7,663 research outputs found
An Intuitionistic Formula Hierarchy Based on High-School Identities
We revisit the notion of intuitionistic equivalence and formal proof
representations by adopting the view of formulas as exponential polynomials.
After observing that most of the invertible proof rules of intuitionistic
(minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms
corresponding to the high-school identities, we show that one can obtain a more
compact variant of a proof system, consisting of non-invertible proof rules
only, and where the invertible proof rules have been replaced by a formula
normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent calculus of
Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the
non-invertible proof rules as strict inequalities between exponential
polynomials; a careful combinatorial treatment is given in order to establish
this fact.
Finally, we extend the exponential polynomial analogy to the first-order
quantifiers, showing that it gives rise to an intuitionistic hierarchy of
formulas, resembling the classical arithmetical hierarchy, and the first one
that classifies formulas while preserving isomorphism
On equations over sets of integers
Systems of equations with sets of integers as unknowns are considered. It is
shown that the class of sets representable by unique solutions of equations
using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in
T} and with ultimately periodic constants is exactly the class of
hyper-arithmetical sets. Equations using addition only can represent every
hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can
also be represented by equations over sets of natural numbers equipped with
union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in
T, \: m \geqslant n}. Testing whether a given system has a solution is
-complete for each model. These results, in particular, settle the
expressive power of the most general types of language equations, as well as
equations over subsets of free groups.Comment: 12 apges, 0 figure
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