Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by V\"a\"an\"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,...,p_n, =(p_1,...,p_(n-1);p_n) intuitively states that the value of p_n is determined by those of p_1,...,p_(n-1). Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man's dependence logic, the language consisting of formulas built from literals and dependence atoms using conjunction, necessity and possibility (i.e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend V\"a\"an\"anen's language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by V\"a\"an\"anen and Sevenster.Comment: 22 pages, full version of CSL 2010 pape

Modal logic NL for common language

Despite initial appearance, paradoxes in classical logic, when comprehension is unrestricted, do not go away even if the law of excluded middle is dropped, unless the law of noncontradiction is eliminated as well, which makes logic much less powerful. Is there an alternative way to preserve unrestricted comprehension of common language, while retaining power of classical logic? The answer is yes, when provability modal logic is utilized. Modal logic NL is constructed for this purpose. Unless a paradox is provable, usual rules of classical logic follow. The main point for modal logic NL is to tune the law of excluded middle so that we allow for a sentence and its negation to be both false in case a paradox provably arises. Curry's paradox is resolved differently from other paradoxes but is also resolved in modal logic NL. The changes allow for unrestricted comprehension and naive set theory, and allow us to justify use of common language in formal sense

Modal logic NL for common language

Despite initial appearance, paradoxes in classical logic, when comprehension is unrestricted, do not go away even if the law of excluded middle is dropped, unless the law of noncontradiction is eliminated as well, which makes logic much less powerful. Is there an alternative way to preserve unrestricted comprehension of common language, while retaining power of classical logic? The answer is yes, when provability modal logic is utilized. Modal logic NL is constructed for this purpose. Unless a paradox is provable, usual rules of classical logic follow. The main point for modal logic NL is to tune the law of excluded middle so that we allow for a sentence and its negation to be both false in case a paradox provably arises. Curry's paradox is resolved differently from other paradoxes but is also resolved in modal logic NL. The changes allow for unrestricted comprehension and naive set theory, and allow us to justify use of common language in formal sense

Non-dual modal operators as a basis for 4-valued accessibility relations in Hybrid logic

The modal operators usually associated with the notions of possibility and necessity are classically duals. This paper aims to defy that duality in a paraconsistent environment, namely in a Belnapian Hybrid logic where both propositional variables and accessibility relations are four-valued. Hybrid logic, which is an extension of Modal logic, incorporates extra machinery such as nominals – for uniquely naming states – and a satisfaction operator – so that the formula under its scope is evaluated in the state whose name the satisfaction operator indicates. In classical Hybrid logic the semantics of negation, when it appears before compound formulas, is carried towards subformulas, meaning that eventual inconsistencies can be found at the level of nominals or propositional variables but appear unrelated to the accessibility relations. In this paper we allow inconsistencies in propositional variables and, by breaking the duality between modal operators, inconsistencies at the level of accessibility relations arise. We introduce a sound and complete tableau system and a decision procedure to check if a formula is a consequence of a set of formulas. Tableaux will be used to extract syntactic models for databases, which will then be compared using different inconsistency measures. We conclude with a discussion about bisimulation

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

Negation and Dichotomy

The present contribution might be regarded as a kind of defense of the common sense in logic. It is demonstrated that if the classical negation is interpreted as the minimal negation with n = 2 truth values, then deviant logics can be conceived as extension of the classical bivalent frame. Such classical apprehension of negation is possible in non- classical logics as well, if truth value is internalized and bivalence is replaced by bipartition