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. 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. 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. 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. 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. 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. 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. 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. 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. and R are not independent

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. 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. 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. THANK YOU