NNML offer a partial solution to the problem of omniscience

• 2B does not follow from 2A and 2A→ 2B• rejection of the rule of monotonicity: A→ B implies

2A→ 2B

• Deontic logic• 2A interpreted as “it is obligatory that A”• NNML offers solutions to some paradoxes

• Ross’ paradox• gentle-murder paradox





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Systems considered

E := CPL + A→ B B → ARE2A→ 2B

• Extensions:

M I 2(A ∧ B)→ 2A

C I 2A ∧2B → 2(A ∧ B)

N I 2>E

EM

EC EN

EMC EMN

ECN

EMCN (K)





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HYPNOConclusions


Bi-neighbourhood semantics

Bi-neighbourhood modelsTriplesM = 〈W ,N ,V 〉• W is a non-empty set of worlds (states)• V is a valuation function• N is a bi-neighbourhood function W −→ P(P(W)× P(W))

such that (α, β) ∈ N (w) implies α ∩ β = ∅Intuition: α and β provide independent positive and negative evi-dence for A

Truth of 2Aw ∈ [2A] iff there is (α, β) ∈ N (w) such that

for all u ∈ α, u ∈ [A], and for all v ∈ β, v 6∈ [A]





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Bi-neighbourhood semantics

Model properties

C-models: (α1, β1), (α2, β2) ∈ N (w) implies(α1 ∩ α2, β1 ∪ β2) ∈ N (w)

N-models: (W , ∅) ∈ N (w) for all w ∈WM-models: (α, β) ∈ N (w) implies β = ∅

CompletenessLogic E(M,C,N) is sound and complete w.r.t. bi-neighbourhood(M,C,N-)models





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Desiderata

• Analytic• Standard

• Finite numbers of rules each one with finite and fixed numberof premises

• Modular• Fixed set of basic rules• Extensions obtained by adding suitable rules expressing

semantic properties

• Terminating• Optimal• Countermodel generation

• Obtain directly a countermodel from a failed proof search





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The Sequent Calculi LSE∗

Labels• World labels x , y , z , ...• Neighbourhood labels a, b, c, ...

Neighbourhood terms• Positive terms [a1, ..., an] = finite sets of neighbourhood labels• Negative terms t, where t is a positive term• Neighbourhood constants τ , τ

Intuition• t and t are the two members of a pair of neighbourhoods• [a1, ..., an] ; a1 ∩ ... ∩ an [a1, ..., an] ; a1 ∪ ... ∪ an τ ; W





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The Sequent Calculi LSE∗

Initial sequents: x : p, Γ ⇒ ∆, x : p x : ⊥, Γ ⇒ ∆ Γ ⇒ ∆, x : >x ∈ t, x : A, t ∀ A, Γ ⇒ ∆

L ∀x ∈ t, t ∀ A, Γ ⇒ ∆

x ∈ t, Γ ⇒ ∆, x : AR ∀

Γ ⇒ ∆, t ∀ A

x ∈ t, x : A, Γ ⇒ ∆L ∃

t ∃ A, Γ ⇒ ∆

x ∈ t, Γ ⇒ ∆, x : A, t ∃ AR ∃

x ∈ t, Γ ⇒ ∆, t ∃ A

[a] ∈ N (x), [a] ∀ A, Γ ⇒ ∆, [a] ∃ AL2

x : 2A, Γ ⇒ ∆

t ∈ N (x), Γ ⇒ ∆, x : 2A, t ∀ A t ∈ N (x), t ∃ A, Γ ⇒ ∆, x : 2AR2

t ∈ N (x), Γ ⇒ ∆, x : 2A

Mt ∈ N (x), y ∈ t, Γ ⇒ ∆

τ ∈ N (x), Γ ⇒ ∆Nτ

Γ ⇒ ∆Nτ

x ∈ τ, Γ ⇒ ∆

[a1, ..., an ] ∈ N (x), [a1] ∈ N (x), ..., [an ] ∈ N (x), Γ ⇒ ∆C

[a1] ∈ N (x), ..., [an ] ∈ N (x), Γ ⇒ ∆

x ∈ [a1], ..., x ∈ [an ], Γ ⇒ ∆dec

x ∈ [a1, ..., an ], Γ ⇒ ∆

x ∈ [a1], Γ ⇒ ∆ ... x ∈ [an ], Γ ⇒ ∆dec

x ∈ [a1, ..., an ], Γ ⇒ ∆

Application conditions:

