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Home Technology News Newly Discovered Earth-Like Planet May Have the Right Conditions for Life
Newly Discovered Earth-Like Planet May Have the Right Conditions for Life
adminApr 21, 2017Technology NewsComments Off on Newly Discovered Earth-Like Planet May Have the Right Conditions for Life
With every new exoplanet discovery, we seem to get that much closer to finding evidence of extraterrestrial life. A new exoplanet identified in orbit of a nearby star could be the best place to look yet. Astronomers have confirmed a star 39 light-years away plays host to a planet that is much more Earth-like than any we've been able to spot in the past. The planet, known as LHS 1140b, is larger than Earth, but it's rocky and could well have liquid water on its surface.
According to a new publication in Nature, the international team of astronomers with the Mearth project used an array of eight telescopes at Cerro Tololo Inter-American Observatory in Chile to find LHS 1140b. Luckily, this planet passes in front of its parent star frequently — like many other planets we've spotted, it orbits close to a small, cool star. This allowed the team to characterize it to a degree not possible with other discoveries. For example, the mass of all those planets orbiting Trappist-1 are mere estimates, and we know very little about the composition of the exoplanet in orbit of Proxima Centauri.
What we know about LHS 1140b is that it's about 40 percent larger than Earth with 6.6 times the mass. Thus, gravity on LHS 1140b would be three times greater than Earth's. So if you weigh 175 pounds here, you'd tip the scales at 525 pounds. It would be more accurate to call this a Super-Earth, 32 of which have been discovered in recent years. LHS 1140b might not be a pleasant place for humans to hang out, but it's well within the range of a rocky planet, not a gas giant like Jupiter or Neptune. Scientists also expect LHS 1140b could have liquid water on its surface simply based on its distance from the star.
The Mearth array used to find LHS 1140b.
A trip of 39 light-years isn't something humanity is capable of right now, but that's still in our galactic neighborhood. Because it transits its star so frequently, there is hope scientists will be able to measure LHS 1140b's atmosphere. If we find both oxygen and carbon, that could mean something is alive out there. Astronomers are already placing LHS 1140b at the top of their lists when new telescopes like the James Webb Space Telescope come online. This instrument will launch in October 2018, allowing astronomers to get a closer look at a variety of objects, including exoplanets like LHS 1140b. There are also some ground-based telescope projects that could allow for closer examination of LHS 1140b.
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Researchers Recommend New Approaches to Sway the Opinion of Climate Deniers
Those who want to change the views of climate deniers, can start by respecting and acknowledging the beliefs of people. That was one of the four suggestions provided by a Stanford University scientist in a review of the psychology behind why a certain section of people deny climate change in spite of their knowledge or having access to the facts.
A new Stanford-led paper reviewed the psychological motivations of "motivated denial," in which people know or have access to the facts, but nevertheless deny them. Image Credit: Shutterstock.com
Rejecting the impacts of climate change acts as a barrier, specifically when implementing the actions needed to reduce the worst impacts of climate change—such as increased heatwaves and droughts, stronger hurricanes, and rising seas.
But the scientists identified that those who reject human-induced climate change can be influenced via conversations that appeal to their reframing solutions, different identities, or even accept their view on climate.
I think in the climate change sphere there's this thinking of, 'there's the deniers over there, let's just not even engage with them—it's not worth it. A lot of the tactics and strategies start from the point that something is wrong with the climate deniers, rather than trying to acknowledge that they have a belief and opinion and it matters. But I think there is an opportunity to keep trying to understand one another, especially now.
Gabrielle Wong-Parodi, Study Lead Author and Behavioral Scientist, Stanford University
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The study was reported in the Current Opinion in Environmental Sustainability journal on January 8th, 2020.
The scientists focused more on the so-called "motivated denial"—knowing the facts or having access to them but still rejecting them. For some people, accepting the fact that humans are responsible for causing climate change, questions their self-worth, poses a threat to financial institutions, and is followed by an enormous sense of responsibility.
While it may appear to be futile to influence the deniers of climate change, the investigators have nevertheless come up with four methods that have been described in peer-reviewed studies from past years. These approaches can be highly effective and have been described below:
Decreasing the ideological divide by integrating the Earth's purity, instead of focusing on how people harm or care for it
Reframing solutions to climate change are one of the ways to preserve the social system and work toward its longevity and stability
Motivating people to clearly discuss their views and values on climate change before engaging with climatic details
Having discussions regarding scientific consensus about climate change with reliable people
According to Wong-Parodi, the fourth method is the most desirable, since not much research has been performed in this area compared to the others —and appears to have a great deal of potential for behavior change.
When people face climate change, their self-affirmation is challenged, because it would make them consider their contribution to the issue, thereby threatening their sense of integrity and triggering self-defense.
A good portion of people who deny climate change recognize that there is some change, but the change is so threatening because it basically could affect your quality of life. It could affect your income. It could affect a number of different things that you care about.
A few initial studies recommended that discussions should rather embrace their views, instead of attempting to get around the denial of climate change and peoples identities.
According to Wong-Parodi, one should never try to ignore who people are, but instead consider their opinions so that they can be addressed and the discussion can move on to behavioral changes—such as searching for solutions that correspond with their values and do not pose a threat to a persons quality of life or sense of identity.
I think we often forget that people can have many identities–there might be a political identity, but there is also an identity as a mother, or an identity as a friend or an identity as a student. You can elicit other identities when you're talking about climate change that may be more effective.
Wong-Parodi is also a fellow at the Stanford Woods Institute for the Environment.
Other co-authors of the study include Irina Feygina, an independent practitioner in Brooklyn, New York.
The study was supported by the National Socio-Environmental Synthesis Center with funding from the U.S. National Science Foundation.
Source: https://www.stanford.edu/

redpajama_set_name: RedPajamaCommonCrawl


The Wild and Wicked Beach Bash is coming to Gulf Place Public Beach on Oct. 19. Locals and visitors are invited to wear their best Halloween costume and participate in this horrifyingly fun event featuring a Haunted Hustle 5K, Witches on Wheels, and live music from Doctor Zarr's Amazing Funk Monster. For more information about the Wild & Wicked Beach Bash, please call 251-968-9824 or visit www.gulfshoresal.gov/halloween.


text:
\section{Introduction}

Automated Theorem Provers (ATPs)~\cite{DBLP:books/el/RobinsonV01} can be in principle
used to attempt the proof of any provable mathematical conjecture.
The standard ATP approaches have
so far relied primarily on fast implementation of manually designed search procedures and heuristics.
However, using machine learning for guidance in the vast action spaces of the ATP calculi is a
natural choice that has been recently shown to significantly improve over the
unguided systems~\cite{KaliszykUMO18,JakubuvU19}.

The common procedure of a first-order ATP system -- saturation-style
or tableaux -- is the following. The ATP starts with a set of first
order axioms and a conjecture. The conjecture is negated and the
formulas are Skolemized and clausified.
The objective is then to derive a contradiction from the set of clauses,
typically using some form of resolution and related inference rules.
The Skolemization as well as introduction of new definitions during
the clausification results in the introduction of many new function and predicate
symbols.

When guiding the proving process by statistical machine learning,
the state of the prover and the
formulas, literals, and clauses, are typically encoded  to vectors of
real numbers. This has been so far mostly done with hand-crafted features resulting
in large sparse vectors~\cite{DBLP:conf/ijcai/KaliszykUV15,hammers4qed,abs-1108-3446,UrbanVS11,KaliszykU15,JakubuvU17a}, possibly reducing their dimension afterwards~\cite{ChvalovskyJ0U19}.
Several experiments with neural
networks have been made recently, in particular based on 1D convolutions,
RNNs \cite{GollerK96}, TreeRNNs~\cite{ChvalovskyJ0U19}, and GraphNNs \cite{DuvenaudMABHAA15}. Most of the approaches, however, cannot
capture well the idea of a variable occurring multiple times in the
formula and to abstract from the names of the variables.
These issues were first
addressed in FormulaNet~\cite{DBLP:conf/nips/WangTWD17} but even that
architecture relies on knowing the names of function and predicate symbols. This makes it
unsuitable for handling the large number of problem-specific function and predicate
symbols introduced during the clausification.\footnote{The ratio of such symbols in real-world clausal datasets is around 40\%, see Section~\ref{symbol-experiments}.}

The same holds for large datasets of ATP problems
where symbol names are not used consistently, such as the TPTP library~\cite{Sutcliffe10}.

In this paper, we make further steps towards the abstraction of
mathematical clauses, formulas and proof states. We present a network that is invariant not only
under renaming of variables, but also under renaming of arbitrary function and predicate
symbols. It is also invariant under replacement of the symbols by their negated versions.
This is achieved by a novel conversion of the input formulas into a hypergraph, followed by a
particularly designed
graph neural network (GNN) capable of maintaining the invariance under negation.
We experimentally demonstrate in three case studies
that the network works well on data
coming from automated theorem proving tasks.

The paper is structured as follows. We first formally describe our network architecture in
Section~\ref{architecture}, and discuss its invariance properties
in Section~\ref{invariance}. We describe an experiment using the
network for guiding \systemname{leanCoP} in Section~\ref{leancop-example}, and
two experiments done on a fixed dataset in
Section~\ref{deepmath-experiments}. Section~\ref{results} contains the
results of these three experiments.

\section{Network Architecture for Invariant Embedding}
\label{architecture}

This section describes the design and details of the proposed neural architecture for invariant embeddings.
 The architecture gets as its input a set of clauses $\mathbf C$.
It outputs an embedding for each of the clauses in $\mathbf C$, each literal and subterm and each function and predicate
symbol present in $\mathbf C$. The process consists of initially
constructing a hypergraph out of the given set of clauses, and then
several message passing layers on the hypergraph. In Section~\ref{hypergraph} we first explain the construction of a hypergraph from the input clauses. The details of the message passing are explained in Section~\ref{messagepassing}
.

