Continuation. Lecture 24: Continuations, continued. Computations. What is a continuation?
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1 Continuation Lecture 24: Continuations, continued CS 181O Spring 2016 Kim Bruce Meaning of linguistic context of expression called its continuation Expressions can have lots of continuations Hence we ll make the continuation a parameter of the meaning. We can think of providing an argument to a function or a function to an argument! What is a continuation? Computations Continuation is provided to an expression so can get meaning of sentence. Continuation is function type returning type t Continuation of NP is of type e t Continuation of intransitive verb is (e t) t Continuation of transitive verb is (e e t) t Meaning functions will now take continuations as an argument to get meaning. Computations are functions that take a continuation and give a meaning (of type r) type Cont a r = a -> r type Comp a r = Cont a r -> r Examples: meaning of NP: (e t) t meaning of IV, CN: ((e t) t) t meaning of TV: ((e e t) t) t meaning of ADJ:?
2 Continuation-Passing Semantics Application? Make all meaning take a continuation parameter k Constant [[c]] λk. k c cpsconst:: a -> Comp a r cpsconst c = \ k -> k c Trick to check work: Get the original meaning by applying to identity function (cpsconst c) (\x -> x) = (\ k -> k c) (\x -> x) = (\x -> x) c = c [[Dorothy cheered]] [[Dorothy]] = λk:e t. k dorothy:: Comp e t where Comp e t = (e t) t [[cheered]] = λk :(e t) t. k cheered:: Comp (e t) t where Comp (e t) t = ((e t) t) t [[Dorothy cheered]]: Comp t t where Comp t t = (t t) t [[Dorothy cheered]] = λk: t t.??? CpsApply CpsApply cpsapply m n = λk. n (λb. m (λa. k (a b))) result is a function that takes a continuation k. To use k, must: evaluate n with a continuation that takes the value b of n, and then evaluates m with a continuation that takes the value a of m, and finally applies k to the result of evaluating (a b) Watch out, there is an alternative later!! intsent_cps (Sent np vp) = cpsapply (intvp_cps vp) (intnp_cps np) Given continuation k: Compute intnp_cps np, call it b Compute intvp_cps vp, call it a Apply k to (a b) Work out intsent_cps(sent Dorothy Cheered)
3 Example Example intsent_cps(sent Dorothy Cheered) = cpsapply (intvp_cps Cheered) (intnp_cps Dorothy) = cpsapply (λk :(e t) t. k cheered) (λk:e t. k dorothy) = (λk. (λk:e t. k dorothy) ((λb. (λk :(e t) t. k cheered) (λa. k (a b)))) = (λk. (λk:e t. k dorothy) ((λb.((λa. k (a b)) cheered)))) = (λk. (λk:e t. k dorothy) ((λb.(k (cheered b))))) = (λk.k (cheered dorothy)) intsent_cps(sent Dorothy Cheered) = cpsapply (intvp_cps Cheered) (intnp_cps Dorothy) = cpsapply (λk :. k cheered) (λk. k dorothy) = (λk. (λk. k dorothy) ((λb. (λk. k cheered) (λa. k (a b)))) = (λk. (λk. k dorothy) ((λb.((λa. k (a b)) cheered)))) = (λk. (λk. k dorothy) ((λb.(k (cheered b))))) = (λk.k (cheered dorothy)) More CPS What about quantifiers? [[everyone]] = λk. x ((Person x) k x) [[someone]] = λk. x ((Person x) k x) What is scope of x? Includes k! Abstract to quantifiers: [[every]] = λk λp. k(λq. x ((Q x) P x) [[some]] = λk λp. k(λq. x (Q x) P x) [[the]] = λk λp. k(λq. x (( Q x) P x) [[no]] = λk λp. k(λq. x ((Q x) P x) Example [[every person]] = (λk λp. k(λq. x ((Q x) P x))))(λk. k Person) = (λp. (λk. k Person)(λQ. x ((Q x) P x)) = (λp.(λq.( x (Q x) P x)person) = (λp.(( x (Person x) P x) has type (e t) t Same value as everyone, as expected!
