On the Computational Complexity of IntuitionisticHybrid Modal Logic

Edward Hermann HauslerMario BenevidesValeria de Paiva

Alexandre Rademaker

PUC-RioUFRJ

Univ. Birmingham - UKFGV - IBM Research

Basic Motivation

Some factsI Description Logic is among the best logical frameworks to

represent knowledge.I Powerful language expression plus decidability (TBOX PSPACE,

TBOX+ABOX EXPTIME).I Deontic logic approach to legal knowledge representation brings

us paradoxes (contrary-to-duty paradoxes);I ALC, as a basic DL, might be considered to legal knowledge

representation if it can deal with the paradoxes;I Considering a jurisprudence basis, classical ALC it is not

adequate to our approach.

Basic Motivation

Our approach

I An intuitionistic version of ALC tailored to represent legalknowledge.

I Dealing with the paradoxes.I A proof-theoretical basis to legal reasoning and explanation.I PSPACE completeness of iALC.

iALC and ALC have the same logical language

I Binary (Roles) and unary (Concepts) predicate symbols, R(x , y)and C(y).

I Prenex Guarded formulas (∀y(R(x , y)→ C(y)),∃y(R(x , y) ∧ C(y))).

I Essentially propositional (Tboxes), but may involve reasoning onindividuals (Aboxes), expressed as “i : C”.

I Semantics: Provided by a structure I = (∆I ,I , ·I) closed underrefinement, i.e., y ∈ AI and x I y implies x ∈ AI . “¬” and “v”must be interpreted intuitionistically .

I It is not First-order Intuitionistic Logic. It is a genuine IntuitionisticHybrid logic (IHK).

Using iALC to formalize Conflict of Laws in Space

A Case StudyPeter and Maria signed a renting contract. The subject of the contract is an

apartment in Rio de Janeiro. The contract states that any dispute will go to court inRio de Janeiro. Peter is 17 and Maria is 20. Peter lives in Edinburgh and Maria livesin Rio.

Only legally capable individuals have civil obligations:PeterLiable ContractHolds@RioCourt , shortly, pl cmpMariaLiable ContractHolds@RioCourt , shortly, ml cmpConcepts, nominals and their relationshipsBR is the collection of Brazilian Valid Legal StatementsSC is the collection of Scottish Valid Legal StatementsPILBR is the collection of Private International Laws in BrazilABROAD is the collection of VLS outside BrazilLexDomicilium is a legal connection:

Legal Connections The pair 〈pl, pl〉 is in LexDomicilium

Non-Logical Axiom Sequents

The sets ∆, of concepts, and Ω, of iALC sequents representing theknowledge about the case

∆ =ml : BR pl : SC pl cmp

ml cmp pl LexDom pl

Ω =PILBR ⇒ BR

SC⇒ ABROAD∃LexD1.L1 . . . t ∃LexDom.ABROAD t . . . ∃LexDk.Lk ⇒ PILBR

In Sequent Calculus

∆⇒ pl : SCΩ

pl : SC⇒ pl : Acut

∆⇒ pl : A ∆⇒ pl LexD pl∃ − R

∆⇒ pl : ∃LexD.A

∃LexD.A⇒ ∃LexD.At-R

∃LexD.A⇒ PILBR

Ω

PILBR ⇒ BRcut

∃LexD.A⇒ BRinc − R

∆⇒ pl : BR

∆⇒ ml : BR

Π

∆⇒ pl : BR

Ω

ml : BR, pl : BR⇒ cmp : BRcut

∆,ml : BR⇒ cmp : BRcut

∆⇒ cmp : BR

The logic IHK

The language of IHK is described by the following grammar.

ϕ ::=p | ni | ⊥ | > | ¬ϕ | ϕ1 ∧ ϕ2 | ϕ1 ∨ ϕ2 | ϕ1 → ϕ2 | 2 | 3ϕ | @niϕ

LetM = 〈W ,,R, Vww∈W 〉 be a Kripke model for IHML, w ∈ W and α be an IHML formula.

1. W is the (non-empty) set of worlds, partially ordered by ;

2. R ⊆ W ×W ;

3. for each w , Vw is a function from Φ to 2W , such that, if w v then Vw (p) ⊆ Vv (p).

The logic IHK

The language of IHK is described by the following grammar.

ϕ ::=p | ni | ⊥ | > | ¬ϕ | ϕ1 ∧ ϕ2 | ϕ1 ∨ ϕ2 | ϕ1 → ϕ2 | 2 | 3ϕ | @niϕ

LetM = 〈W ,,R, Vww∈W 〉 be a Kripke model for IHML, w ∈ W and α be an IHML formula.The satisfaction relation,M,w , i |= α, is defined inductively as follows:

A M,w , i |= p iff i ∈ Vw (p);

B M,w , i 6|= ⊥ andM,w , i |= >Nom M,w , i |= nj iff i = j

C M,w , i |= α ∧ β, iff,M,w , i |= α andM,w , i |= β

D M,w , i |= α ∨ β, iff,M,w , i |= α orM,w , i |= β

NEG M,w |= ¬α, iff, for all w ′, w ≤ w ′,M,w ′ 6|= α

IMP M,w , i |= α1 → α2 iff for all v , w v , ifM, v , i |= α1 thenM, v , i |= α2;

IBOX M,w , i |= 2α iff for all v , w v , for all k ∈ W , if iRk thenM, v , i |= α;

IDIA M,w , i |= 3α iff there is k ∈ W , iRk andM, k , i |= α

Anchor M,w , i |= @njα iffM,w , j |= α

and R are not independent

Metatheorems on IHK

I IHK is sound and complete regarded IHK frames.I IPL ⊂ IHK (lower bound is PSPACE)I Alternating Polynomial Turing-Machine to find out winner-strategy

on the SAT-Game adapted from Areces2000 (upper-bound isPSPACE).

IHK is PSPACE-complete

SATIHK ⊂ PSPACE

I One wants fo verify whether Γ→ γ is satisfiable.I Γ→ γ is satisfiable, if and only if, (uθ∈Γθ)→ γ is satisfiable in a model of Γ. A

game is defined on Γ ∪ γI ∃loise starts by playing a list L0, . . . , Lk of Γ ∪ γ-Hintikka I-sets, and two

relations R and 2 on them.I ∃loise loses if she cannot provide the list as a pre-model.I ∀belard chooses a set from the list and a formula inside this set.I ∃loise has to verify/extend the (pre)-model in order to satisfy the formula.I Γ ∪ γ is satisfiable, iff, ∃loise has a winning strategy.

∆-Hintikka I-set is a maximal prime consistent set of subformulas from ∆.

IHK is PSPACE-complete

1. Ladner proved that Sat for K , S4 and KD are PSPACE-complete;2. Using Gödel translation it is proved that IPL is PSPACE-complete;3. Wolter and Zakharyaschev proved that IK is translated in S4⊗ K ,

so IK is PSPACE-complete;4. Every logic containing IK is PSPACE-Hard;5. IK ⊆ IHK . No known translation of Hybrid language into ordinary

Modal Logic;6. We prove that IHK is in PSPACE using the method of

Areces 2000 extended to Intuitionistic Modal models;

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