\subsection{Hypergraph Construction}
\label{hypergraph}
When converting the clauses to the graph, we aim to capture as much relevant structure as possible. We roughly convert the tree structure of the terms to a circuit by sharing variables, constants and also bigger terms. The graph will be also interconnected through special nodes representing function symbols.
Let $n_{\mathbf c}$ denote the number of clauses, and let the clauses be
$\mathbf C = \{C_1, \ldots, C_{n_{\mathbf c}}\}$. Similarly, let
$\mathbf S = \{S_1, \ldots, S_{n_{\mathbf s}}\}$ denote all the function and
predicate symbols occurring at least once in the given set of clauses,
and $\mathbf T = \{T_1, \ldots, T_{n_{\mathbf t}}\}$ denote all the
subterms and literals occurring at least once in the given set of
clauses. Two subterms are considered to be identical (and therefore
represented by a single node) if they are constructed the same way
using the same functions and variables. If $T_i$ is a negative
literal, the unnegated form of $T_i$ is not automatically added to
$\mathbf T$ but all its subterms are.

The sets $\mathbf C, \mathbf S, \mathbf T$ represent the nodes of our
hypergraph. The hypergraph
will also contain two sets of edges: Binary edges
$E_{\mathbf{ct}} \subset \mathbf C\times\mathbf T$ between clauses and
literals, and 4-ary oriented labeled edges
$E_{\mathbf st} \subset \mathbf S\times\mathbf T\times(\mathbf T\cup\{T_0\})^2\times\{1,-1\}$.
Here $T_0$ is a specially created and added term node disjoint from all actual terms and serving in the arity-related encodings described below. The label is present at the last position of the 5-tuple.
The set $E_{\mathbf{ct}}$ contains all
the pairs $(C_i, T_j)$ where $T_j$ is a literal contained in $C_i$. Note that this encoding makes the order of the literals in the clauses irrelevant, which corresponds to the desired semantic behavior.

The set $E_{\mathbf{st}}$ is constructed by the following
procedure applied to every
literal or subterm $T_i$ that is not a variable. If $T_i$ is a
negative literal, we set $\sigma = 1$, and interpret $T_i$ as
$T_i = \neg S_j(t_1,\ldots,t_n)$, otherwise we set $\sigma = -1$
and interpret $T_i$ as $T_i = S_j(t_1,\ldots,t_n)$, where $S_j\in\mathbf S$,  $n$ is the arity of $S_j$ and
$t_1,\ldots,t_n\in\mathbf T$. If $n=0$, we add
$(S_j,T_i,T_{0},T_{0},\sigma)$ to $E_{\mathbf{st}}$. If
$n=1$, we add $(S_j,T_i,t_1,T_{0},\sigma)$ to $E_{\mathbf{st}}$.
And finally, if $n\geq 2$, we extend $E_{\mathbf{st}}$ by
all the tuples $(S_j,T_i,t_k,t_{k+1},\sigma)$ for $k=1,\ldots,n-1$.

This encoding is used instead of just $(S_j,T_i,t_k,\sigma)$
to (reasonably) maintain the order of function and predicate arguments.
For example, for two non-isomorphic (i.e., differently encoded) terms $t_1$ and $t_2$, $t_1<t_2$ will be encoded differently than $t_2<t_1$.
Note that even this encoding does not capture the complete information about the argument order.
For example, the term $f(t_1,t_2,t_1)$ would be encoded the same way as $f(t_2,t_1,t_2)$.
We consider such information loss acceptable.
Further note that the sets $E_{\mathbf{ct}}$, $E_{\mathbf{st}}$, and the derived sets labeled $F$ (explained below) are in fact multisets in our implementation. We present them using the set notations here for readability.

\subsection{Message Passing}
\label{messagepassing}
Based on the hyperparameters $L$ (number of layers), and $d_{\mathbf c}^i$,
$d_{\mathbf s}^i$, $d_{\mathbf t}^i$ for $i=0,\ldots,L$ (dimensions of
vectors), we construct vectors
$c_{i,j}\in \mathbb{R}^{d^i_{\mathbf c}}$, $s_{i,j}\in \mathbb{R}^{d^i_{\mathbf s}}$,
and $t_{i,j}\in \mathbb{R}^{d^i_{\mathbf t}}$. First we initialize
$c_{0,j}$, $s_{0,j}$ and $t_{0,j}$ by learned constant vectors for every type
of clause, symbol, or term. By a ``type'' we mean
an attribute based on the underlying task, see
Section~\ref{leancop-example} for an example. To preserve invariance
under negation (see Section~\ref{invariance}), we initialize all
predicate symbols to the zero vector.

After the initialization, we
propagate through
$L$ message-passing layers. The
$i$-th layer will output vectors $c_{i,j}$, $s_{i,j}$ and $t_{i,j}$.
The values in the last layer, that is $c_{L,j}$, $s_{L,j}$ and
$t_{L,j}$, are considered to be the output of the network.
The basic idea of the message passing layer is to propagate information from a node to all its neighbors related by $E_{\mathbf{ct}}$ and $E_{\mathbf{st}}$ while recognizing the ``direction'' in which the information came.
After this, we
reduce the incoming data to a finite dimension using a
reduction function (defined below)
and
propagate through standard
neural layers.\footnote{Mostly implemented using the ReLU activation function.}
The symbol nodes $s_{i,j}$ need particular care, because they can represent two predicate symbols at once: if $s_{i,j}$ represents a predicate symbol $P$, then $-s_{i,j}$ represents the predicate symbol $\neg P$. To preserve the polarity invariance, the symbol nodes
are
treated slightly differently.

In the following we first provide the formulas describing the computation. The
symbols used in them are explained afterwards.

\begin{align*}
c_{i+1,j} &= \ReLU(
  B^i_{\mathbf c} + M^i_{\mathbf c}\cdot c_{i,j} +
  M^i_{\mathbf{ct}}\cdot \mathop{\mathrm{red}}\limits_{a\in F^j_{\mathbf{ct}}}(t_{i,a})
)\cr
x_{i}^{a,b,c} &=
  B^i_{\mathbf{ts}} +
  M^i_{\mathbf{ts},1}\cdot t_{i,a} +
  M^i_{\mathbf{ts},2}\cdot t_{i,b} +
  M^i_{\mathbf{ts},3}\cdot t_{i,c}
  \cr
s_{i+1,j} &= \tanh(
  M^i_{\mathbf s}\cdot s_{i,j} +
  M^i_{\mathbf{ts}}\cdot\mathop{\mathrm{red'}}\limits_{(a,b,c,g)\in
  F^j_{\mathbf{st}}}(g\cdot x_{i}^{a,b,c})
)\cr
y_{i,d}^{a,b,c,g} &=
  B^i_{\mathbf{st}} +
  M^{1,d}_{\mathbf{st},i}\cdot t_{i,a} +
  M^{2,d}_{\mathbf{st},i}\cdot t_{i,b} +
  M^{3,d}_{\mathbf{st},i}\cdot s_{i,c}\cdot g
  \cr
z_{i,j,d} &= M^i_{\mathbf{st,d}}\cdot \kern -3pt
  \mathop{\mathrm{red}}\limits_{(a,b,c,g)\in F^j_{\mathbf{ts},d}}(
  \ReLU(y_{i,d}^{a,b,c,g})
)\cr
v_{i,j} &= M^i_{\mathbf{tc}}\cdot \mathop{\mathrm{red}}\limits_{a\in F^j_{\mathbf{tc}}}(c_{i,a})\cr
t_{i+1,j} &= \ReLU(
  B^i_{\mathbf t} +
  M^i_{\mathbf t}\cdot t_{i,j} + v_{i,j} + \kern -3pt
  \sum_{d\in\{1,2,3\}} z_{i,j,d}
)\cr
\end{align*}

Here, all the $B$ symbols represent learnable vectors (biases), and all the
$M$ symbols represent learnable matrices. Their sizes are listed in Fig.~\ref{matrice-sizes}.

\begin{figure}
\hbox to\hsize{\hss$
B^i_{\mathbf c} : d^{i+1}_{\mathbf c}
$\hss$
B^i_{\mathbf{ts}} : d^{i+1}_{\mathbf s}
$\hss$
B^i_{\mathbf{st}} : d^{i+1}_{\mathbf t}
$\hss$
B^i_{\mathbf t} : d^{i+1}_{\mathbf t}
$\hss}
\hbox to\hsize{\hss
\vtop{\halign{\hss$#{}$&${}#$\hss\cr
M^i_{\mathbf c}         &: d^{i+1}_{\mathbf c}\times d^i_{\mathbf c}\cr
M^i_{\mathbf{ct}}       &: d^{i+1}_{\mathbf c}\times 2d^i_{\mathbf t}\cr
M^i_{\mathbf s}         &: d^{i+1}_{\mathbf s}\times d^i_{\mathbf s}\cr
}}\hss\vtop{\halign{\hss$#{}$&${}#$\hss\cr
M^i_{\mathbf t}         &: d^{i+1}_{\mathbf t}\times d^i_{\mathbf t}\cr
M^i_{\mathbf{tc}}       &: d^{i+1}_{\mathbf t}\times 2d^i_{\mathbf c}\cr
M^i_{\mathbf{ts}}       &: d^{i+1}_{\mathbf s}\times 2d^{i+1}_{\mathbf s}\cr
}}\hss\vtop{\halign{\hss$#{}$&${}#$\hss\cr
M^i_{\mathbf{ts},j}     &: d^{i+1}_{\mathbf s}\times d^i_{\mathbf t}\cr
M^i_{\mathbf{st},j}     &: d^{i+1}_{\mathbf t}\times 2d^{i+1}_{\mathbf t}\cr
M^{k,j}_{\mathbf{st},i} &: d^{i+1}_{\mathbf t}\times d^i_{\mathbf t}\cr
}}\hss}
    \centering
    \caption{\label{matrice-sizes}Sizes of learnable biases and matrices for $i=0,\ldots,L-1$ and $j,k\in\{1,2,3\}$.}
\end{figure}

By a reduction
operation $\mathop{\mathrm{red}}\limits\nolimits_{i\in I}(u_i)$, where all $u_i$ are vectors of
the same dimension $n$, we mean the vector of dimension $2n$ obtained
by concatenation of $\max_{i\in I}(u_i)$ and $\avg_{i\in I}(u_i)$. The
maximum and average operation are performed point-wise. We also
use another reduction operation $\mathop{\mathrm{red'}}\limits$ defined in the same way except
taking $\max_{i\in I}(u_i)+\min_{i\in I}(u_i)$ instead of just
maximum. This makes $\mathop{\mathrm{red'}}\limits$ commute with multiplication by $-1$.
If a reduction operation obtains an empty input (the indexing set is
an empty set), the result is the zero vector of the expected size.

