Prosem: Rechtschreibfehler korrigiert und Vergleich von CYK mit Earley ergänzt.

prosem/prosempaper.tex

@@ -195,7 +195,7 @@
 	It's the dream of many Science-Fiction fans: A fully sentient AI. Let's ignore for a moment all the odds that are against it (morality, physics, etc.) and concentrate on one aspect that is mandatory for even much less ambitious dreams. Imagine a computer game in which you can talk natural language to the NPC counterparts so that they react appropriately to it. Well maybe that is still too ambitious. What about writing what you want to say? In that case the computer needs to understand what you are writing so that it can react to it.
 	
 	This process of understanding natural language contains multiple methods. The first one is the syntactic parsing, the second one the semantic analysis. Syntactic parsing relies on a grammar that describes the set of possible input, also called syntax. The syntax specifies what are allowed sentence structures and how these are built.
-	The semantic analysis relies on the semantics of a given input. That means what the given input means. An example: ``You run around the bush''. The semantic meaning of this sentence is that you are running around a bush. The pragmatics define what is the intended meaning of an input. In this example it's not that you run around the bush but actually that you take a long time to get to the point in a discussion. It's a so called idiom. This difference between semantic meaning, where just the sentence as it is written is considered, and pragmatic meaning, where the intended meaning is considered, generates ambiguity that is easy for humans to resolve but difficult for computers. But even the pragmatics in this example are ambigious, because it depends on the context what it actually means. If two persons are walking around in a forest and one starts running around the bush, the sentence of this example, would have the semantic meaning as it's pragmatic meaning.
+	The semantic analysis relies on the semantics of a given input. That means what the given input means. An example: ``You run around the bush''. The semantic meaning of this sentence is that you are running around a bush. The pragmatics define what is the intended meaning of an input. In this example it's not that you run around the bush but actually that you take a long time to get to the point in a discussion. It's a so called idiom. This difference between semantic meaning, where just the sentence as it is written is considered, and pragmatic meaning, where the intended meaning is considered, generates ambiguity that is easy for humans to resolve but difficult for computers. But even the pragmatics in this example are ambiguous, because it depends on the context what it actually means. If two persons are walking around in a forest and one starts running around the bush, the sentence of this example, would have the semantic meaning as it's pragmatic meaning.
 	
 	On top of that the semantic meaning itself isn't always clear either. Sometimes words have multiple meanings, so that even the semantic meaning can have different possible interpretations.
 	
@@ -224,7 +224,7 @@
 		
 		The CYK does only work with grammars in the Chomsky Normal Form (CNF). Every context-free grammar can be converted to CNF without loss in expressiveness. Therefore this restriction does no harm but simplifies the parsing. For information on how context-free grammars can be converted to CNF, refer to Jurafsky\cite{Jurafsky2009b}.
 		
-		CYK requires $\mathcal{O}(n^{2}m)$ space for the $P$ table (a table with probabilities), where ``$m$ is the number of nonterminal symbols in the grammar''\cite[p.~893]{Russel2010}, and uses $\mathcal{O}(n^{3}m)$ time. ``$m$ is constant for a particular grammar, [so it] is commonly described as $\mathcal{O}(n^{3})$''\cite[p.~893]{Russel2010}. There is no algorithm that is better than CYK for general context-free grammars\cite{Russel2010}. 
+		CYK requires $\mathcal{O}(n^{2}m)$ space for the $P$ table (a table with probabilities), where ``$m$ is the number of nonterminal symbols in the grammar''\cite[p.~893]{Russel2010}, and uses $\mathcal{O}(n^{3}m)$ time. ``$m$ is constant for a particular grammar, [so it] is commonly described as $\mathcal{O}(n^{3})$''\cite[p.~893]{Russel2010}. But these values are of no value if there is no benchmark. How good is $\mathcal{O}(n^{3})$ in comparison? To give a better idea of the relations, here a small comparison to the ``Earley Algorithm''\cite[p.~477]{Jurafsky2009b}. The Earley algorithm performs better with all unambiguous grammars.\cite{Li} It has the same upper bound in time but in most cases it is quicker. Furthermore it has a space complexity of $\mathcal{O}(n)$ which is definitely better than CYK.\cite{Li} For ambiguous grammars though the Earley algorithm uses more space than CYK and the real space used is dependent on the length of the input.\cite{Li} In time complexity the CYK algorithm can only compete with Earley if ambiguous grammars are used.\cite{Li} But CYK is still of use for parsing of natural language, because natural language grammars are always ambiguous. Therefore there is no algorithm that is better than CYK for general context-free grammars.\cite{Russel2010}
 		
 		But how does CYK work? CYK doesn't examine all parse trees. It just examines the most probable one and computes the probability of that tree. All the other parse trees are present in the $P$ table and could be enumerated with a little work (in exponential time). But the strength and beauty of CYK is, that they don't have to be enumerated. CYK defines ``the complete state space defined by the `apply grammar rule' operator''\cite[p.~894]{Russel2010}. You can search just a part of this space with $A^{*}$ search.\cite{Russel2010} ``With the $A^{*}$ algorithm [...] the first parse found will be the most probable''\cite[p.~895]{Russel2010}.