3,033 research outputs found
On lengths of proofs in non-classical logics
AbstractWe give proofs of the effective monotone interpolation property for the system of modal logic K, and others, and the system IL of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 (3) (2007) 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 (2007) 72–90]; here, we give considerably simplified proofs, as well as some generalisations
On the complexity of the disjunction property in intuitionistic and modal logics
In this paper we study the complexity of disjunction property for Intuitionistic Logic, the modal logics S3, S4.1, Grzegorczyk Logic, Godel-Lob Logic and the intuitionistic counterpart of the modal logic K. For S4 we even prove the feasible interpolation theorem and we provide a lower bound for the length of proofs. The techniques we use do not require to prove structural properties of the calculi in hand, such as the Cut-elimination Theorem or the Normalization Theorem. This is a key-point of our approach, since it allows us to treat logics for which only Hilbert-style characterizations are known
The model checking problem for intuitionistic propositional logic with one variable is AC1-complete
We show that the model checking problem for intuitionistic propositional
logic with one variable is complete for logspace-uniform AC1. As basic tool we
use the connection between intuitionistic logic and Heyting algebra, and
investigate its complexity theoretical aspects. For superintuitionistic logics
with one variable, we obtain NC1-completeness for the model checking problem.Comment: A preliminary version of this work was presented at STACS 2011. 19
pages, 3 figure
Mass problems and intuitionistic higher-order logic
In this paper we study a model of intuitionistic higher-order logic which we
call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the
category of sheaves of sets over the topological space consisting of the Turing
degrees, where the Turing cones form a base for the topology. We note that our
Muchnik topos interpretation of intuitionistic mathematics is an extension of
the well known Kolmogorov/Muchnik interpretation of intuitionistic
propositional calculus via Muchnik degrees, i.e., mass problems under weak
reducibility. We introduce a new sheaf representation of the intuitionistic
real numbers, \emph{the Muchnik reals}, which are different from the Cauchy
reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice
principle} and a \emph{bounding principle} where range over Muchnik
reals, ranges over functions from Muchnik reals to Muchnik reals, and
is a formula not containing or . For the convenience of the
reader, we explain all of the essential background material on intuitionism,
sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems,
Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an
English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page
Semantic A-translation and Super-consistency entail Classical Cut Elimination
We show that if a theory R defined by a rewrite system is super-consistent,
the classical sequent calculus modulo R enjoys the cut elimination property,
which was an open question. For such theories it was already known that proofs
strongly normalize in natural deduction modulo R, and that cut elimination
holds in the intuitionistic sequent calculus modulo R. We first define a
syntactic and a semantic version of Friedman's A-translation, showing that it
preserves the structure of pseudo-Heyting algebra, our semantic framework. Then
we relate the interpretation of a theory in the A-translated algebra and its
A-translation in the original algebra. This allows to show the stability of the
super-consistency criterion and the cut elimination theorem
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