We construct sets $F^{j_{\mathbf c}}_{\mathbf{ct}}$ and $F^{j_{\mathbf t}}_{\mathbf{tc}}$
based on $E_{\mathbf{ct}}$, and
$F^{j_{\mathbf s}}_{\mathbf{st}}$ and
$F^{j_{\mathbf t}}_{\mathbf{ts},d}$ based on $E_{\mathbf{st}}$, where
$j_{\mathbf c}=1,\ldots,{n_{\mathbf c}}$, $j_{\mathbf s}=1,\ldots,{n_{\mathbf s}}$,
$j_{\mathbf t}=1,\ldots,{n_{\mathbf t}}$, and $d=1,2,3$.
Informally, the set $\mathbf F^j_{\mathbf {xy}}$ contains the indices related to type $\mathbf y$ for message passing, given the $j$-th receiving node of type $\mathbf x$.
Formally:
\begin{align*}
F^{j}_{\mathbf{ct}} &= \{a : (C_j, T_a)\in E_{\mathbf{ct}}\}\cr
F^{j}_{\mathbf{tc}} &= \{a : (C_a, T_j)\in E_{\mathbf{ct}}\}\cr
F^{j}_{\mathbf{st}} &= \{(a,b,c,g) : (S_j, T_a, T_b, T_c, g)\in E_{\mathbf{st}}\}\cr
F^{j}_{\mathbf{ts},1} &= \{(a,b,c,g) : (S_c, T_j, T_a, T_b, g)\in E_{\mathbf{st}}\}\cr
F^{j}_{\mathbf{ts},2} &= \{(a,b,c,g) : (S_c, T_a, T_j, T_b, g)\in E_{\mathbf{st}}\}\cr
F^{j}_{\mathbf{ts},3} &= \{(a,b,c,g) : (S_c, T_a, T_b, T_j, g)\in E_{\mathbf{st}}\}\cr
\end{align*}

Since $E_{\mathbf{st}}$ can contain a dummy node $T_0$ on the third and fourth positions, following $F^{j}_{\mathbf{st}}$ or $F^{j}_{\mathbf{ts},d}$ in the message passing layer may lead us to a non-existing vector $t_{i,0}$. In that case, we just take the zero vector of dimension $d^i_{\mathbf t}$.

After $L$ message passing layers, we obtain the embeddings $c_{L,j}$, $s_{L,j}$, $t_{L,j}$ of the clauses $C_j$, symbols $S_j$ and terms and literals $T_j$ respectively.

\section{Invariance Properties}
\label{invariance}

By the design of the network, it is apparent that the output is
invariant under the names of the symbols. Indeed, the names are used only
for determining which symbol nodes and term nodes should be the same
and which should be different.

It is also worth noticing that the network is invariant under
reordering of literals in clauses, and under reordering of clauses.
More precisely, if we reorder the clauses
$C_1, \ldots, C_{n_{\mathbf c}}$, then the values
$c_{i,1}, \ldots, c_{i,n_{\mathbf c}}$ are reordered accordingly, and
the values $s_{i,j}, t_{i,j}$ do not change if they still correspond
to the same symbols and terms (they could be also rearranged in
general). This property is clear from the fact that there is no
ordered processing of the data, and the only way how literals are
attributed to clauses is through graph edges which are also unordered.

Finally, the network is also designed to preserve the symmetry under
negation. More precisely, consider replacing every occurrence of a predicate
symbol $S_x$ by the predicate symbol $\neg S_x$ in every clause $C_x$,
and every literal $T_i$. Then the vectors $c_{i,j}$, $t_{i,j}$ do not
change, the vectors
$s_{i,j}$ do not change either for all
$j \neq x$, and the vector $s_{i,x}$ is multiplied by $-1$.

We show this by induction on the layer $i$. For layer $0$,
this is apparent since the $S_x$ is a predicate symbol, so
$s_{0,x} = \vec 0 = -s_{0,x}$.
Now, let us assume that the claim is true for a layer $i$.
We follow the computation of the next layer. The symbol
vectors $s_i$ are not used at all in the computation of $c_{i+1,j}$,
so $c_{i+1,j}$ remains the same. For $s_{i+1,j}$ where $j\neq x$, we
don't use $s_{i,x}$ in the formula, and the signs have not changed in
$F^j_{\mathbf{st}}$. Therefore $s_{i+1,j}$ remains the same.
When computing $t_{i+1,j}$, we multiply every $s_{i,c}$ with the
appropriate sign (denoted $g$ in the formula). Since we have replaced every
occurrence of $S_i$ by $\neg S_i$ and kept the other symbols,
the sign $g$ is multiplied by $-1$ if and only if $c = x$, and
therefore the product does not change.
Finally, when computing $s_{i+1,x}$, we follow the formula below:
$$
s_{i+1,x} = \tanh(
  M^i_{\mathbf s}\cdot s_{i,x} +
  M^i_{\mathbf{ts}}\cdot\mathop{\mathrm{red'}}\limits_{(a,b,c,g)\in
  F^x_{\mathbf{st}}}(g\cdot x_{i}^{a,b,c})
)
$$
where $x_{i}^{a,b,c}$ depends only on values $t_{i,j}$, and therefore
was not changed. We can rewrite the formula as
$$
s_{i+1,x} = -\tanh(
  M^i_{\mathbf s}\cdot (-s_{i,x}) +
  M^i_{\mathbf{ts}}\cdot\mathop{\mathrm{red'}}\limits_{(a,b,c,g)\in
  F^x_{\mathbf{st}}}((-g)\cdot x_{i}^{a,b,c})
)
$$
This is because $\tanh$, addition, matrix multiplication, and the
reduction function $\mathop{\mathrm{red'}}\limits$ are compatible with multiplication by $-1$.
In fact, except $\tanh$ they are all linear, thus compatible with
multiplication by any constant, and $\tanh$ is an odd function.
The second formula for $s_{i+1,x}$ can be also seen as a formula
for minus the value of the
original $s_{i+1,x}$ since $-s_{i,x}$ is the original value of
$s_{i,x}$, and $(-g)$ is the original value of $g$. Therefore
$s_{i+1,x}$ was multiplied by $-1$.

\section{Guiding a Connection Tableaux Prover}
\label{leancop-example}

One of the most important
uses
of machine learning in theorem
proving is guiding the inferences
done by
the automating theorem
provers. The first application of our proposed model is to guide the
inferences performed by the \systemname{leanCoP} prover~\cite{OB03}. This line of work
follows our previous experiments with this prover using the XGBoost system
for guidance~\cite{KaliszykUMO18}. In this section, we first give a
brief description of the
\systemname{leanCoP} prover, then we explain how we fit our network to the \systemname{leanCoP} prover, and
finally discuss the interaction between the network and the Monte-Carlo
Tree Search that we use.

The \systemname{leanCoP} prover attempts to prove a given first-order logic problem by first
transforming its negation to a clausal form and then finding a set of instances of
the input clauses that is unsatisfiable. \systemname{leanCoP} proves the
unsatisfiability by building a connection tableaux, i.e. a tree,\footnote{In some implementations
this is a rooted forest, as there can be multiple literals in the start clause.}
where
every node contains a literal of the following properties.
\begin{itemize}
  \item The root of the tree is an instance of a given initial clause.
  \item The children of every non-leaf node are an instance of an input clause

    (we call such clauses
    axioms). Moreover, one of the child literals must be complementary
    to the node.
  \item Every leaf node is complementary to an element of its path.
\end{itemize}

\begin{figure}
  \includegraphics[width = \hsize]{tableaux.pdf}
\caption{Example of a closed connection tableaux, adapted from~\cite{Letz1994ControlledIO}.}\label{tableaux}
\end{figure}

The tree is built during the proof process which involves
automatic computation of substitutions using unification. Therefore the only
decisions that have to be made are ``which axiom should be used for
which node?''. In particular, \systemname{leanCoP} starts with the initial
clause and in every step, it selects the left-most unclosed (open) leaf. If the
leaf can be unified with an element of the path, the unification is
applied. Otherwise, the leaf has to be unified with a literal in an
axiom, and a decision, which literal in which axiom to use, has to be
made. The instance of the axiom is then added to the tree and the
process continues until the entire tree is closed (i.e., the prover
wins, see Fig. \ref{tableaux}), or there is no remaining available move (i.e., the prover
loses). As most branches are infinite, additional limits are introduced
and the prover also loses if such a limit is reached.
In our experiments, we use a version of the prover with two additional
optimizations: lemmata and regularization, originally proposed by Otten~\cite{DBLP:journals/aicom/Otten10}.

\begin{figure}
  \centerline{\includegraphics[width = 8cm]{tableaux2.pdf}}
  \caption{A state in the \systemname{leanCoP} solving process}\label{tabstate}
\end{figure}

To guide the proof search (Fig.~\ref{tabstate}), i.e. to select the next action,
we use Monte Carlo Tree Search with policy and value, similar to the AlphaZero~\cite{silver2017mastering} algorithm. This
means that the trainable model should take a \systemname{leanCoP} state as its
input, and return estimated value of the state, i.e., the probability that the
prover will win, and the action logits, i.e., real numbers assigned to every
available action. The action probabilities are then computed from action
logits using the softmax function.

