9,760 research outputs found
Focused labeled proof systems for modal logic
International audienceFocused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules from Gentzen's original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system G3K of Negri for the modal logic K and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in G3K corresponds to a bipole—a pair of a positive and a negative phases—in LKF. Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs
A focused framework for emulating modal proof systems
International audienceSeveral deductive formalisms (e.g., sequent, nested sequent, labeled sequent, hyperse-quent calculi) have been used in the literature for the treatment of modal logics, and some connections between these formalisms are already known. Here we propose a general framework, which is based on a focused version of the labeled sequent calculus by Negri, augmented with some parametric devices allowing to restrict the set of proofs. By properly defining such restrictions and by choosing an appropriate polarization of formulas, one can obtain different, concrete proof systems for the modal logic K and for its extensions by means of geometric axioms. In particular, we show how to use the expressiveness of the labeled approach and the control mechanisms of focusing in order to emulate in our framework the behavior of a range of existing formalisms and proof systems for modal logic
A general proof certification framework for modal logic
One of the main issues in proof certification is that different theorem provers, even when designed for the same logic, tend to use different proof formalisms and to produce outputs in different formats. The project ProofCert promotes the usage of a common specification language and of a small and trusted kernel in order to check proofs coming from different sources and for different logics. By relying on that idea and by using a classical focused sequent calculus as a kernel, we propose here a general framework for checking modal proofs. We present the implementation of the framework in a prolog-like language and show how it is possible to specialize it in a simple and modular way in order to cover different proof formalisms, such as labeled systems, tableaux, sequent calculi and nested sequent calculi. We illustrate the method for the logic K by providing several examples and discuss how to further extend the approach
Ecumenical modal logic
The discussion about how to put together Gentzen's systems for classical and
intuitionistic logic in a single unified system is back in fashion. Indeed,
recently Prawitz and others have been discussing the so called Ecumenical
Systems, where connectives from these logics can co-exist in peace. In Prawitz'
system, the classical logician and the intuitionistic logician would share the
universal quantifier, conjunction, negation, and the constant for the absurd,
but they would each have their own existential quantifier, disjunction, and
implication, with different meanings. Prawitz' main idea is that these
different meanings are given by a semantical framework that can be accepted by
both parties. In a recent work, Ecumenical sequent calculi and a nested system
were presented, and some very interesting proof theoretical properties of the
systems were established. In this work we extend Prawitz' Ecumenical idea to
alethic K-modalities
Constructive Provability Logic
We present constructive provability logic, an intuitionstic modal logic that
validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting
logical reflection over provability. Two distinct variants of this logic, CPL
and CPL*, are presented in natural deduction and sequent calculus forms which
are then shown to be equivalent. In addition, we discuss the use of
constructive provability logic to justify stratified negation in logic
programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl
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
On the equivalence of game and denotational semantics for the probabilistic mu-calculus
The probabilistic (or quantitative) modal mu-calculus is a fixed-point logic
de- signed for expressing properties of probabilistic labeled transition
systems (PLTS). Two semantics have been studied for this logic, both assigning
to every process state a value in the interval [0,1] representing the
probability that the property expressed by the formula holds at the state. One
semantics is denotational and the other is a game semantics, specified in terms
of two-player stochastic games. The two semantics have been proved to coincide
on all finite PLTS's, but the equivalence of the two semantics on arbitrary
models has been open in literature. In this paper we prove that the equivalence
indeed holds for arbitrary infinite models, and thus our result strengthens the
fruitful connection between denotational and game semantics. Our proof adapts
the unraveling or unfolding method, a general proof technique for proving
result of parity games by induction on their complexity
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