x is fresh in R ∀ and L ∃, a is fresh in L2, and x occurs in the conclusion of Nτ .





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Examples of derivations

Axiom M..., y : A, y : B, y : [a], [a] : A ∧ B ⇒ y : A, ...

L∧..., y : A ∧ B, y : [a], [a] : A ∧ B ⇒ y : A, ...

L ∀..., y : [a], [a] : A ∧ B ⇒ y : A, ...

R ∀..., [a] : A ∧ B ⇒ [a] : A, ...

M..., y : [a], y : A, [a] : x ⇒ ...

L ∃..., [a] : A, [a] : x ⇒ ...

R2[a] : x, [a] : A ∧ B ⇒ x : 2A, [a] : A ∧ B

L2x : 2(A ∧ B) ⇒ x : 2A





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Example of failed proof search and countermodel

Axiom M in LSE∗

A cl.

A′ cl. y : [a], y : p, [a] : x, [a] : p ∧ q ⇒ x : 2p, [a] : p ∧ q, y : qL∧

y : [a], y : p, [a] : x, [a] : p ∧ q ⇒ x : 2p, [a] : p ∧ q, y : p ∧ qR ∃

y : [a], y : p, [a] : x, [a] : p ∧ q ⇒ x : 2p, [a] : p ∧ qL ∃

[a] : x, [a] : p ∧ q, [a] : p ⇒ x : 2p, [a] : p ∧ qR2

[a] : x, [a] : p ∧ q ⇒ x : 2p, [a] : p ∧ qL2

x : 2(p ∧ q) ⇒ x : 2p

saturated branch B

Bi-neighbourhood countermodelMW = {x, y} N (x) = {(α[a], α[a]

)} α[a] = ∅ V (p) = {y}N (y) = ∅ α

[a]= {y} V (q) = ∅

We have M, x |= 2(p ∧ q) and M, x 6|= 2p





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Results

AXIOMATICSYSTEMS E∗

SEMANTICSBi-neighbourhood

models

PROOF SYSTEMSLabelled calculi

Decision procedures

S.&C. Cut elim.

S. & C.

F.m.p. Saturation





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PRONOM

Basic ideas• inspired by the “lean” methodology of leanTAP• set of clauses, each one implementing a sequent rule or an

axiom of LSE• proof search provided for free by DFS of Prolog• predicateterminating_proof_search(Gamma,Delta,Derivation).• when it fails PRONOM executes the predicatebuild_saturate_branch(Gamma,Delta,Model).





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PRONOM

Clause for axiomterminating_proof_search(Neigh,Gamma,Delta,tree(axiom),_,_,_):-

member([X,A],Gamma),member([X,A],Delta),!.





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PRONOM

Clause for R2terminating_proof_search(Neigh,Gamma,Delta,

tree(rbox,LeftTree,RightTree),RBox,RExist,LAll):-member([X,box A],Delta),member([X,SpOfX],Neigh),member(T,SpOfX),\+member([X,A,T],RBox),!,terminating_proof_search(Neigh,Gamma,[[forall,T,0,A]|Delta],

LeftTree,[[X,A,T]|RBox],RExist,LAll),terminating_proof_search(Neigh,[[exists,T,1,A]|Gamma],Delta,

RightTree,[[X,A,T]|RBox],RExist,LAll).