To process the \systemname{leanCoP} state with our network, we first need to
convert it to a list of clauses. If there are $A$ axioms, and a
path of length $P$, we give the network the $A+P+1$ clauses: every
axiom is a clause and every element in the path is a clause consisting of
one literal. The last clause given to the network consists of all
the unfinished goals, both under the current path and in earlier
branches. This roughly corresponds to the set of clauses from which we
aim to obtain the contradiction. The initial labels of the clauses can be
therefore of 3 types: a clause originating from a goal, a member of a path, or an
axiom. Each of these types represent a learnable initial vector of
dimension 4.

The symbols can be of two types: predicates and functions, their
initial value is represented by a single real number: zero for
predicates, and a learnable number for functions. For term nodes,
variables in different axioms are always considered to be different,
and they are also considered to be different from the variables in the
tableaux (note that unification performs variable renaming). Variables in the tableaux are shared among the path and the
goals. Every term node can be of four types: a variable in an axiom,
a variable in the tableaux, a literal, or another term. The term nodes have
initial dimension 4.

Afterwards, we propagate through five message passing layers, with
dimensions $d_{\mathbf c}^i = d_{\mathbf c}^i = 32$,
$d_{\mathbf s}^i = 64$, obtaining $c_{5,j}$, $s_{5,j}$ and $t_{5,j}$.
Then we consider all the $c_{5,j}$ vectors, apply
a hidden layer of size 64 with ReLU activation to them, apply
the $\mathop{\mathrm{red}}\limits$ reduction and use one more hidden layer of size 64 with ReLU
activation. The final value
is then computed by a linear
layer with sigmoid activation.

Given the general setup, we now describe how we compute the logit for an action corresponding to the use of axiom
$C_i$, and complementing its literal $T_j$ with the current goal. Let
$C_k$ represents the clause of all the remaining goals. We concatenate
$c_{5,i}$, $t_{5,j}$ and $c_{5,k}$, process it with a hidden layer
of size 64 with ReLU activation, and then use a linear output layer
(without the activation function).

With the \systemname{leanCoP} prover, we perform four solving and training
iterations. In every solving iteration, we attempt to solve every problem in
the dataset, generating training data in the meantime. The training
data are then used for training the network, minimizing the
cross-entropy with the target action probabilities and the MSE of the
target value of every trained state. Every solving iteration therefore
produces the target for action policy, and for value estimation, that are used
for the following training.

The solving iteration number 0 (we also call it ``bare prover'') is
performed differently from the
following ones. We use the prover without guidance, performing random
steps with the step limit 200 repeatedly within a time limit.
For every proof we find, we run the random solver again from every
partial point in the proof, estimating the probabilities that the
particular actions will lead to a solution. This is our training data
for action probabilities. In order to get training data for value, we take all the states which we
estimated during the computation of action probabilities. If the
probability of finding a proof is non-zero in that state, we give it value
1, otherwise, we give it value 0.

Every other solving iteration is based on the network guidance in an
MCTS setting, analogously to AlphaZero~\cite{silver2017mastering} and
to the \systemname{rlCoP} system~\cite{KaliszykUMO18}. In order to decide
on the action, we first built a game tree of size 200 according to the PUCT formula
$$
U(s,a) = \log\left(\frac{1+N(s)+c_{\mathrm{base}}}{c_{\mathrm{base}}}\right)\cdot\frac{\sqrt{N(s)}}{1+N(s,a)},
$$
where the prior probabilities and values are given by the network, and
then we select the most visited node (performing a bigstep). This contrasts to the previous
experiment with a simpler clasifier~\cite{KaliszykUMO18} where
every decision node is given 2000 new expansions (in addition to the
expansions already performed on the node).
Additionally a limit of game steps of 200 has been added.
The target probabilities of any state in every bigstep is proportional
the number of visit counts of the appropriate actions in the tree
search. The target value in these states is boolean depending on
whether the proof was ultimately found or not.

\section{DeepMath Experiments}
\label{deepmath-experiments}
DeepMath is a dataset developed for the
first deep learning experiments with premise selection~\cite{IrvingSAECU16ju} on the Mizar40
problems~\cite{KaliszykU13b}. Unlike other datasets such as
HOLStep~\cite{DBLP:conf/iclr/KaliszykCS17}, DeepMath contains first-order formulas
which makes it more suitable for our network. We used the dataset for
two experiments -- \emph{premise selection} (Section~\ref{premise-experiments}) and recovering symbol names from the structure, i.e. \emph{symbol guessing} (Section~\ref{symbol-experiments}).

\subsection{Premise Selection}
\label{premise-experiments}
DeepMath contains 32524 conjectures, and a balanced list of positive
and negative premises for each conjecture. There are on average 8
positive and 8 negative premises for each conjecture. The task we consider first is to
tell apart the positive and negative premises.

For our purposes, we randomly divided the conjectures into 3252 testing
conjectures and 29272 training conjectures. For every conjecture,
we clausified the negated conjecture together with
all its (negative and positive) premises, and gave them all as input to the
network (as a set of clauses). We kept the hyperparameters
from the \systemname{leanCoP} experiment. There are two differences. First, there
are just two types of clause nodes: negated conjectures and premises.
Second, we consider just one type of variable nodes.

To obtain the output, we reduce (using the $\mathop{\mathrm{red}}\limits$ function introduced
 in Section~\ref{architecture}) the clause nodes belonging to the conjecture and we do the same
also for each premise. The the two results are concatenated and pushed through a
hidden layer of size 128 with ReLU activation. Finally,
an output layer (with sigmoid activation) is applied to obtain the
estimated probability of the
premise being positive (i.e., relevant for the conjecture).

\subsection{Recovering Symbol Names from the Structure}
\label{symbol-experiments}
In addition to the standard premise selection task, our setting is
also suitable for defining and experimenting with a novel interesting
task: \emph{guessing the names of the symbols} from the structure of the formula.
In particular, since the network has no information about the names of the symbols,
it is interesting to see how much the trained system can correctly
guess the exact names of the function and predicates symbols based just on the problem
structure.

One of the interesting uses
is for \emph{conjecturing by
  analogies}~\cite{GauthierKU16}, i.e., creating new conjectures by
detecting and following alignments of various mathematical theories
and concepts. Typical examples include the alignment between the
theories of additive and multiplicative groups, complex and real
vector spaces, dual operations such as join and meet in lattices, etc.
The first systems used for alignment detection have been so far
manually engineered~\cite{GauthierK19}, whereas in our setting such
alignment is just a byproduct of the structural learning.

There are two ways how a new unique
symbol can arise during the clausification process.
Either as a Skolem function, or as a new definition (predicate)
that represents parts of the original formulas.
We performed two experiments based on how
such new symbols are handled. We either ignore them, and train the neural
network on the original (labeled) symbols only, or we give to all the
new symbols the common labels \texttt{skolem} and
\texttt{def}. Table~\ref{tab:sym-distr} shows the frequencies of
the five most common symbols in the DeepMath dataset after the
clausification. Note that the newly introduced skolems and definitions
account for almost 40\% of the data.

\begin{table}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{cccccc}
TPTP name & \texttt{def} & \texttt{skolem} & \texttt{=} & \texttt{m1\_subset\_1} & \texttt{k1\_zfmisc\_1} \\
Mizar name & N/A & N/A & \texttt{=} & \texttt{Element} & \texttt{bool} \\
Frequency & 21.5\% & 17.3\% & 2.0\% & 1.7\% & 1.2\%\cr
\end{tabular}
\caption{\label{tab:sym-distr} The most common symbols in the clausified DeepMath.}
\end{table}

\section{Experimental Results}
\label{results}
\subsection{Guiding \systemname{leanCoP}}
We evaluate our neural guided \systemname{leanCoP} against rlCoP~\cite{KaliszykUMO18}. Note however, that for both systems we
use 200 playouts per MCTS decision so the \systemname{rlCoP} results presented here are different from~\cite{KaliszykUMO18}. We start with a set of \systemname{leanCoP} states with their values and action probabilities coming from the 4595 training
problems
solved with the bare random prover.

After training on this set, the MCTS guided
by our network manages to solve 11978 training (160.7\% more) and 1322 (159.2\% more) testing problems, in total 13300 problems (160.5\% more -- Fig.~\ref{invcop}). This is
in total
49.1\%
more than \systemname{rlCoP} guided by XGBoost  which in the same setup and with the same limits
solves 8012 training problems, 908 testing problems, and 8920 problems in total.
The improvement in the first iteration over XGBoost on the training and testing set is 49.5\% and 45.6\% respectively.

