1,078 research outputs found
Proof Theory of Finite-valued Logics
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics
Elimination of Cuts in First-order Finite-valued Logics
A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information
Neutrality and Many-Valued Logics
In this book, we consider various many-valued logics: standard, linear,
hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We
survey also results which show the tree different proof-theoretic frameworks
for many-valued logics, e.g. frameworks of the following deductive calculi:
Hilbert's style, sequent, and hypersequent. We present a general way that
allows to construct systematically analytic calculi for a large family of
non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and
p-adic valued logics characterized by a special format of semantics with an
appropriate rejection of Archimedes' axiom. These logics are built as different
extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's,
Product, and Post's logics). The informal sense of Archimedes' axiom is that
anything can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and it
is not well-founded and well-ordered. On the base of non-Archimedean valued
logics, we construct non-Archimedean valued interval neutrosophic logic INL by
which we can describe neutrality phenomena.Comment: 119 page
Enriched MU-Calculi Module Checking
The model checking problem for open systems has been intensively studied in
the literature, for both finite-state (module checking) and infinite-state
(pushdown module checking) systems, with respect to Ctl and Ctl*. In this
paper, we further investigate this problem with respect to the \mu-calculus
enriched with nominals and graded modalities (hybrid graded Mu-calculus), in
both the finite-state and infinite-state settings. Using an automata-theoretic
approach, we show that hybrid graded \mu-calculus module checking is solvable
in exponential time, while hybrid graded \mu-calculus pushdown module checking
is solvable in double-exponential time. These results are also tight since they
match the known lower bounds for Ctl. We also investigate the module checking
problem with respect to the hybrid graded \mu-calculus enriched with inverse
programs (Fully enriched \mu-calculus): by showing a reduction from the domino
problem, we show its undecidability. We conclude with a short overview of the
model checking problem for the Fully enriched Mu-calculus and the fragments
obtained by dropping at least one of the additional constructs
Tuples of disjoint NP-sets
Disjoint NP-pairs are a well studied complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k ≥ 2. We define subclasses of the class of all disjoint k-tuples of NP-sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint NP-pairs exist if and only if complete disjoint k-tuples of NP-sets exist for all k ≥ 2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems
Mixin Composition Synthesis based on Intersection Types
We present a method for synthesizing compositions of mixins using type
inhabitation in intersection types. First, recursively defined classes and
mixins, which are functions over classes, are expressed as terms in a lambda
calculus with records. Intersection types with records and record-merge are
used to assign meaningful types to these terms without resorting to recursive
types. Second, typed terms are translated to a repository of typed combinators.
We show a relation between record types with record-merge and intersection
types with constructors. This relation is used to prove soundness and partial
completeness of the translation with respect to mixin composition synthesis.
Furthermore, we demonstrate how a translated repository and goal type can be
used as input to an existing framework for composition synthesis in bounded
combinatory logic via type inhabitation. The computed result is a class typed
by the goal type and generated by a mixin composition applied to an existing
class
- …