4 Transitive Verbs Scope Reversal inttv_cps Helped = cpsconst help = λk.k helped where helped: e e t First argument is object, second is subject! Can use different apply function: cpsapply :: Comp (a -> b) r -> Comp a r -> Comp b r cpsapply' m n = λk. m (λa. n (λb. k (a b))) Compared to original: cpsapply m n = λk. n (λb. m (λa. k (a b))) What does it mean in practice Everyone helped someone. From the Text: From the Text: Using cpsapply: [[helped someone]] = cpsapply ([[helped]])(someone]]) = λk.([[someone]](λn. [[helped]](λm.k (m n)))) = λk.((λk. x. ((Person x) (k x))(λn. [[helped]](λm.k (m n)))) = λk.( x. ((Person x) (λn. [[helped]](λm.k (m n)))x)) = λk.( x. ((Person x) ([[helped]](λm.k (m x)))) = λk.( x. ((Person x) ((λk. k help)(λm.k (m x)))) = λk.( x. ((Person x) (λm.k (m x))help)) = λk.( x. (Person x) (k (help x))) [[helped someone]] = λk.( y. (Person y) (k (help y))) [[everyone helped someone]] = cpsapply([[helped someone]])([[everyone]]) = λk.([[everyone]](λb. [[helped someone]](λa.k (a b)))) = λk.(λk. x. ((Person x) k x)(λb. [[helped someone]](λa.k (a b)))) = λk.( x. ((Person x) (λb. [[helped someone]](λa.k (a b))x)) = λk.( x. ((Person x) ([[helped someone]](λa.k (a x)))) = λk.( x. ((Person x) (λk.( y. (Person y) (k (help y))))(λa.k (a x)))) = λk.( x. ((Person x) ( y. (Person y) ((λa.k (a x)) (help y))))) = λk.( x. ((Person x) ( y. (Person y) (k ((help y) x)))))
5 From the Text: From the Text: Using cpsapply : [[helped someone]] = cpsapply ([[helped]])([[someone]]) = λk.([[helped]](λa.[[someone]](λb.k (a b)))) = λk.(λk. k help)(λa.[[someone]](λb.k (a b)))) = λk.(λa.[[someone]](λb.k (a b)))help) = λk.([[someone]](λb.k (help b))) = λk.(λk. x. ((Person x) (k x))(λb.k (help b))) = λk.( x. ((Person x) ((λb.k (help b)) x))) = λk.( x. (Person x) (k (help x))) Exactly as before! [[helped someone]] = λk.( y. (Person y) (k (help y))) [[everyone helped someone]] = cpsapply ([[helped someone]])([[everyone]]) = λk.([[helped someone]](λa.[[everyone]](λb.k (a b)))) = λk.(λk.( y. (Person y) (k (help y))))(λa. [[everyone]](λb.k (a b)))) = λk.( y. (Person y) (λa. [[everyone]](λb.k (a b))(help y)) = λk.( y. (Person y) ([[everyone]](λb.k ((help y) b)))) = λk.( y. (Person y) (λk. x. ((Person x) (k x)) (λb.k ((help y) b)))) = λk.( y. (Person y) ( x. ((Person x) ((λb.k ((help y) b))x))) = λk.( y. (Person y) ( x. ((Person x) (k ((help y) x))))) Bottom Line More continuations cpsapply expands subject first, with object expanded inside. cpsapply does opposite Allows us to capture both expressions of quantifiers. Can be helpful in handling coordination Already know how to make sense of sentential operators: and, or, not Interpreted in predicate logic with,, But they also appear as operators on other grammatical features
6 Coordination Meaning via Continuations NP: John and Mary went to the store John went to the store and Mary went to the store V: Mary danced and sang all night Mary danced all night and Mary sang all night Adj: The ball was big and red VP: John kicked the ball and ran down the field John kicked the ball and John ran down the field Ann baked and Betty ate all the cookies. What is context around conjunctive phrase? Mary danced and sang all night k = λx. Mary x all night k (danced and sang) = k(danced) and k(sang) intcon_cps And = λk λm λn. k(m) k(n) intcon_cps Or = λk λm λn. k(m) k(n) Still issues Chris and Betty met at the fair Chris met at the fair Betty met at the fair???? Different meaning of and Individuals or group Questions?
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