Lecture 16: van Benthem's "Semantic Automata" (1984).
Generalized Quantifiers
-
What are generalized quantifiers?
every man in Montague Grammar is a term (ling. NP)
denoting a generalized quantifier, type <<e,t>,t>
(extensional version).
 or 
every is a quantifier (van Benthem) or a determiner
(Barwise and Cooper), type <<e,t>,<<e,t>,t>>.
(Note: van Benthem's quantifier = Barwise and Cooper's determiner.)
"flattened" version <<e,t>,<e,t>,t>: relation
between sets
 iff 
More generally: for  ,
think of  as a Common
Noun Phrase (e.g. man), 
as a Verb Phrase (e.g. walks); 
together corresponds to a term or NP.
-
Constraints on natural language quantifiers (pp. 2-3)
- EXTension
If  , then
iff 
In principle, the extension of a quantifier can vary with
the total domain of individuals  .
This constraint says that individuals in
that are in neither
nor are irrelevant.
What would a counterexample look like? For instance a "globally
proportional" reading of "many" according to which
 iff  ,
i.e. "the proportion of
that are is greater
that the proportion of individuals that are
in the universe as whole."
- QUANTity
If  and 
is some permutation of ,
then iff  .
This constraint distinguishes a narrow class of quantifiers
from determiners like John's which violate it.
iff
 iff  ,
e.g. iff 
A permutation of
is a map  which
is one-to-one and onto.
Example
Let  and
iff 
because iff
 .
But not iff
 because
is not equivalent to  .
- CONServativity
iff 
Examples:
- every man walks iff every man is a man who walks
- most men walk iff most men are men who walk
Potential counterexample:
- If only were a quantifier, it would be one: only men
walk (false)
only men are men who walk (true).
- Cumulative effect
Truth of depends on  ,
 , i.e. the common
noun set and the intersection of the set and the VP set.
In what follows, van Benthem uses 
for  , 
for .
- Tree of numbers
pairs: ,
Example: let  ,
 . "1,3"
means: There are 4 men; 1 man doesn't walk, 3 men walk.
-
at least one man walks:  :
all pairs ,,
such that  .
-
every man walks: 
-
exactly 2 men walk: 
- Monotonicity
Upward Monotonicity: If
and  , then  .
Example:
- some man walks
some man moves
Any node to the right of a node in 
is itself in .
Downward Monotonicity: If
and  then .
Example:
- no man moves
no man walks
Any node to the left of a node in
is itself in .
Exercise: Check that every, some, at
least two are upward monotone, not every, no,
at most two are downward monotone, exactly two is
neither.
- Persistence
If and 
then  .
Examples:
- some is persistent: some man walks
some person walks
- every is not persistent: every man walks
every person walks
Any node in has
its entire generated downward subtree in .
- First-order definable
is first-order
definable iff there is some sentence 
in monadic first-order predicate logic with identity with unary
predicates and
such that for all
universes with
,
iff  .
Examples:
- Corresponding tree pattern:
In general: arbitrary distribution of +/- above some level, below
that all values propagate uniformly from values in some row, as
indicated.
Examples:
Quantifiers and Automata
(pp. 4-6)
- A " - Automaton"
Input: A sequence of members of a set ,
marked as to whether they are members of or
not.
Representation as a sequence of 
and  ,
representing a member of  ,
a member of  .
Output: Yes or no, representing whether or not
is true for the sequence thus far observed.
Examples:
-
The all-automaton should accept
all and only those sequences consisting only of s.
Note: 
marks the initial state, I use 
to represent accepting states, so here the initial state is
also the only accepting state.
For all men walk, the s
represent men who walk, the s
represent men who don't walk.
Compare the tree of numbers for every.
- The at least one-automaton
should accept any string containing at least one .
Compare the tree of numbers for
at least one.
- The exactly two-automaton
should accept strings containing exactly two s.
Compare the tree of numbers for
exactly two.
-
The not all-automaton should accept sequences containing
at least one .
Note: This is the complementary automaton to the all-machine,
i.e. accepting/non-accepting states are reversed.
-
The no-automaton should accept all and only those sequences
consisting only of s.
This one is similarly related to at least one.
Note that in the tree of numbers, not all and all
have the same geometry, with + and - reversed, similarly for
at least one and no.
- Turning tree of numbers into an automaton.
Example:
In tree of numbers, a position corresponds to " 
s and 
s in the input so far",
i.e. to  ,  .
- A configuration
corresponds to 
(acceptance forever after).
- A configuration
corresponds to 
(rejection forever after).
- A configuration
corresponds to  .
- A configuration
corresponds to  .
- A configuration
corresponds to  .
- A configuration
corresponds to  .
How to collapse states to find the simplest automaton:
(not fully general; this applies well to tree of numbers
case, though.) Look at the downward triangle generated by some node
m,n (as a pattern of +/-). If it's identical to the triangle pattern
generated by the node above and to the right, it should correspond
to the same state as that one. If it's identical to the node up
and left, same state as that one. Min. number of states = number
of distinct triangle patterns.
Somebody check if that's true
Example:
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