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Counter-model

Basic ideas• predicate build_saturate_branch• build an open, saturated branch• clauses are the same as the predicate

terminating_proof_search, however rules introducing a branchin a backward proof search are implemented by pairs of(disjoint) clauses, each one representing an attempt to buildan open saturated branch





HYPNOConclusions


PRONOM

Clauses for R2build_saturate_branch(Neigh,Gamma,Delta,Model,RBox,RExist,LAll):-

member([X,box A],Delta),member([X,SpOfX],Neigh),member(T,SpOfX),\+member([X,A,T],RBox),build_saturate_branch(Neigh,Gamma,[[forall,T,0,A]|Delta],Model,

[[X,A,T]|RBox],RExist,LAll).build_saturate_branch(Neigh,Gamma,Delta,Model,RBox,RExist,LAll):-

member([X,box A],Delta),member([X,SpOfX],Neigh),member(T,SpOfX),\+member([X,A,T],RBox),build_saturate_branch(Neigh,[[exists,T,1,A]|Gamma],Delta,Model,

[[X,A,T]|RBox],RExist,LAll).





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Web application

http://193.51.60.97:8000/pronom/


http://193.51.60.97:8000/pronom/




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HYPNOConclusions



Basic ideas• internal calculi• direct interpretation of a sequent into a NNML formula• easier to describe a decision procedure• block 〈Σ〉, where Σ is a multiset of formulas of L• sequent: pair Γ⇒ ∆, where Γ is a multiset of formulas and

blocks, and ∆ is a multiset of formulas





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Basic ideas• hypersequent S1 | ... | Sn, where S1, ...,Sn are sequents• single sequents interpreted as:

i(A1, ...,An, 〈Σ1〉, ..., 〈Σm〉 ⇒ B1, ...,Bk) =

=∧i≤n

Ai ∧∧j≤m

2∧

Σj →`≤k B`

• validity of a sequent S in a bi-neighbourhood modelMM |= S : for all w ∈M,M,w i(S)

• ifM |= S for some S ∈ H, H is valid inM• H is valid if it is valid in all models of that kind





HYPNOConclusions



initG | p, Γ⇒ ∆, p

⊥LG | ⊥, Γ⇒ ∆

>RG | Γ⇒ ∆,>

G | A→ B, Γ⇒ ∆,A G | B,A→ B, Γ⇒ ∆→LG | A→ B, Γ⇒ ∆

G | A, Γ⇒ ∆,A→ B,B→RG | Γ⇒ ∆,A→ B

G | 〈A〉,2A, Γ⇒ ∆2L

G | 2A, Γ⇒ ∆

G | 〈Σ〉, Γ⇒ ∆,2B | Σ⇒ BM2R

G | 〈Σ〉, Γ⇒ ∆,2B

G | 〈Σ〉, Γ⇒ ∆,2B | Σ⇒ B G | 〈Σ〉, Γ⇒ ∆,2B | B V Σ2R

G | 〈Σ〉, Γ⇒ ∆,2B

G | A⇒ BV1

G | AV B

G | A⇒ B G | AV ΣV2 |Σ| ≥ 1

G | AV B,Σ

G | 〈>〉, Γ⇒ ∆N

G | Γ⇒ ∆

G | 〈Σ,Π〉, 〈Σ〉, 〈Π〉, Γ⇒ ∆C

G | 〈Σ〉, 〈Π〉, Γ⇒ ∆

Figure: Rules of HE? .Gian Luca POZZATO 13 febbraio 2020 Proof theory for NNMLs 27 / 46




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Implementation

Basic ideas• Same ideas of PRONOM• Prolog implementation inspired by leanTAP• hypersequent represented with a Prolog list whose elements

are Prolog terms:• singleSeq(Gamma,Delta,Additional)• each one represents a sequent Γ⇒ ∆ in the hypersequent• Additional: Prolog list for termination

• Two main predicates for finding either a derivation or acountermodel





HYPNOConclusions


Clause for axiomterminating_proof_search(Hyper,tree(axiom,PrintableHyper,no,no)):-

member(singleSeq([Gamma,Delta],_),Hyper),member(P,Gamma), member(P,Delta),!,extractPrintableSequents(Hyper,PrintableHyper).





