Matching sentences by meaning

This is kind of classic information retrieval task. You should represent each sentence as a vector and all you need to do is to compute the similarity between two vectors. I would avoid the word "meaning" but rather use word-cooccurence. On top of my mind it comes Semantic Vectors and a book which you may find helpful is Introduction to Information Retrieval, look for vector space modeling. And the Wikipedia article on Vector Space Model I guess other people in this community will probably come with other helpful comments. Успех!

A nice library to look at is gensim. In particular, look at the tutorial. The example he uses is quite similar to what you want to do.

Vector space models will only work on near-exact semantic matches unless you do some creative filtering.

In natural language processing, there is a task is known as textual entailment. The goal is to infer whether or not something is true given prior information. While that might not sound exactly like what you're looking for, many approaches to this task have developed excellent text similarity metrics that you could probably use. Look at the Recognizing Textual Entailment challenges for many of the solutions people have come up with.

An interesting gem I heard about yesterday is Explicit Semantic Analysis, by Gabrilovich et al. The idea is to construct a "concept vector" for each word which is a bag of wikipedia articles where this word appears. This gives you a word similarity, which you can then average and get document similarity. It seems to outperform many latent methods by virtue of leveraging this already built huge knowledge base in an aggregate way without overtrusting any single article.