16 research outputs found
Coherence in Modal Logic
A variety is said to be coherent if the finitely generated subalgebras of its
finitely presented members are also finitely presented. In a recent paper by
the authors it was shown that coherence forms a key ingredient of the uniform
deductive interpolation property for equational consequence in a variety, and a
general criterion was given for the failure of coherence (and hence uniform
deductive interpolation) in varieties of algebras with a term-definable
semilattice reduct. In this paper, a more general criterion is obtained and
used to prove the failure of coherence and uniform deductive interpolation for
a broad family of modal logics, including K, KT, K4, and S4
Ruitenburg's Theorem via Duality and Bounded Bisimulations
For a given intuitionistic propositional formula A and a propositional
variable x occurring in it, define the infinite sequence of formulae { A \_i |
i1} by letting A\_1 be A and A\_{i+1} be A(A\_i/x). Ruitenburg's Theorem
[8] says that the sequence { A \_i } (modulo logical equivalence) is ultimately
periodic with period 2, i.e. there is N 0 such that A N+2
A N is provable in intuitionistic propositional calculus. We
give a semantic proof of this theorem, using duality techniques and bounded
bisimulations ranks
Fixed-point elimination in the intuitionistic propositional calculus
It is a consequence of existing literature that least and greatest
fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic
models of the Intuitionistic Propositional Calculus-always exist, even when
these algebras are not complete as lattices. The reason is that these extremal
fixed-points are definable by formulas of the IPC. Consequently, the
-calculus based on intuitionistic logic is trivial, every -formula
being equivalent to a fixed-point free formula. We give in this paper an
axiomatization of least and greatest fixed-points of formulas, and an algorithm
to compute a fixed-point free formula equivalent to a given -formula. The
axiomatization of the greatest fixed-point is simple. The axiomatization of the
least fixed-point is more complex, in particular every monotone formula
converges to its least fixed-point by Kleene's iteration in a finite number of
steps, but there is no uniform upper bound on the number of iterations. We
extract, out of the algorithm, upper bounds for such n, depending on the size
of the formula. For some formulas, we show that these upper bounds are
polynomial and optimal
Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond
Ruitenburg\u2019s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ulti- mately periodic if f fixes all the generators but one. More precisely, there is N 65 0 such that f^N+2 = f^N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators
The automorphism group of the Fra\"iss\'e limit of finite Heyting algebras
Roelcke non-precompactness, non-simplicity, and non-amenability of the
automorphism group of the Fra\"iss\'e limit of finite Heyting algebras are
examined among others.Comment: Submitted to Journal of Symbolic Logic; 15 page
Proof Theory for Intuitionistic Strong L\"ob Logic
This paper introduces two sequent calculi for intuitionistic strong L\"ob
logic : a terminating sequent calculus based
on the terminating sequent calculus for intuitionistic
propositional logic and an extension of the
standard cut-free sequent calculus without structural rules for
. One of the main results is a syntactic proof of the
cut-elimination theorem for . In addition, equivalences
between the sequent calculi and Hilbert systems for are
established. It is known from the literature that is complete
with respect to the class of intuitionistic modal Kripke models in which the
modal relation is transitive, conversely well-founded and a subset of the
intuitionistic relation. Here a constructive proof of this fact is obtained by
using a countermodel construction based on a variant of . The
paper thus contains two proofs of cut-elimination, a semantic and a syntactic
proof.Comment: 29 pages, 4 figures, submitted to the Special Volume of the Workshop
Proofs! held in Paris in 201