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# On the Computational Complexity of Intuitionistic Hybrid Modal Logic

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### On the Computational Complexity of Intuitionistic Hybrid Modal Logic

1. 1. On the Computational Complexity of Intuitionistic Hybrid Modal Logic Edward Hermann Hausler Mario Benevides Valeria de Paiva Alexandre Rademaker PUC-Rio UFRJ Univ. Birmingham - UK FGV - IBM Research
2. 2. Basic Motivation Some facts Description Logic is among the best logical frameworks to represent knowledge. Powerful language expression plus decidability (TBOX PSPACE, TBOX+ABOX EXPTIME). Deontic logic approach to legal knowledge representation brings us paradoxes (contrary-to-duty paradoxes); ALC, as a basic DL, might be considered to legal knowledge representation if it can deal with the paradoxes; Considering a jurisprudence basis, classical ALC it is not adequate to our approach.
3. 3. Basic Motivation Our approach An intuitionistic version of ALC tailored to represent legal knowledge. Dealing with the paradoxes. A proof-theoretical basis to legal reasoning and explanation. PSPACE completeness of iALC.
4. 4. iALC and ALC have the same logical language Binary (Roles) and unary (Concepts) predicate symbols, R(x, y) and C(y). Prenex Guarded formulas (∀y(R(x, y) → C(y)), ∃y(R(x, y) ∧ C(y))). Essentially propositional (Tboxes), but may involve reasoning on individuals (Aboxes), expressed as “i : C”. Semantics: Provided by a structure I = (∆I, I, ·I) closed under reﬁnement, i.e., y ∈ AI and x I y implies x ∈ AI. “¬” and “ ” must be interpreted intuitionistically . It is not First-order Intuitionistic Logic. It is a genuine Intuitionistic Hybrid logic (IHK).
5. 5. Using iALC to formalize Conﬂict of Laws in Space A Case Study Peter 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 in Rio de Janeiro. Peter is 17 and Maria is 20. Peter lives in Edinburgh and Maria lives in Rio. Only legally capable individuals have civil obligations: PeterLiable ContractHolds@RioCourt, shortly, pl cmp MariaLiable ContractHolds@RioCourt, shortly, ml cmp Concepts, nominals and their relationships BR is the collection of Brazilian Valid Legal Statements SC is the collection of Scottish Valid Legal Statements PILBR is the collection of Private International Laws in Brazil ABROAD is the collection of VLS outside Brazil LexDomicilium is a legal connection: Legal Connections The pair pl, pl is in LexDomicilium
6. 6. Non-Logical Axiom Sequents The sets ∆, of concepts, and Ω, of iALC sequents representing the knowledge about the case ∆ = ml : BR pl : SC pl cmp ml cmp pl LexDom pl Ω = PILBR ⇒ BR SC ⇒ ABROAD ∃LexD1.L1 . . . ∃LexDom.ABROAD . . . ∃LexDk.Lk ⇒ PILBR
7. 7. In Sequent Calculus ∆ ⇒ pl : SC Ω pl : SC ⇒ pl : A cut ∆ ⇒ pl : A ∆ ⇒ pl LexD pl ∃ − R ∆ ⇒ pl : ∃LexD.A ∃LexD.A ⇒ ∃LexD.A -R ∃LexD.A ⇒ PILBR Ω PILBR ⇒ BR cut ∃LexD.A ⇒ BR inc − R ∆ ⇒ pl : BR ∆ ⇒ ml : BR Π ∆ ⇒ pl : BR Ω ml : BR, pl : BR ⇒ cmp : BR cut ∆, ml : BR ⇒ cmp : BR cut ∆ ⇒ cmp : BR
8. 8. 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 | @ni ϕ Let M = W, , R, {Vw }w∈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).
9. 9. 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 | @ni ϕ Let M = W, , R, {Vw }w∈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 |= ⊥ and M, w, i |= Nom M, w, i |= nj iff i = j C M, w, i |= α ∧ β, iff, M, w, i |= α and M, w, i |= β D M, w, i |= α ∨ β, iff, M, w, i |= α or M, w, i |= β NEG M, w |= ¬α, iff, for all w , w ≤ w , M, w |= α IMP M, w, i |= α1 → α2 iff for all v, w v, if M, v, i |= α1 then M, v, i |= α2; IBOX M, w, i |= 2α iff for all v, w v, for all k ∈ W, if iRk then M, v, i |= α; IDIA M, w, i |= 3α iff there is k ∈ W, iRk and M, k, i |= α Anchor M, w, i |= @nj α iff M, w, j |= α
10. 10. and R are not independent
11. 11. Metatheorems on IHK IHK is sound and complete regarded IHK frames. IPL ⊂ IHK (lower bound is PSPACE) Alternating Polynomial Turing-Machine to ﬁnd out winner-strategy on the SAT-Game adapted from Areces2000 (upper-bound is PSPACE).
12. 12. IHK is PSPACE-complete SATIHK ⊂ PSPACE One wants fo verify whether Γ → γ is satisﬁable. Γ → γ is satisﬁable, if and only if, ( θ∈Γθ) → γ is satisﬁable in a model of Γ. A game is deﬁned on Γ ∪ {γ} ∃loise starts by playing a list {L0, . . . , Lk } of Γ ∪ {γ}-Hintikka I-sets, and two relations R and on them. ∃loise loses if she cannot provide the list as a pre-model. ∀belard chooses a set from the list and a formula inside this set. ∃loise has to verify/extend the (pre)-model in order to satisfy the formula. Γ ∪ γ is satisﬁable, iff, ∃loise has a winning strategy. ∆-Hintikka I-set is a maximal prime consistent set of subformulas from ∆.
13. 13. 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;
14. 14. THANK YOU