Subsequent iterations are also much better than for rlCoP, with
slower progress already in third iteration (note that rlCoP also loses problems, starting with
6th iteration).
The evaluation ran 100 provers in parallel on multiple CPUs
communicating with the network running on a GPU.
Receiving the queries from the prover
takes
on average 0.1 s, while
the message-passing layers alone take around 0.12 s per batch.  The
current main speed issue turned out to be the communication overhead
between the provers and the network. The average inference step in 100
agents and one network inference took on average 0.57 sec.

\begin{figure}
    \centering

\begin{tabular}{ l | r r r r r }
invariant net guided & bare & iter. 1 & iter. 2 & iter. 3 \cr
\hline
overall  & 5105 & 13300 & 14042 & 14002\cr
training & 4595 & 11978 & 12648 & 12642\cr
testing  &  510 &  1322 &  1394 &  1360\cr
\hline
rlCoP & bare & iter. 1 & iter. 2 & iter. 3 \cr
\hline
overall  & 5105 & 8920 & 10030 & 10959 \cr
training & 4595 & 8012 & 9042 & 9874 \cr
testing  &  510 & 908 & 988 & 1085 \cr
\end{tabular}
\caption{Comparison of the number of problems solved by leanCoP guided by the invariant-preserving GNN and by XGBoost.}
    \label{invcop}
\end{figure}

\bigskip

\begin{figure}
    \centering
    \includegraphics[width=.9\linewidth]{plot_premsel_pp.pdf}
    \caption{Testing and training accuracy on the premise selection task on the DeepMath dataset.}
    \label{fig:prem_sel}
\end{figure}

\subsection{Premise Selection}
In the first DeepMath experiment with Boolean classification, we obtained testing accuracy of around $80\%$. We trained the network in 100 epochs on minibatches of size 50. A stability issue can be spotted around the epoch 60 from which the network quickly recovered.
We cannot compare the results to the standard methods since the dataset is by design hostile to them -- the negatives samples are based on the KNN, so KNN has accuracy even less than $50\%$. Simpler neural networks were previously tested on the same dataset~\cite{DBLP:journals/corr/abs-1807-10268}
reaching accuracy $76.45\%$ .

\subsection{Recovering Symbol Names from the Structure}
For guessing of symbol names, we used minibatches consisting only of
10 queries, and trained the network for 50 epochs. When training and
evaluating on the labeled symbols only, the testing accuracy reached
$65.27\%$ in the last epoch. Note that this accuracy is measured on
the whole graph, i.e., we count both the symbols of the conjecture and of the
premises. When training and evaluating also on the
\texttt{def} and \texttt{skolem} symbols, the testing accuracy reached
$78.4\%$ in the last epoch -- see Fig.~\ref{fig:symbol_guess_all}.

\begin{figure}
\centering
  \centering
    \includegraphics[width=.9\linewidth]{plot_symbol_guessing_labeled_pp.pdf}
\caption{Testing and training accuracy on the label guessing task on the DeepMath dataset.}
    \label{fig:symbol_guess_labeled}
  \end{figure}

\begin{figure}
  \centering
    \includegraphics[width=.9\linewidth]{plot_symbol_guessing_all_pp.pdf}
    \caption{Testing and training accuracy on the label guessing task including labels \texttt{def} and \texttt{skolem} on the DeepMath dataset.}
    \label{fig:symbol_guess_all}
\end{figure}

We evaluate the symbol
guessing (without considering  \texttt{def} and \texttt{skolem}) in more detail on the 3252 test problems and their conjectures.
In particular, for each of these problems and each conjecture symbol,
the evaluation with the trained network gives a list of candidate
symbol names ranked by their probability.  We first compute the number
of cases where the most probable symbol name as suggested by the
trained network is the correct one.  This happens in 22409 cases out
of 32196, i.e., in 70\% cases.\footnote{This differs from the
  testing accuracy of $65.27\%$
  mentioned above, because we only consider the conjecture symbols here.}
A perfect naming of all symbols is achieved for 544 conjectures, i.e.,
in 16.7\% of the test cases. Some of the most common analogies
measured as the common symbol-naming mistakes done on the test
conjectures are shown in Table~\ref{analogies}.
\begin{table}
\begin{center}
\begin{scriptsize}
  \caption{\label{analogies} Some of the common analogies}
\centering
  \begin{tabular}{lllll}
    \toprule
count &   original Mizar symbol &  Mizar analogy \\
\midrule
   129& Relation-like&Function-like\\
     69&void&empty\\
     53&Abelian&add-associative\\
     47&total&-defined\\
     45&0&1\\
     40&+&*\\
     39&reflexive&transitive\\
     38&Function-like&FinSequence-like\\
     33&-&+\\
     31&trivial&empty\\
     28& $>=$&=\\
     27&associative&transitive\\
     26&infinite&Function-like\\
     25&empty&degenerated\\
     24&real&natural\\
     23&sigma-multiplicative&compl-closed\\
     20&REAL&COMPLEX\\
     18&transitive&reflexive\\
     18&RelStr&TopStruct\\
     18&Category-like&transitive\\
     17&=&c=\\
     16&initial&infinite\\
     16&[Graph-like]&Function-like\\
     16&associative&Group-like\\
     16&0&\{\}\\
     16&/&*\\
     15&add-associative&associative\\
     15&-&*\\
     13&width&len\\
     13&integer&natural\\
     13&in&c=\\
     12 &$\cap$&$\cup$\\
     11 &c=&$>=$\\
     10 &with\_infima&with\_suprema\\
     10 &ordinal&natural\\
      9 &closed&open\\
      8 &sup&inf\\
      8 &Submodule&Subspace\\
      7 &Int&Cl\\
\bottomrule
  \end{tabular}
\end{scriptsize}
\end{center}
\end{table}

We briefly analyze some of the analogies produced by the network predictions.
In theorem
\texttt{XBOOLE\_1:25}\footnote{\url{http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/xboole_1\#T25}}
below, the trained network's best guess correctly labels the symbols
as binary intersection and union (both with probability ca 0.75).
Its second best guess is however also quite probable ($p=0.2$),
swapping the union and intersection. This is quite common, probably
because dual theorems about these two symbols are frequent in the
training data. Interestingly, the second best guess results also in an
provable conjecture, since it easily follows from
\texttt{XBOOLE\_1:25} just by symmetry of equality.
\begin{lstlisting}[language=Mizar,basicstyle=\ttfamily\scriptsize]
theorem :: XBOOLE_1:25
for X, Y, Z being set holds ((X /\ Y) \/ (Y /\ Z)) \/ (Z /\ X) = ((X \/ Y) /\ (Y \/ Z)) /\ (Z \/ X)

second guess:
for X, Y, Z being set holds ((X \/ Y) /\ (Y \/ Z)) /\ (Z \/ X) = ((X /\ Y) \/ (Y /\ Z)) \/ (Z /\ X)
\end{lstlisting}
In theorem
\texttt{CLVECT\_1:72}\footnote{\url{http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/clvect_1\#T72}}
the trained network has consistently decided to replace the symbols
defined for complex vector spaces with their analogs defined for real
vector spaces (i.e., those symbols are ranked higher). This is most likely because of the large theory of real
vector spaces in the training data, even though the exact theorem
\texttt{RLSUB\_1:53}\footnote{\url{http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/rlsub_1\#T53}} was not among the training data. This again means that the trained network has produced \texttt{RLSUB\_1:53} as a new (provable) conjecture.
\begin{lstlisting}[language=Mizar,basicstyle=\ttfamily\scriptsize]
theorem :: CLVECT_1:72
for V being ComplexLinearSpace for u, v being VECTOR of V
for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )

theorem :: RLSUB_1:53
for V being RealLinearSpace for u, v being VECTOR of V
for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )
\end{lstlisting}

Finally, we show below two examples. The first one illustrates on theorems
\texttt{LATTICE4:15}\footnote{\url{http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/lattice4\#T15}} and
\texttt{LATTICE4:23}\footnote{\url{http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/lattice4\#T23}}
the network finding well-known dualities of concepts in lattices (join vs. meet, upper-bounded vs. lower-bounded and related concepts). The second one is an example of a discovered analogy between division and subtraction operations on complex numbers, i.e, conjecturing
\texttt{MEMBER\_1:130}\footnote{\url{http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/member_1\#T130}} from
\texttt{MEMBER\_1:77}\footnote{\url{http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/member_1\#T77}}.

\begin{lstlisting}[language=Mizar,basicstyle=\ttfamily\scriptsize]
theorem :: LATTICE4:15
for 0L being lower-bounded Lattice
for B1, B2 being Finite_Subset of the carrier of 0L holds
(FinJoin B1) "\/" (FinJoin B2) = FinJoin (B1 \/ B2)

similar to:
theorem Th23: :: LATTICE4:23
for 1L being upper-bounded Lattice
for B1, B2 being Finite_Subset of the carrier of 1L holds
(FinMeet B1) "/\" (FinMeet B2) = FinMeet (B1 \/ B2)

theorem :: MEMBER_1:77
for a, b, s being complex number holds
{a,b} -- {s} = {(a - s),(b - s)}

similar to:
theorem :: MEMBER_1:130
for a, b, s being complex number holds
{a,b} /// {s} = {(a / s),(b / s)}
\end{lstlisting}

\section{Related Work}

Early work on combining machine learning with automated theorem
proving includes,
e.g.,~\cite{DBLP:conf/ogai/ErtelSS89,DenzingerFGS99,DBLP:books/daglib/0002958}.
Machine learning over large formal corpora created from ITP
libraries~\cite{Urban06,MengP08,holyhammer} has been used for premise
selection~\cite{Urb04-MPTP0,US+08-long,KuhlweinLTUH12-long,IrvingSAECU16ju},
resulting in strong \emph{hammer} systems for selecting relevant facts
for proving new conjectures over large formal
libraries~\cite{abs-1108-3446,BlanchetteGKKU16,hh4h4}. More recently,
machine learning has also started to be used to guide the internal
search of the ATP systems. In saturation-style provers this has been
done by feedback loops for strategy
invention~\cite{blistr,JakubuvU18a,SchaferS15} and by using supervised
learning~\cite{JakubuvU17a,LoosISK17} to select the next given
clause~\cite{Overbeek:1974:NCA:321812.321814}. In the simpler
connection tableau systems such as \systemname{leanCoP}~\cite{OB03} used here, supervised
learning has been used to choose the next tableau extension
step~\cite{UrbanVS11,KaliszykU15}, using Monte-Carlo guided proof
search~\cite{FarberKU17} and reinforcement
learning~\cite{KaliszykUMO18} with fast non-deep learners. Our main evaluation is done in this setting.

