set theory / predicate logic math language problem

• Dec 6th 2010, 09:03 PM
jsmithson
set theory / predicate logic math language problem
Hi folks!

I'm seeking help with some math in a scientific paper. I'm having trouble understanding the mathematical language used in the paper. I understand some of it, but I can't seem to find a reasonable explanation of some of the symbols and hence the meaning of the math escapes me.

I refer you to the attachments to this forum post. They are screen grabs from the paper.

In definition 1 my understanding is this:

K = set of (referents , 0 element set) PIPE append ( 0 element set, y ) to K

I understand all this except for the PIPE (vertical stroke). Some math symbol references describe this as "such as" but that doesn't seem right here. Is the pipe used to separate the equation from the append operation and that in English it means "separately"?

Points 1 and 3 in the where clauses I understand however I'm stuck on 2. The PIPE character appears again:

y = R(x1,...,xn) PIPE not K PIPE if K1 then K2

What is the meaning of the PIPE and therefore what is the meaning of where clause 2?

~~~

In definition 4 I'm also having trouble. I understand the second part of clause 1 (ie. "if and only if the domain of g equals the union of domain of f and U") however the first part I'm lost.

Also there are parts of clause 2 I don't understand such as the first part and the circle in the second part.

~~~

Any help in translating this stuff into English would be much appreciated!

I'm happy read some reference material to understand this stuff but I don't even know what it's called. What kind math is this? Where can I learn more about it?

Some notes about the paper: DRS means discourse representation structure; the math in the paper contains set theory and predicate logic (both of which I am OK with).

For your interest, the paper is called "Segmented Discourse Representation Theory" and is about a computer program being able to understand natural language.
• Dec 7th 2010, 12:21 AM
emakarov
Definition 1 uses something like Backus–Naur Form, though BNFs usually use ::= instead of := . This means that K is either an ordered pair $\langle U,\emptyset\rangle$ or the result of the $\oplus$ operation defined in point 3. Along with K, $\gamma$ is another nonterminal symbol. I would render this grammar as follows, with $\oplus$ defined separately.

$\begin{array}{rcl}
K & ::= & \langle U,\emptyset\rangle\mid K\oplus\langle\emptyset,\gamma\rangle\\
\gamma & ::= & R(x_1,\dots,x_n)\mid\neg K\mid K\Rightarrow K
\end{array}$

Concerning Definition 4, apparently the interpretation $[\![K]\!]_M$ of K is a binary relation on functions. Also, $[\![\gamma]\!]_M$ is defined as a relation on functions. The five points of the definition correspond to five variants of K and $\gamma$ in the grammar above. The circle $\circ$ is the composition of relations.