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SimplifiedLogicForm.md

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#Brief guide to simplified logical form

Consider the utterance:

Pixar, the company which Disney acquired, made Ratatouille.

DepLambda will produce the following neo-Davidsonian representation:

(lambda $0:<a,e> (exists:ex $1:<a,e> (and:c (exists:ex $2:<a,e> (and:c (p_EVENT_w-9-made:u $0) 
(p_TYPE_w-10-ratatouille:u $2 (p_EVENT.ENTITY_arg2:b $0 $2))) (and:c (and:c (p_TYPE_w-1-pixar:u $1) 
(p_EVENT_w-1-pixar:u $1) (p_EVENT.ENTITY_arg0:b $1 $1)) (and:c (and:c (and:c (p_TYPE_w-4-company:u $1) 
(p_EVENT_w-4-company:u $1) (p_EVENT.ENTITY_arg0:b $1 $1)) (p_EMPTY:u $1))
(exists:<<a,e>,<t,t>> $3:<a,e> (exists:ex $4:<a,e> (and:c (exists:ex $5:<a,e> (and:c (exists:ex $6:<a,e> 
(and:c (p_EVENT_w-7-acquired:u $3) (p_EMPTY:u $6) (p_EVENT.ENTITY_arg2:b $3 $6))) 
(p_EQUAL:<<a,e>,<<a,e>,t>> $1 $5) (p_EVENT.ENTITY_arg2:b $3 $5))) (p_TYPE_w-6-disney:u $4) 
(p_EVENT.ENTITY_arg1:b $3 $4)))))) (p_EVENT.ENTITY_arg1:b $0 $1))))

This can be a bit hard to read -- new existential variables are introduced ($1, $2, etc), the variables (and predicate symbols) are adorned with type information, predicate names are complex (constructed, e.g., from kind information such as EVENT.ENTITY and the actual predicate name, arg0). Further, there are equality constraints ((p_EQUAL:<<a,e>,<<a,e>,t>> $1 $5)) which should really be simplified away, and predications arising from the way in which the formula was constructed ((p_EMPTY:u $6)) which have no logical import and should be eliminated.

DepLambda takes care of these issues. It walks the form above and produces the following simplified conjunction of atomic formulas:

[["company(3:e)","made(8:e)","arg1(6:e , 5:m.disney)","pixar(3:e)","arg2(8:e , 9:m.ratatouille)",
"acquired(6:e)","arg2(6:e , 0:m.pixar)","arg0(3:e , 0:m.pixar)","company(3:s , 0:m.pixar)",
"arg1(8:e , 0:m.pixar)"]]

Here is what is going on. The neo-Davidsonian representation typically starts with a lambda expression, following by a logical formula that consists of interleaved existentials and conjunctions of unary and binary atomic formulas. The simplified logical form gets rid of the existentials by exploiting the idea that each variable typically occurs as the sole argument of a "defininig" predication, which is also typically associated with a word in the utterance. Hence each variable occurrence can be replaced by a Skolem constant whose name carries the index (minus 1) of the word in the utterance it stands for, and the type of this occurrence. The type (:x, :s, :e, :m) is obtained from the kind information in the predication in which the variable has the "defining" occurrence, and the predication in which this constant occurs. (Recall that in DepLambda variables take as values pairs of individuals, one of type "event" and one of type "individual" and the context of use is intended to determine which of these is implicitly selected.)

Thus:

  • 3:e is the constant 3 of type event ... the denotation of the word company in the utterance -- note the index is 4 (3+1) since the comma is counted
  • 0:m.pixar is the constant 0 which stands for the named entity pixar in the utterance (index 1)
  • Thought not ilustrated here, 11:x would stand for the constant 11 of type individual that is not a named entity.
  • 3:s is the "type" associated with constant 3 ... see below for how such a term is used.

Now we can read the atomic formulas in the simplified logical form above:

  • company(3:e) asserts that there is an event 3 which is a "being a company" event. This is a consequence of the predication (p_EVENT_w-4-company:u $1) in the lambda expression. Similarly for made(8:e),pixar(3:e) and acquired(6:e).
  • arg1(6:e,5:m.disney) asserts that the arg1 of 6:e ("acquired") is 5:m:disney ("Disney"). This is a consequence of the predications (p_EVENT.ENTITY_arg1:b $3 $4), (p_EVENT_w-7-acquired:u $3) and (p_TYPE_w-6-disney:u $4) in the lambda form. Similarly for arg2(8:e,9:m.ratatouille), arg2(6:e,0:m.pixar),arg0(3:e,0:m.pixar) and arg1(8:e,0:m.pixar).
  • company(3:s,0:m.pixar) asserts that constant 0 (i.e. "pixar") is a company (represented by index 3). This is a consequence of (p_EVENT_w-1-pixar:u $1) and (p_TYPE_w-4-company:u $1) in the lambda form.

Thus we can paraphrase the simplified logical form as: there is a company event (3:e), a make event (8:e), a pixar event (3:e), an acquired event (6:e). Further, Disney is the acquirer (arg1(6:e, 5:m.disney)) and Pixar the acquired (arg2(6:e,0:m.pixar)), Ratatouille was made (arg2(8:e,9:m.ratatouille)) by Pixar (arg1(8:e,0:m.pixar)), and Pixar (the named entity) participates in the Pixar event (arg0(3;e, 0:m.pixar)).

It takes a little bit of practice to start reading this, but this notation is certainly simpler and more compact than the unvarnished neo-Davidsonian representation!