Deep neural networks for classification of mathematical formulae were first
introduced in the DeepMath experiments~\cite{IrvingSAECU16ju}
with 1D convolutional networks and LSTM networks.
For higher-order logic, the HolStep~\cite{DBLP:conf/iclr/KaliszykCS17} dataset
was extracted from
interactive theorem prover HOL Light. 1D convolutional neural networks, LSTM, and their combination were proposed as baselines for the dataset.
On this dataset a Graph-based neural network was for a first time applied to theorem proving in the FormulaNet~\cite{DBLP:conf/nips/WangTWD17} work.
FormulaNet, like our work, also represents identical
variables by a single nodes in a graph, being therefore invariant
under variable renaming. Unlike our network, FormulaNet glues
variables only and not more complex terms.
FormulaNet is not designed
specifically for first-order logic, therefore it lacks invariance under
negation and possibly reordering of clauses and literals.
The
greatest difference is however that our network
abstracts over the symbol names
while FormulaNet learns them individually.

A different invariance property was proposed in a network for propositional calculus by Selsam et
al. \cite{SelsamLBLMD19}. This network is invariant under negation,
order of clauses, and order of literals in clauses, however this
is restricted to propositional logic, where no quantifiers and variables
are present.
In the first-order setting, Kucik and
Korovin~\cite{DBLP:journals/corr/abs-1807-10268} performed experiments with basic neural
networks with one hidden layer on the DeepMath dataset.
Neural networks reappeared in state-of-the-art saturation-based proving (E prover) in the work of Loos et al.~\cite{slgicscklpar17}. The considered models included CNNs, LSTMs, dilated convolutions, and tree models.
The first practical comparison of neural networks, XGBoost and Liblinear in guiding E prover was done by Chvalovsky et al. \cite{ChvalovskyJ0U19}.

An alternative to connecting an identifier with all the formulas about it, is to perform definitional embeddings. This has for the first time been done in the context of theorem proving in DeepMath \cite{IrvingSAECU16ju}, however in a non-recursive way. A fully recursive, but non-deep name-independent encoding has been used and evaluated in HOLyHammer experiments~\cite{ckju-mcs-hh}. Similarity between concepts has been discovered using alignments, see e.g.~\cite{DBLP:conf/lpar/GauthierK15}.
Embeddings of particular individual logical concepts have been considered as well, for example polynomials \cite{AllamanisCKS17} or equations \cite{abs-1803-09123}.

\section{Conclusion}

We presented a neural network for processing mathematical formulae invariant under symbol names, negation and ordering of clauses and their literals, and we demonstrated its learning capabilities in
three automated reasoning tasks.
In particular, the network improves over the previous version of \systemname{rlCoP}
guided by XGBoost by 45.6\% on the test set in the first iteration of
learning-guided proving. It also outperforms earlier methods on the
premise-selection data, and establishes a strong baseline for symbol
guessing. One of its novel uses proposed here and allowed by this
neural architecture is creating new conjectures by detecting and
following alignments of various mathematical theories and
concepts. This task turns out to be a straightforward application of the structural
learning performed by the network.

Possible future work includes for example
integration with state-of-the-art saturation-style provers.
An interesting next step is also evaluation on a heterogeneous dataset such as TPTP where
symbols are not used consistently and learning on multiple libraries --
e.g. jointly on HOL and HOL Light as done previously
by~\cite{DBLP:conf/lpar/GauthierK15} using a hand-crafted alignment
system.

\section{Acknowledgements}
Ol\v{s}\'{a}k and Kaliszyk were supported by the ERC Project \emph{SMART} Starting Grant no. 714034.
Urban was supported by the \textit{AI4REASON} ERC
Consolidator grant number 649043, and by the Czech project
AI\&Reasoning CZ.02.1.01/0.0/0.0/15\_003/0000466 and the European
Regional Development Fund.

\input{leancop-graph-nn.bbl}
\end{document}
When converting the clauses to the graph, we aim to capture as much relevant structure as possible. We roughly convert the tree structure of the terms to a circuit by sharing variables, constants and also bigger terms. The graph will be also interconnected through special nodes representing function symbols.

redpajama_set_name: RedPajamaArXiv


We are growing our London office and so we are looking for aspirational designers at all levels to join us.
At the core of our shared culture is the belief in Light as Craft, by which we mean a tailored approach to all of our projects and an investment in the time we take to create our designs.
You will need to be proficient in AutoCAD, Adobe Creative Suite (in particular Photoshop / InDesign), industry standard calculation software and Microsoft Office. You should also have at least basic sketching skills and be able to communicate well in English.
In exchange, we can offer you a stimulating, friendly, and supportive working environment with a focus on growth and development. Being part of ÅF Lighting means you would have access to a unique network of skills and training through the newly established ÅF Academy.
Applicants must be eligible to work in the UK. Independent applications only, no agencies, please.


Here's a simple, minimal implementation of AJAX in Drupal using jQuery.
First create a menu item callback.
Next create the callback function. To test the menu item and callback function, go to http://[YOURSITE]/ajax; you should get the text "wee!" without any theming.
Next create a page that has a button on it with an ID of "ajaxButton". The following jQuery will add an onClick event to the button.


Ape Municipality (Latvie: Apes novads) is a municipality in Vidzeme, Latvie. The municipality wis formit in 2009 bi mergin Gaujiena pairish, Vireši pairish, Trapene pairish an Ape toun wi its kintraside territory the admeenistrative centre bein Ape.
This page wis last eeditit on 19 November 2013, at 18:33.


We are looking for someone with great customer service, a positive attitude, great problem solving abilities and the potential for growth.
We have a vacancy on our team that are the go to for any IT queries. You would become a valued member of the IT Service Team providing 24/7 support to our business and customers. If you are a kind, helpful, friendly, tech savvy and smart individual, you could be the perfect fit!
Flexibility to work in a rotational roster that covers days/evenings/nights over 7 days.
Listed in the NZX Top 10, we are one of the most successful and fastest growing companies in NZ. Employing 4500+ staff across healthcare, construction and corporate fields in both NZ and Melbourne, we are varied and diverse! We have a 'grow our own' philosophy, invest in developing our people and promote a culture of kindness!
If you're keen to join the IT Service team, simply apply online now.


After a week of taking meals on the fly, enduring the everyday traffic, and dealing with work deadlines and business meetings, I'm sure we all want one thing during the weekend. In chorus, we would all say, "Rest!" Instead of staying in your house, why not give yourself a little luxury to indulge in peace, somewhere away from the hustle and bustle of the city?
Good thing there are many massage centres and physiotherapy clinics in Singapore that can address your sweet escape cravings over the weekend.
Get all the muscle tensions, muscle spasms, inflammation, back aches, and stiffness down the drain through the massage therapists' magic. They use kneads, rubs, and strokes that point to affected muscles to increase blood circulation throughout the body. Even massage spas use the discipline of physiotherapy in Singapore. You sure won't get your money to waste.
This type of massage targets the muscles seated in the deeper layers of the skin. Therapists usually use slow but hard strokes, and it is perfect for those who experience muscle strains. Those people who have problems in posture or who are recovering from a minor injury can largely benefit from this type of massage.
You'll get to different layers of seventh heaven from the long, slow strokes with circular movements of Swedish massage. It is very popular in the United States because this massage therapy is gentle and relaxing, just right for people who want to drain all the body's tension away.
This type of massage has a large background of science behind it. Therapists press and target the body's acupuncture points. They hold the pressure down for about five to eight seconds to improve the blood circulation and help the body regain balance. It is common to actually feel sore after a Shiatsu massage because pressure points have been stimulated, but this kind of body reaction is just normal. You'll feel better and more energized after a night's sleep. Muscles have been loosened, and strains have been released.
This type of massage is just like doing a yoga, only that the massage therapist does all the hard work and stretching. According to people who frequent spas, this is the most invigorating of all other types of massage. There are a number of compressions and stretches that aim to increase your body's flexibility and range of motion.
Yes, adulting life is an unstoppable force. Surely, you have friends who face the same, real struggle. Hence, they might have been into massage centres at least a couple of time. You may ask them where their proffered spa is. Otherwise, you can ask your local physiotherapist where to get a good massage.
Having a degree or certificate from a program that is accredited or approved by the Commission on Massage Training Accreditation (COMTA) or the National Certification Board of Therapeutic Massage and Bodywork is a good indicator that the therapist knows what she or he is doing. For your peace of mind, better turn to licensed or certified ones.


