3,076 research outputs found
Provability Logic and the Completeness Principle
In this paper, we study the provability logic of intuitionistic theories of
arithmetic that prove their own completeness. We prove a completeness theorem
for theories equipped with two provability predicates and
that prove the schemes and for
. Using this theorem, we determine the logic of fast provability
for a number of intuitionistic theories. Furthermore, we reprove a theorem
previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the
-provability logic of Heyting Arithmetic
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
On Bisimulations for Description Logics
We study bisimulations for useful description logics. The simplest among the
considered logics is (a variant of PDL). The others
extend that logic with inverse roles, nominals, quantified number restrictions,
the universal role, and/or the concept constructor for expressing the local
reflexivity of a role. They also allow role axioms. We give results about
invariance of concepts, TBoxes and ABoxes, preservation of RBoxes and knowledge
bases, and the Hennessy-Milner property w.r.t. bisimulations in the considered
description logics. Using the invariance results we compare the expressiveness
of the considered description logics w.r.t. concepts, TBoxes and ABoxes. Our
results about separating the expressiveness of description logics are naturally
extended to the case when instead of we have any sublogic
of that extends . We also provide results
on the largest auto-bisimulations and quotient interpretations w.r.t. such
equivalence relations. Such results are useful for minimizing interpretations
and concept learning in description logics. To deal with minimizing
interpretations for the case when the considered logic allows quantified number
restrictions and/or the constructor for the local reflexivity of a role, we
introduce a new notion called QS-interpretation, which is needed for obtaining
expected results. By adapting Hopcroft's automaton minimization algorithm and
the Paige-Tarjan algorithm, we give efficient algorithms for computing the
partition corresponding to the largest auto-bisimulation of a finite
interpretation.Comment: 42 page
Lindstrom theorems for fragments of first-order logic
Lindstr\"om theorems characterize logics in terms of model-theoretic
conditions such as Compactness and the L\"owenheim-Skolem property. Most
existing characterizations of this kind concern extensions of first-order
logic. But on the other hand, many logics relevant to computer science are
fragments or extensions of fragments of first-order logic, e.g., k-variable
logics and various modal logics. Finding Lindstr\"om theorems for these
languages can be challenging, as most known techniques rely on coding arguments
that seem to require the full expressive power of first-order logic. In this
paper, we provide Lindstr\"om theorems for several fragments of first-order
logic, including the k-variable fragments for k>2, Tarski's relation algebra,
graded modal logic, and the binary guarded fragment. We use two different proof
techniques. One is a modification of the original Lindstr\"om proof. The other
involves the modal concepts of bisimulation, tree unraveling, and finite depth.
Our results also imply semantic preservation theorems.Comment: Appears in Logical Methods in Computer Science (LMCS
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
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