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Clause for 2R

terminating_proof_search(Hyper,tree(rbox,PrintableHyper,Sub1,Sub2)):-select(singleSeq([Gamma,Delta],Additional),Hyper,NewHyper),member(Sigma,Gamma),is_list(Sigma),member(box B,Delta),list_to_ord_set(Sigma,SigmaOrd),\+member(apdR(SigmaOrd,B),Additional),!,terminating_proof_search([singleSeq([Sigma,[B]],[])|[singleSeq([Gamma,Delta],[apdR(SigmaOrd,B)|Additional])|NewHyper]],Sub1),

terminating_proof_search([singleSeq([[],[B => Sigma]],[])|[singleSeq([Gamma,Delta],[apdR(SigmaOrd,B)|Additional])|NewHyper]],Sub2),

extractPrintableSequents(Hyper,PrintableHyper).





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Countermodel constructionbuild_saturated_branch(Hyper,Model):-

select(singleSeq([Gamma,Delta],Additional),Hyper,NewHyper),member(Sigma,Gamma),is_list(Sigma),member(box B,Delta),list_to_ord_set(Sigma,SigmaOrd),\+member(apdR(SigmaOrd,B),Additional),build_saturated_branch([singleSeq([Sigma,[B]],[])|

[singleSeq([Gamma,Delta],[apdR(SigmaOrd,B)|Additional])|NewHyper]],Model).

build_saturated_branch(Hyper,Model):-select(singleSeq([Gamma,Delta],Additional),Hyper,NewHyper),member(Sigma,Gamma),is_list(Sigma),member(box B,Delta),list_to_ord_set(Sigma,SigmaOrd),\+member(apdR(SigmaOrd,B),Additional),build_saturated_branch([singleSeq([[],[B => Sigma]],[])|

[singleSeq([Gamma,Delta],[apdR(SigmaOrd,B)|Additional])|NewHyper]],Model).





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Implementation

• Last clause:

Countermodelbuild_saturated_branch(Hyper,model(Hyper)):-\+instanceOfAnAxiom(Hyper).





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Web application

http://193.51.60.97:8000/HYPNO/


http://193.51.60.97:8000/HYPNO/




HYPNOConclusions












HYPNOConclusions



Our contribution• sequent and hypesequent calculi for NNMLs• first programs providing both proof-search and countermodel

generation for the whole cube of NNML

Future Works• use of free variables for term instantiation• implement an automated transformation of bi-neighbourhood

countermodels into standard neighbourhood models





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Best student paper award

Gian Luca Pozzato





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Description Logics

Description Logics• Important formalisms of knowledge representation• Two key advantages:

• well-defined semantics based on first-order logic• good trade-off between expressivity and complexity

• at the base of languages for the semantic (e.g. OWL)

Knowledge bases

• Two components:

• TBox=inclusion relations among concepts (e.g.Dog v Mammal)

• ABox= instances of concepts and roles = properties andrelations among individuals (e.g. Dog(saki))





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Description Logics




Knowledge bases• Two components:







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Description Logics











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Description Logics











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Description Logics

DLs with nonmonotonic features• need of representing prototypical properties and of reasoning

about defeasible inheritance• handle defeasible inheritance needs the integration of some

kind of nonmonotonic reasoning mechanism• DLs + MKNF• DLs + circumscription• DLs + default

• all these methods present some difficulties ...