Triglav Park is located in a picturesque setting where all things Slovenian are celebrated. Recognizing our rich history, sense of family, faith life and cultural significance, the vision for the current location in the Town of Norway was for Triglav Park to be a second home to the many Slovenian refugees who came to Wisconsin. In the 50 years since, generations have found serenity, history and pride at Triglav.
Your support helps us to carry the Triglav legacy of service and culture through our next 50 years, ensuring that the Slovenian spirit of giving continues on. We can be very proud of Triglav Park, a place that many have come to enjoy whether they are Slovenian in ancestry or spirit!


text:
UHB NHS Foundation Trust Research and InnovationThe Scar Free Foundation Birmingham Burns Research CentreCentre for Burns Research partnersBirmingham Children's Hospital
The Birmingham Paediatric Burns Centre is located at the Birmingham Children's Hospital (BCH) in a state-of-the-art, nine-bedded unit adjacent to a 22-bed Paediatric critical care unit.
Investment in 2010/11 is supporting the expansion of the this critical care facility to 31 beds, and surgical leadership is provided by the world class team of dedicated burns surgeons at Queen Elizabeth Hospital Birmingham as part of a close working relationship betwene the two hospitals.
Birmingham Children's Hospital runs a paediatric Wellcome Clinical Research Facility, which is managed in partnership with the University of Birmingham and University Hospitals Birmingham NHS Foundation Trust (UHB).
BCH receives Experimental Cancer Medicine Centre funding, hosts the West Midlands Medicines for Children Research Network, and an NIHR funded research capital bid 3tMRI scanner will come into operation within the next year.
Click here to visit the website of Birmingham Children's Hospital NHS Foundation Trust


text:
Q: How can i solve npm error on install -g yarn? i want to run this command and install yarn
npm install -g yarn
but i get this error
npm ERR! code ENOENT
npm ERR! syscall spawn bash
npm ERR! path C:\Users\DigiMax\AppData\Roaming\npm\node_modules\yarn
npm ERR! errno -4058
npm ERR! enoent spawn bash ENOENT
npm ERR! enoent This is related to npm not being able to find a file.
npm ERR! enoent

npm ERR! A complete log of this run can be found in:
npm ERR!     C:\Users\DigiMax\AppData\Local\npm-cache\_logs\2021-07-03T15_03_24_683Z-debug.log

how can i solve this problem?

A: The quickest solution is npm cache clean and npm cache verify works in most cases.
According to docs.npmjs you should check your npm version and update if needed or reinstall it at worst.
Please, let us know if it worked

A: This problem solve by "running VSCode as administrator"

redpajama_set_name: RedPajamaStackExchange


Since DisplayObject is an abstract class, you cannot instantiate it directly, but have to use one of its subclasses instead. There are already a lot of them available, and most of the time they will suffice.
Have a look at the Quad class for a sample implementation of the 'getBounds' method. For a sample on how to write a custom render function, you can have a look at this article in the Starling Wiki.
When you override the render method, it is important that you call the method 'finishQuadBatch' of the support object. This forces Starling to render all quads that were accumulated before by different render methods (for performance reasons). Otherwise, the z-ordering will be incorrect.
The filter that is attached to the display object. The starling.filters package contains several classes that define specific filters you can use. Beware that a filter should NOT be attached to different objects simultaneously (for performance reasons). Furthermore, when you set this property to 'null' or assign a different filter, the previous filter is NOT disposed automatically (since you might want to reuse it).


text:
filed: May 17, 2017 • South Dakota
Hughes County to study windmill rules as California company plans two wind farms south of Harrold
Credit: By Stephen Lee | Capital Journal | www.capjournal.com ~~
Jan Krull and her husband, Ross, run 200 Black Angus cows on 2,000 acres of grass south of Harrold, South Dakota and she is all for wind energy.
"I love windmills," she said Monday in the Hughes County courthouse in Pierre.
"We have an old wooden one on the ranch. It doesn't work anymore." But the 20-foot tall weathered old windmill is a sort of ornament of history on the prairie that she treasures.
Krull also is looking ahead to windmills of a more modern design. Ones with towers soaring 220 feet or more above the grass, with blades that are 140 feet long, maybe.
She and Ross Krull and several other landowners attended the Hughes County Commission meeting on Monday for a public hearing.
They are two of several ranchers and farmers in eastern Hughes County and western Hyde County who have leased land to Infinity Renewables, a Santa Barbara, California company planning to develop two wind farms south of Harrold.
The Hughes County Commission heard recently about Infinity making deals with landowners and had one landowner contact the Commission worried about whether county ordinances protected landowners well, said County Manager Kevin Hipple.
Krull retired two years ago from a job with the state Department of Transportation in which she acquired right-of-way easements from landowners so the state could facilitate transportation.
So she knows about negotiating for such rights.
"We have leased them six quarters," she said of her being on the other side of the negotiating table with Dave Mabene, land agent for Infinity Renewables, who was doing the buying in leasing 960 acres from the Krulls for the company to place the big wind turbines.
On Monday, after a short public hearing in front of a packed room, the five-man Commission voted to put a six-month moratorium on any wind-energy development until county officials can study and probably tweak the ordinances.
They were written eight or nine years ago and things have changed, said Commission Chairman Norm Weaver.
Looking at what other counties are doing, especially increasing the "setbacks," or distances between the tall turbines and any homes, roads or other things already around, is a good idea, said Weaver and other commissioners.
Commissioner Tom Tveit said he's informed by the experience of a relative who has a cattle operation in North Dakota and had a wind-energy company build a big turbine right in the middle of a new and expensive cattle gate the man had just installed.
"That's not being a good neighbor," Tveit said.
However, he also does not want to stand in the way of economic development and wants the county to move fast on updating its zoning ordinances for wind farms, Tveit said, as he moved to authorize the six-month moratorium.
Mabene told the Commission that Infinity Renewables is cool with that.
"Six months is not going to really make a difference to us," he told the Commission. "We will not be able to move on these projects within a six-month time frame."
He and Christine White, a project developer for Infinity, attended the public hearing to find out how they can help, Mabene said.
"We are here to hear what everyone has to say. We are not necessarily opposed to a moratorium if that's what you think you need to do to make reasonable changes," he said. "We want to offer ourselves up to be a part of that. . .. We've got experience from Dakota down to Texas in wind development and got a lot of miles under our belt. We're here to offer assistance."
Started up several years ago as Infinity Wind, the company's founders say they wanted to focus on the startup process of wind farms. Especially, they said, in acquiring land right-of-way, or easements, for the wind turbines, and in building relationships with landowners, nearby residents and local government officials to avoid the conflicts that often arise in such developments.
Infinity has worked on several projects in several states in the middle section of the country, said White.
They include the Sunflower wind farm of 52 turbines that is near Hebron, North Dakota, 60 miles west of Bismarck on the north side of Interstate 94. Sunflower began generating power a few months ago. Infinity is eager to develop two projects south of Harrold because "this is where the wind blows," White said of the Dakotas.
In 2016, the 974 wind turbines standing and turning – on windy days, anyway – across South Dakota, generated 3.1 million megawatt hours of electrical power, said Darren Kearney, a utilities analyst for the state's Public Utilities Commission.
That was 36 percent of the "plated capacity," or total potential power of the wind generators if they turned all the time.
Wind turbines are getting more efficient and now many of the newer and bigger ones average 45 percent or better of their plated capacity, Kearney said.
In 2015, 26 percent of the electrical power generated in the state came from wind; while 50 percent came from hydroelectric dams, 16 percent came from coal and 8 percent came from natural gas, Kearney said.
Last year, 30 percent of the electrical power generated in the state came from wind turbines and 39 percent from hydroelectric sources.
Talking to landowners on Monday, Mebane said Infinity Renewables – which changed to a more expansive name when it added solar energy projects a few years ago – tries to take into account the concerns of landowners and residents near their sites, even if the landowners are not receiving any easement like their neighbors who do lease land to Infinity.
"The last thing we want to do is create bad blood between anybody," he said.
So Infinity "sets back" wind turbines 1,400 feet from any residence, farther than many government regulations require, he said.
Many complaints are voiced about the giant windmills, across the state and nationally, Lee McCurrin, the county's new planning and zoning director, told the Commission.
The issues often raised include the noise, the flickering shadows for anyone living close to one, as well as dangers to birds and bats and other reported effects of the huge electrical generating plants.
"I am really not concerned," Jan Krull said. Infinity seems to have the right ideas about how to design a wind farm to avoid many of the reported problems, she said.
Meanwhile, the money paid to landowners for leasing the land to Infinity is a good deal, she said.
White said the company doesn't make public what it pays landowners. Krull said she and her husband are satisfied with the deal she negotiated with Infinity. It's especially timely for ranchers and farmers dealing with lowering prices for crops and cattle compared to a few years ago.
"Right now, Hughes County is very dry," she said.
That means the same amount of pasture and rangeland can't support as many cattle, meaning "we might have to reduce our herds."
As for the money for the wind-turbine easements, "I think some of these ranchers are looking at it as part of their retirement," she said. "Anyway, it can help us get through tough times and get us to a better day."
Source: By Stephen Lee | Capital Journal | www.capjournal.com


