Lecture 13: Semantics IIIProfessor Robert C. [email protected]/9.611J SP11The Menu Bar• Administrivia: Lab 4 due; Lab 5/6 out• What does a more sophisticated lambda calculusto SQL mapping look like?• Events, models, and determiners6.863J/9.611J SP11We’ll do increasingly ‘more challenging’sentences1. Kathy respects Fong2. Red car in Palo Alto3. Overpriced house in Palo Alto4. Kathy runs in Palo Alto5. What does Kathy like6. Who does Kathy like7. Which cars did Kathy like8. How many red cars in Palo Alto does Kathy like9. Did Kathy see the red car in Palo Alto6.863J/9.611J SP11Paired syntactic-semantic rules• Nonterminal : semantic translationtranslation rules say how to associatesemantic representations with syntacticrepresentations• In general, head-nonhead syntacticcomposition corresponds to functionapplication6.863J/9.611J SP11New lexicon6.863J/9.611J SP11Build a decorated tree, with anintermediate ‘semantic’language6.863J/9.611J SP11An aside about ‘types’• You may have noticed that we associated ‘types’ to theobjects in our Palo Alto universe - this is to make sure thatfunctions apply to objects of the right sort• There’s a standard notation for defining types in this sort oflambda calculus logic, starting with an inductive base• There are two base types:1. Ind, equivalently, e (for ‘individual’) is the ‘type’ of abasic entity, like rocky(but: we will alter this below!)2. Bool, equivalently, t (for ‘truth value) is the ‘type’ of aformula, which can be either true or false• We can now form complex types for function expressions outof these; e.g., <e,t>, is the type of expressions that describefunctions from entities to truth values, eg, unary predicateslike sleeps or runs. (Eg., in our Palo Alto notation, we haverunIND →BOOL)6.863J/9.611J SP11The complications• What about intransitive verbs, e.g, ‘sleeps’?• This is a function mapping to truth values, ie,sleeps(x) is true/false; so sleeps(x) is of type<e,t>6.863J/9.611J SP11More types• What about transitive verbs?• Remember, we can think of the VP as forming averb of ‘1 argument’, once the object is found… so aVP, eg, respects Fong is just like an intransitiveverb• So a VP with a named object must be of the sametype as a instransitive verb, namely, <e, t>• And now the subject is of type e, so, the functionmapping from Subject NPs to VPs must be of type:<t, <e, t>>This is why we have written respectsIND→IND →BOOL(Putting aside for now the question of more complex subjecttypes & the lambda calculus…)6.863J/9.611J SP11More types: names for things• What about names?• Recall: we change ‘rocky’ from being just a plain oldconstant, to be the ‘set of all predicates that are trueof rocky’ - by using the lambda calculus• What about types and the lambda calculus?• We’ll say that if α is of type τ and x is a variable of type e,then λx.α is of type <e, τ>• So we can generalize (abstract over) all the predicatesapplicable to ‘rocky’ and say that ‘rock’ just denotes the set ofall sets satisfying all the things true of rocky, ie, λP.P(rocky)• This ‘type raising’ of names from <e> to <e, t> will turn outto match up well with what we need to describe logicaldeterminers like ‘a’, ‘every’, etc.6.863J/9.611J SP11More about types: intransitive verbs• So, if we have λx.sleeps @rocky; (‘rocky’ is of type e for themoment)• Suppose we want to use one lambda abstract as the argument ofanother lambda abstract?• We can’t directly use λy.y@rocky@(λx.sleeps(x))• Why? Because λ y.y only applies to arguments of type e and thelambda form λx.sleeps(x) is of type <e, t> (as befits a function)• Recall the hack: we need to abstract over types to get a higherorder type - use the idea of a ‘function over functions’, since aunary function P is of type <e,t>, we have that the wholeabstract λ P.P is of type <<e,t>, t>, so we have:λP.P@rocky@(λx.sleeps(x)); now we can substitute for P toget the form we want: λx.sleeps(x)@rocky which reduces to sleeps(rocky)• All this shows is that our function applications are kosher6.863J/9.611J SP11More about types: transitive verbs• The basic intuition: is that a verb of 2 arguments(the subject and the object), will behave like anintransitive verb (a verb of 1 argument), as soon aswe fill in the object• So, this means that complete VPs (with the objectNP filled in) really should have the same type asan ordinary intransitive verb, ie, <e, t> or IND→BOOL6.863J/9.611J SP11Sequence of function applications (betareductions) tells us how to compose the semanticrepresentation6.863J/9.611J SP11Abbreviate some notation…(long form)6.863J/9.611J SP11So we can abbreviate the whole sentencesemantics this way…Which leads to an immediate SQL treatment:Ok, what’s next???What about prepositions, adjectives, adverbs,…6.863J/9.611J SP11New, improved grammar, with semanticaugmentation6.863J/9.611J SP11New lexicon6.863J/9.611J SP11Let’s try it outred car in Palo AltoN'AdjN'N"NcarredPPPinNPPalo Altosyntaxβ(α)αββ(α)βαβ(α)βαsemanticsNow add lexical entries and start doing functionapplication…Desired Goal:6.863J/9.611J SP11Adding lexical entriesN'AdjN'N':carN:carcarredPPP: λyλPλx(P(x)∧in'(y)(x) inNP:PAPalo Altosyntaxβ(α)αββ(α)βαβ(α)βαAdj: λP(λxP(x)∧red'(x))Now do function applications (beta reductions)6.863J/9.611J SP11Beta reduction 1: PP ‘in Palo Alto’NB: Church-Rosser theorem assures us we could dothese reductions in any order and get same result atthe end – good for computation6.863J/9.611J SP11β-reduction 2: ‘car [in Palo Alto]’6.863J/9.611J SP11β-reduction 3: ‘red [car in Palo Alto]’Hah, it works!!We’re done!!! Well, almost…lexicon6.863J/9.611J SP11But…isn’t there another parse…?6.863J/9.611J SP11Parse #2 - what are its semantics?N'AdjN'carredPPPinNPPalo Altosyntaxβ(α)αββ(α)βαβ(α)βαN'NWill this have the same semantics?What does β-reduction construct?6.863J/9.611J SP11After the beta dust settles…It’s the same semantics as the first parse!!… why???Because: red is an intersective adjective(there are other adjectives that are non-intersective)6.863J/9.611J SP11More on adjectives• Intersective:Phrases like [blue suit] are easy tounderstand because we can intersect the setof blue things with the set of suits and getthe set of blue suits. Such adjectives arecalled intersective
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