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Description Logics

DLs with typicality• Non-monotonic extensions of Description Logics for reasoning

about prototypical properties and inheritance with exceptions• Basic idea: to extend DLs with a typicality operator T• T (C ) singles out the “most normal” instances of the concept C

• semantics of T defined by a set of postulates that are arestatement of Lehmann-Magidor axioms of rational logic

Basic notions• A KB comprises assertions T (C) v D

• T (Dog) v Affectionate means “normally, dogs are affectionate”

• T is nonmonotonic

• C v D does not imply T (C ) v T (D)





HYPNOConclusions


Description Logics

DLs with typicality• Non-monotonic extensions of Description Logics for reasoning

about prototypical properties and inheritance with exceptions• Basic idea: to extend DLs with a typicality operator T• T (C ) singles out the “most normal” instances of the concept C• semantics of T defined by a set of postulates that are a

restatement of Lehmann-Magidor axioms of rational logic

Basic notions• A KB comprises assertions T (C) v D

• T (Dog) v Affectionate means “normally, dogs are affectionate”

• T is nonmonotonic

• C v D does not imply T (C ) v T (D)





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Description Logics

ExampleT (Pig) v ¬FireBreathingT (Pig u Pokemon) v FireBreathing

Reasoning

• ABox:

• Pig(tepig)

• Expected conclusions:

• ¬FireBreathing(tepig)





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Description Logics


Reasoning• ABox:

• Pig(tepig)

• Expected conclusions:

• ¬FireBreathing(tepig)





HYPNOConclusions


Description Logics


Reasoning• ABox:

• Pig(tepig)

• Expected conclusions:• ¬FireBreathing(tepig)





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Description Logics

ExampleT (Pig) v ¬FireBreathingT (Pig t Pokemon) v FireBreathing

Reasoning• ABox:

• Pig(tepig),Pokemon(tepig)

• Expected conclusions:• FireBreathing(tepig)





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Description Logics

Semantics• = 〈∆I , <, .I 〉

• additional ingredient: preference relation among domainelements

• < is an irreflexive, transitive, modular and well-foundedrelation over ∆:• for all S ⊆ ∆I , for all x ∈ S , either x ∈ Min<(S) or∃y ∈ Min<(S) such that y < x

• Min<(S) = {u : u ∈ S and @z ∈ S s.t. z < u}• Semantics of the T operator: (T (C ))I = Min<(C I )





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Description Logics

Rational closure• Preference relation among models of a KB

• M1 < M2 if M1 contains less exceptional (not minimal)elements

• M minimal model of KB if there is no M ′ model of KB suchthat M ′ < M

• Minimal entailment

• KB |=min F if F holds in all minimal models of KB

• Nonmonotonic logic

• KB |=min F does not imply KB’ |=min F with KB’ ⊃ KB

• Corresponds to a notion of rational closure of KB





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Description Logics





• KB |=min F if F holds in all minimal models of KB

• Nonmonotonic logic

• KB |=min F does not imply KB’ |=min F with KB’ ⊃ KB

• Corresponds to a notion of rational closure of KB





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Description Logics





• KB |=min F if F holds in all minimal models of KB• Nonmonotonic logic

• KB |=min F does not imply KB’ |=min F with KB’ ⊃ KB• Corresponds to a notion of rational closure of KB





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Description Logics

Probabilities for Concept combination• Further extension: p :: T (C ) v D

• p ∈ (0.5, 1) degree of belief• prototype of concepts CH and CM

• distributed semantics• selection of typicality inclusion to be considered as true V

different scenarios• probability distribution over scenarios

• DISPONTE semantics + heuristics from cognitive semanticsfor concept combination• Prototype of the combined concept CH u CM





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Description Logics

Application to RaiPlay (with Centro Ricerche RAI)• Recommender system for RaiPlay• New genres obtained by combining basic ones• Glass box approach• Prototypes from data


Documents

Proof-search and countermodel generation for non-normal modal logics › gncs › wp-content › uploads › sites › … · Non-Normal Modal Logics Bi-neighbourhood semantics Sequent