2,664 research outputs found
Characterizing the NP-PSPACE Gap in the Satisfiability Problem for Modal Logic
There has been a great of work on characterizing the complexity of the
satisfiability and validity problem for modal logics. In particular, Ladner
showed that the validity problem for all logics between K, T, and S4 is {\sl
PSPACE}-complete, while for S5 it is {\sl NP}-complete. We show that, in a
precise sense, it is \emph{negative introspection}, the axiom \neg K p \rimp K
\neg K p, that causes the gap. In a precise sense, if we require this axiom,
then the satisfiability problem is {\sl NP}-complete; without it, it is {\sl
PSPACE}-complete.Comment: 6 page
Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)
We produce a decidable classical normal modal logic of internalised
negation-complete and thus disjunctive non-monotonic interactive proofs (LDiiP)
from an existing logical counterpart of non-monotonic or instant interactive
proofs (LiiP). LDiiP internalises agent-centric proof theories that are
negation-complete (maximal) and consistent (and hence strictly weaker than, for
example, Peano Arithmetic) and enjoy the disjunction property (like
Intuitionistic Logic). In other words, internalised proof theories are
ultrafilters and all internalised proof goals are definite in the sense of
being either provable or disprovable to an agent by means of disjunctive
internalised proofs (thus also called epistemic deciders). Still, LDiiP itself
is classical (monotonic, non-constructive), negation-incomplete, and does not
have the disjunction property. The price to pay for the negation completeness
of our interactive proofs is their non-monotonicity and non-communality (for
singleton agent communities only). As a normal modal logic, LDiiP enjoys a
standard Kripke-semantics, which we justify by invoking the Axiom of Choice on
LiiP's and then construct in terms of a concrete oracle-computable function.
LDiiP's agent-centric internalised notion of proof can also be viewed as a
negation-complete disjunctive explicit refinement of standard KD45-belief, and
yields a disjunctive but negation-incomplete explicit refinement of
S4-provability.Comment: Expanded Introduction. Added Footnote 4. Corrected Corollary 3 and 4.
Continuation of arXiv:1208.184
PSPACE Reasoning for Graded Modal Logics
We present a PSPACE algorithm that decides satisfiability of the graded modal
logic Gr(K_R)---a natural extension of propositional modal logic K_R by
counting expressions---which plays an important role in the area of knowledge
representation. The algorithm employs a tableaux approach and is the first
known algorithm which meets the lower bound for the complexity of the problem.
Thus, we exactly fix the complexity of the problem and refute an
ExpTime-hardness conjecture. We extend the results to the logic Gr(K_(R \cap
I)), which augments Gr(K_R) with inverse relations and intersection of
accessibility relations. This establishes a kind of ``theoretical benchmark''
that all algorithmic approaches can be measured against
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