Congratulations to our Fall 2015 Inaugural Scholarship Winners!
Thank for to all the Fall Semester Scholarship applicants and the many strong submissions that we had.
Spring Semester 2016 Scholarship Applications now being accepted! Please read application rules below.
JobApplicationGuide is pleased to announce our first annual scholarship program. As our online reputation continues to grow, it is our goal to give back to the community of people seeking higher education. Effective immediately JobApplicationGuide will be giving away a minimum of $1000 in scholarships annually. It is our goal to grow this annual scholarship fund as we continue to grow as an organization.
For our Spring 2016 scholarship awards we will again be giving away two $500 scholarships. Entrees must be received before December 15, 2015. Scholarships will be announced no later than Jan 5th, 2015 in anticipation of the scholarship award being paid out before the beginning of 2016 Spring semester.
$500 High School Achievement Scholarship.
The High School Achievement Scholarship will be given to one high school student who has displayed high achievements in grades and extracurricular activities during their senior year in high school. Applicants should also have a specific financial situation that needs assistance.
5. Detailed description of your high school experience, why you deserve this scholarship and what the college experience will mean to you.
6. At least 1 recommendation letter (one page) from mentor, teacher, coach or active member of your local community.
All submissions should be sent to scholarships@www.jobapplicationguide.com.
Our Adult Extended Education Scholarship will be awarded to one adult individual who is returning to college in Fall of 2015 and has not graduated from high school in the last 2 years. Applicants should be seeking to start or complete their college education after having taken time off to pursue real world activities or careers. Applications should also display financial need.
1. Reasons for returning or beginning pursuit of college education.
2. Description of activities or career since in high school.
3. College attending in Spring 2016.
5. Description of why you are seeking scholarship and what college means to you.
A. Legal resident of United States.
B. Must be at least 16 years of age at time of application.
C. For High School Achievement Scholarship must have at least a 3.0 GPA during senior year.
D. Must Submit Application before December 15, 2015.
E. Employees of JobApplicationGuide.com, its divisions, affiliates, subsidiaries, their law firms, accountants, program administrators, promotional or marketing agencies, and their immediate family members and persons living in the same households of each such employee are not eligible.
NO PAYMENT OR PURCHASE OF ANY SORT IS NEEDED OR ALLOWED TO ENTER OR WIN.
Winners will be notified not later than January 5, 2016. Winners will be announced via this page no later than January 15th, 2016.
For further information and all submissions, please email scholarships@www.jobapplicationguide.com.
The office of Scholarship Administrator is located at 2331 West Allview Terrace, Los Angeles, CA, 90068. Scholarship winner correspondence will be sent from this office. In addition, any mail correspondence can also be sent to this address.
All scholarship entries may be subject to verification before award. By entering, participants understand that they must hold harmless Sponsor and Administrator and their directors, subsidiaries, officers, employees, parents, affiliates, and agents from all inconvenience, damage, injury, loss, or irregularity coming from or relating to these Scholarships, even if caused by Sponsor negligence. In addition, materials submitted become the sole property of JobApplicationGuide.com and may not be acknowledged or returned. By submitting, you are agreeing to the rules and these decisions of JobApplicationGuide.com and its staff which are binding and final in all respects to this Scholarship.


means protecting against an unplanned pregnancy.
What Do I Do If I Have An STD?
Even skin-to-skin contact can spread STDs.
What Do I Need To Know About Using Condoms?
How Do I Put A Condom On (For Men And Women)?
step-by-step instructions will guide you.


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text:
Athens United States Decatur Florence General news Coronavirus Infectious diseases Diseases and conditions Health Lung disease State taxes Government taxation and revenue Government finance Government business and finance Business Government and politics State governments Local taxes Local governments Accommodations Travel Lifestyle Road and highway construction Transportation infrastructure Transportation and shipping Industrial products and services Heavy construction industry Construction and engineering Municipal governments Historic sites Sightseeing Leisure travel 2019-2020 Coronavirus pandemic Property taxes Hotel operators Hospitality and leisure industry Consumer services Consumer products and services
Editorial Roundup: Alabama
By The Associated Press - Oct. 21, 2020 11:20 AM EDT
Recent editorials from Alabama newspapers:
The Decatur Daily and The TimesDaily on outdoor trails and historic sites in Alabama:
With the new coronavirus taking its toll on leisure travel and many people stuck at home and starting to go stir crazy, it's important to remember that taking care of one's health means more than taking precautions against COVID-19. It also means looking after one's mental health.
One of the easiest ways is simply to get outdoors. Fortunately, north Alabama has an abundance of walking and biking trails people can enjoy while still being alert for the coronavirus.
Decatur's Dr. Bill Sims Bike Trail runs 14.7 miles from Point Mallard to Rhodes Ferry Park to Wilson Morgan Park, taking riders along some of Decatur's most scenic and historic points.
In Athens, the Swan Creek Greenway Trail starts at the SportsPlex north of Athens High School, connects to the loop track at Athens Middle School, passes Swan Creek Park, and then joins up with Swan Creek to cut through the woods until it ends at U.S. 72.
Farther north in Elkmont, the Richard Martin Rails-to-Trail offers more than 10 miles of trail suitable for walking or horseback riding. Winding through rural woods and past farmland, this trail also includes the site of the Battle of Sulphur Creek Trestle, which was listed on the National Register of Historic Places in 1973.
But one need not stick to official trails. Sometimes a walk though the neighborhood can reveal treasures hiding in plain sight.
The Alabama Historical Commission has recently gone live with an interactive map pinpointing thousands of historic buildings and sites throughout the state. You can find the map at https://ahc.alabama.gov/historicpreservationmap.aspx.
"The map is still a work in progress, but the agency said 42 of the state's 67 counties have been plotted so far, and more than 40,000 points have been mapped into one layer," according to The Associated Press. "About 130,000 documents have been scanned including survey forms, photographs, booklets, and maps."
Decatur and Hartselle are two areas that are already well-represented on the map. Zoom in on Decatur, and you'll see pinpoints lining every inch of the city's historic districts. Click on one at random, and you'll find homes like the J.W. Powell House at 720 Alma St. N.W. Click on a star, and you'll see registered historic sites like the Wayman A.M.E. Church.
"As an easily accessible public record of Alabama historic resources, it also includes documentation of properties receiving state and federal tax credits, cemeteries, Native American sites and historic African-American schools," said Alabama Historical Commission Chairman Eddie Griffith in a statement Monday. "This continuing effort will eventually expand to all of Alabama's counties, and future documentation of historic resources will be added."
Whether organized walking tours, seasonal "ghost tours" or simply going out on your own or with the family, with the Alabama Historical Commission's map on your smartphone, there are plenty of ways to escape to the outdoors before winter sets in.
Online: https://www.decaturdaily.com
Online: https://www.timesdaily.com/
The Dothan Eagle on motorists who fail to observe road signs in Alabama:
Some people would suggest that anyone who would throw caution to the wind and drive around a barrier or ignore a "road closed" sign deserves to encounter whatever hazard they were warned about. Houston County officials deserve credit for wondering if additional strategies might better protect motorists for their own poor decisions, and possibly mitigating the county government's potential liability.
"We can only do so much," County Engineer Barkley Kirkland told commissioners this week. "We do what we can, but when you have barricades up that say 'Road Closed' and dirt's dumped there, people need to have a little sense."
He's right. And he's far more diplomatic than many would be considering the scope of damage left in the wake of Hurricane Sally and the county Road and Bridge Department's mammoth task of road repair.
We'll remind motorists again: Several county roads are closed because flooding and erosion damaged the road beds. Don't be fooled by appearances; county road workers are professionals, and a road they deem unsafe and close until repairs are made is a road drivers should avoid.
There are warning signs, barricades and even mounds of dirt blocking dangerously damaged roads. We urge drivers to heed the warnings and turn back; failing to do so risks vehicle damage, injury, or death.
Online: https://www.dothaneagle.com
The TimesDaily on lodging taxes for short-term rentals in Alabama:
Tourism Director Susann Hamlin wants to collect lodging taxes from the 36 Airbnbs located in Colbert County.
She is not alone in her belief that short-term rentals should pay lodging taxes. After all, it seems logical that an Airbnb should fall under the same rules as hotels, campgrounds and bed and breakfast lodgings.
"When they're not held to the same standards as hotels, with regulations and taxes, it takes money away from cities and counties," she said.
For example, Florence-Lauderdale Tourism officials reported the Airbnbs in Lauderdale County are generating about $15,000 in lodging taxes a year.
Hamlin pointed out the taxing of short-term rentals has been an ongoing issue across Alabama for years. "I'd like them to be more regulated than they are," she said.
That's a growing sentiment in cities all across the nation as more and more tax jurisdictions are making sure short-term rentals are paying lodging taxes, as well as passing rules that require hosts to register their properties, undergo safety and code inspections, and pay registration fees.
In May 2019, Florence raised concerns with 27 Airbnb owners whose properties were located in areas zoned for single-family dwellings.
"The problem is that R1 and R2 (ordinances) don't really address the short-term rentals," Mayor Steve Holt said at the time. "It's got to be addressed."
Following a series of meetings between Airbnb owners and city officials, it was agreed a special permitting process was needed for owners of short-term rental properties, plus a requirement that those owners must purchase a business license. Both would be renewable annually.
City officials also initiated an effort to collect data on short-term rentals, including information on neighborhood preservation.
City Council President Dick Jordan emphasized at the time that "balance is needed" to ensure the concerns of nearby neighbors (traffic, noise and influx of strangers) are factored into decisions to regulate Airbnb rentals.
Creating a level playing field is possible, and having Airbnb properties pay their share of lodging taxes is certainly one way to do that. But regulations should thread the needle between reasonable consumer protections and not making rules or fees so burdensome that home-sharing hosts are unable to make money.
Online: https://www.timesdaily.com


text:
#include "pluginterfaces/base/ftypes.h"
#include <vector>

//------------------------------------------------------------------------
//    EventLogger
//------------------------------------------------------------------------
class EventLogger
{
public:
//------------------------------------------------------------------------
    EventLogger ();

	typedef std::vector<struct LogEvent> Codes;

	void resetLogEvents ();
	const Codes& getLogEvents () const;
	void addLogEvent (Steinberg::int32 logId);
	bool empty () const { return mLogEvents.empty (); }

//------------------------------------------------------------------------
protected:
	Codes mLogEvents;
};

#endif // __eventlogger__

redpajama_set_name: RedPajamaGithub