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Title: Top-Down Neural Model For Formulae

Paper content:
We present a simple neural model that given a formula and a property tries to answer the question whether the formula has the given property, for example whether a propositional formula is always true.
The structure of the formula is captured by a feedforward neural network recursively built for the given formula in a top-down manner.
The results of this network are then processed by two recurrent neural networks.
One of the interesting aspects of our model is how propositional atoms are treated.
For example, the model is insensitive to their names, it only matters whether they are the same or distinct.
In real-world situations a very successful approach, popularized in BID2 , to problem solving is based on a clever combination of fast instinctive (heuristic) reasoning and slow logical reasoning.
The latter is exemplified by abstract logical formulae where only structural properties matter.
If computers are involved, a logical formula is traditionally a syntactic object which is a subject to simple but very fast syntactic manipulations Robinson & Voronkov (2001) .
Hence all but very basic decisions are postponed if possible.
However, this viewpoint is rapidly changing as various AI methods are tested in the field of automated reasoning, in particular machine learning methods.A fundamental problem in using machine learning in automated reasoning is a suitable representation of logical formulae.
A formula as a solely syntactic object is no longer sufficient and we have to exploit its semantic properties.
Various approaches have been proposed for different logical systems.
In this paper we will concentrate on the simplest yet very powerful standard logical system-classical (Boolean) propositional logic.
This paper presents, as far as we know, a novel neural representation of propositional formulae that makes it possible to test whether a given formula has a given property, e.g., whether the formula is always true or not.
Clearly, we try to solve a well-known CONP-complete problem.
However, the fact that the problem is generally hard and requires a non-trivial search does not rule out the possibility that a decent heuristic can be learned, moreover, if only a specific subset of formulae is involved.
In particular, our general goal is to obtain a useful heuristic that can help us in guiding a proof search, where we typically face numerous choice points.Unlike in natural language processing, a parse tree for a formula is available for free.
Although some approaches do not exploit this feature and try to learn the structure of a formula on their own, using usually various recurrent neural networks (RNN), it is more common to take advantage of this knowledge.
Moreover, it seems that the later approach has a significant edge, see BID1 .
Usually propositional atoms, the basic building blocks of propositional formulae, are learned as embeddings and each logical connective is treated as a unique neural network that given the vector representation of its arguments produces a vector that represents an application of the connective on these arguments, e.g., a binary connective takes two vectors of length d, and produces a new one of length d, see BID0 .
This clearly leads to tree recursive neural networks BID5 where the structure of the network follows the parse tree.
Such models are built bottomup and the meaning of the formula is usually the vector produced in the root of the tree.Our model also uses the parse tree of the formula, but the knowledge is propagated in the opposite direction.
We start with a vector (random or learned), representing a property we want to test, and we propagate it from the root to leaves (propositional atoms).
The knowledge propagated to atoms is then processed by recurrent neural networks and a decision is produced.
This makes it possible to ignore completely the names of propositional atoms and concentrate more on structural properties of formulae.The experimental results suggest that the model is more than competitive and beats other known approaches on some benchmarks.
More importantly, our model seems to suffer less if bigger formulae are involved and could be more useful in real world scenarios.The structure of this paper is as follows.
In Section 2 we discuss the architecture of our model in full details and also a dataset on which we will experiment is introduced there.
In Section 3 we discuss an implementation of building blocks of our network, present experimental data, and shortly describe possible interpretations of our model.
Some potential future modifications are briefly mentioned in Section 4.
Few relevant models are mentioned in Section 5 and the paper concludes with Section 6.
We have presented a novel top-down approach to represent formulae by neural networks and showed some preliminary experiments.
They suggest that our approach is competitive with other known approaches and beats them on some benchmarks.
More importantly, it seems that the presented model can deal better with the increasing size of formulae than other approaches.
The model deals only with the structure of formulae and ignores completely for example names of individual atoms, only whether they are the same or distinct matters.
(inaccessible to our model) in processing (p → q) → p we have to deal with more choices along this line of reasoning.

assistant: text: A top-down approach how to recursively represent propositional formulae by neural networks is presented.