65 research outputs found
Are there Hilbert-style Pure Type Systems?
For many a natural deduction style logic there is a Hilbert-style logic that
is equivalent to it in that it has the same theorems (i.e. valid judgements
with empty contexts). For intuitionistic logic, the axioms of the equivalent
Hilbert-style logic can be propositions which are also known as the types of
the combinators I, K and S. Hilbert-style versions of illative combinatory
logic have formulations with axioms that are actual type statements for I, K
and S. As pure type systems (PTSs)are, in a sense, equivalent to systems of
illative combinatory logic, it might be thought that Hilbert-style PTSs (HPTSs)
could be based in a similar way. This paper shows that some PTSs have very
trivial equivalent HPTSs, with only the axioms as theorems and that for many
PTSs no equivalent HPTS can exist. Most commonly used PTSs belong to these two
classes. For some PTSs however, including lambda* and the PTS at the basis of
the proof assistant Coq, there is a nontrivial equivalent HPTS, with axioms
that are type statements for I, K and S.Comment: Accepted in Logical Methods in Computer Scienc
The inhabitation problem for intersection types
In the system lambda ^ of intersection types, without w, the problem as to whether an arbitrary type has an inhabitant, has been shown to be undecidable by Urzyczyn in [9]. For one subsystem of lambda ^, that lacks the ^- introduction rule, the inhabitation problem has been shown to be decidable in Kurata and Takahashi [9]. The natural question that arises is: What other subsystems of lambda ^, have a decidable inhabitation problem? The work in [2], which classifies distinct and inhabitation-distinct subsystems of lambda ^, leads to the extension of the undecidability result to lambda ^ without the (n) rule. By new methods, this paper shows, for the remaining six (two of them trivial) distinct subsystems of lambda ^, that inhabitation is decidable. For the latter subsystems inhabitant finding algorithms are provided
Cancellation laws for BCI-algebra, atoms and p-semisimple BCI-algebras
We derive cancellation laws for BCI-algebras and for p-semisimple BCI- algebras, show that the set of all atoms of a BCI-algebra is a p semisimple BCI-algebra and that in a p-semisimple BCI-algebra and = are the same
The D-Completeness of T→
A Hilbert-style version of an implicational logic can be represented by a set of axiom schemes and modus ponens or by the corresponding axioms, modus ponens and substitution. Certain logics, for example the intuitionistic implicational logic, can also be represented by axioms and the rule of condensed detachment, which combines modus ponens with a minimal form of substitution. Such logics, for example intuitionistic implicational logic, are said to be D-complete. For certain weaker logics, the version based on condensed detachment and axioms (the condensed version of the logic) is weaker than the original. In this paper we prove that the relevant logic T→, and any logic of which this is a sublogic, is D-complete
Natural numbers in illative combinatory logic
In this paper we attempt to develop natural numbers using the second order predicate calculus and the three axioms also used to obtain the set theory of [3]
Solid weak BCC-algebras
We characterize weak BCC-algebras in which the identity is
satisfied only in the case when elements belong to the same branch
Exact Numerical Calculation of the Density of States of the Fluctuating Gap Model
We develop a powerful numerical algorithm for calculating the density of
states rho(omega) of the fluctuating gap model, which describes the low-energy
physics of disordered Peierls and spin-Peierls chains. We obtain rho(omega)
with unprecedented accuracy from the solution of a simple initial value problem
for a single Riccati equation. Generating Gaussian disorder with large
correlation length xi by means of a simple Markov process, we present a
quantitative study of the behavior of rho (omega) in the pseudogap regime. In
particular, we show that in the commensurate case and in the absence of forward
scattering the pseudogap is overshadowed by a Dyson singularity below a certain
energy scale omega^{ast}, which we explicitly calculate as a function of xi.Comment: 4 revtex pages, 3 figure
Thermodynamic Properties of the One-Dimensional Extended Quantum Compass Model in the Presence of a Transverse Field
The presence of a quantum critical point can significantly affect the
thermodynamic properties of a material at finite temperatures. This is
reflected, e.g., in the entropy landscape S(T; c) in the vicinity of a quantum
critical point, yielding particularly strong variations for varying the tuning
parameter c such as magnetic field. In this work we have studied the
thermodynamic properties of the quantum compass model in the presence of a
transverse field. The specific heat, entropy and cooling rate under an
adiabatic demagnetization process have been calculated. During an adiabatic
(de)magnetization process temperature drops in the vicinity of a field-induced
zero-temperature quantum phase transitions. However close to field-induced
quantum phase transitions we observe a large magnetocaloric effect
Individual energy level distributions for one-dimensional diagonal and off-diagonal disorder
We study the distribution of the -th energy level for two different
one-dimensional random potentials. This distribution is shown to be related to
the distribution of the distance between two consecutive nodes of the wave
function.
We first consider the case of a white noise potential and study the
distributions of energy level both in the positive and the negative part of the
spectrum. It is demonstrated that, in the limit of a large system
(), the distribution of the -th energy level is given by a
scaling law which is shown to be related to the extreme value statistics of a
set of independent variables.
In the second part we consider the case of a supersymmetric random
Hamiltonian (potential ). We study first the case of
being a white noise with zero mean. It is in particular shown that
the ground state energy, which behaves on average like in
agreement with previous work, is not a self averaging quantity in the limit
as is seen in the case of diagonal disorder. Then we consider the
case when has a non zero mean value.Comment: LaTeX, 33 pages, 9 figure
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