600.465 - Intro to NLP - J. Eisner 1 Semantics From Syntax to Meaning!600.465 - Intro to NLP - J. Eisner 2 Programming Language Interpreter  What is meaning of 3+5*6?  First parse it into 3+(5*6) + 3 * 5 6 E E F E E E 3 F N 5 N 6 N * +600.465 - Intro to NLP - J. Eisner 3 Programming Language Interpreter  What is meaning of 3+5*6?  First parse it into 3+(5*6)  Now give a meaning to each node in the tree (bottom-up) + 3 * 5 6 E E F E E E 3 F N 5 N 6 N * + 3 5 6 30 33 3 5 6 30 33 add mult600.465 - Intro to NLP - J. Eisner 4 Interpreting in an Environment  How about 3+5*x?  Same thing: the meaning of x is found from the environment (it’s 6)  Analogies in language? + 3 * 5 x 3 5 6 30 33 E E F E E E 3 F N 5 N 6 N * + 3 5 6 30 33 add mult600.465 - Intro to NLP - J. Eisner 5 Compiling  How about 3+5*x?  Don’t know x at compile time  “Meaning” at a node is a piece of code, not a number E E F E E E 3 F N 5 N x N * + 3 5 x mult(5,x) add(3,mult(5,x)) add mult 5*(x+1)-2 is a different expression that produces equivalent code (can be converted to the previous code by optimization) Analogies in language?600.465 - Intro to NLP - J. Eisner 6 What Counts as Understanding? some notions  Be able to translate (a compiler is a translator …)  Good definition? Depends on target language.  English to English? bah humbug!  English to French? reasonable  English to Chinese? requires deeper understanding  English to logic? deepest - the definition we’ll use!  all humans are mortal = x [human(x) mortal(x)]  Assume we have logic-manipulating rules that then tell us how to act, draw conclusions, answer questions …600.465 - Intro to NLP - J. Eisner 7 What Counts as Understanding? some notions  We understand if we can respond appropriately  ok for commands, questions (these demand response)  “Computer, warp speed 5”  “throw axe at dwarf”  “put all of my blocks in the red box”  imperative programming languages  database queries and other questions  We understand a statement if we can determine its truth  If you can easily determine whether it’s true, why did anyone bother telling it to you?  Comparable notion for understanding NP is to identify what it refers to. Useful, but what if it’s out of sight?600.465 - Intro to NLP - J. Eisner 8 What Counts as Understanding? some notions  We understand statement if we know how to determine its truth (in principle!)  Compile it into a procedure for checking truth against the world  “All owls in outer space are bachelors” for every object if x is a owl if location(x)  outerspace if x is not a bachelor return false return true  What if you don’t have an flying robot? (Write the code anyway)  How do you identify owls and bachelors? (Assume library calls)  What if space is infinite, so the procedure doesn’t halt? Same problem for “All prime integers …” (You won’t actually run it) meaning600.465 - Intro to NLP - J. Eisner 9 What Counts as Understanding? some notions  We understand statement if we know how one could (in principle) determine its truth  Compile it into a procedure that checks truth against the world  Better: Compile it into a mathematical formula  x owl(x) ^ outerspace(x)  bachelor(x)  Now you don’t have to worry about running it  Either true or false in the world: a mathematical question!  Statement claims that the world is such that this statement is true.  Auden (1956): “A sentence uttered makes a world appear Where all things happen as it says they do.”  But does this help? Can you check math against the real world?  What are the x’s that x ranges over? Which ones make owl(x) true?  Model the world by an infinite collection of facts and entities  Wittgenstein (1921): “The world is all that is the case. The world is the totality of facts, not of things.”600.465 - Intro to NLP - J. Eisner 10 What Counts as Understanding? some notions  We understand statement if we know how one could (in principle) determine its truth  Compile it into a procedure that checks truth against the world  Better: Compile it into a mathematical formula  x owl(x) ^ outerspace(x)  bachelor(x)  Equivalently, be able to derive all logical consequences  What else is true in every world where this statement is true?  Necessary conditions – let us draw other conclusions from sentence  And what is false in every world where this sentence is false  Sufficient conditions – let us conclude the sentence from other facts  “Recognizing textual entailment” is an NLP task ( competitions!)  John ate pizza. Can you conclude that John opened his mouth?  Knowing consequences lets you answer questions (in principle):  Easy: John ate pizza. What was eaten by John?  Hard: White’s first move is P-Q4. Can Black checkmate?600.465 - Intro to NLP - J. Eisner 11 Lecture Plan  Today:  First, intro to -calculus and logical notation  Let’s look at some sentences and phrases  What logical representations would be reasonable?  Tomorrow:  How can we build those representations?  Another course (AI):  How can we reason with those representations?600.465 - Intro to NLP - J. Eisner 12 Logic: Some Preliminaries Three major kinds of objects 1. Booleans  Roughly, the semantic values of sentences 2. Entities  Values of NPs, e.g., objects like this slide  Maybe also other types of entities, like times 3. Functions of various types  A function returning a boolean is called a “predicate” – e.g., frog(x), green(x)  Functions might return other functions!  Function might take other functions as arguments!600.465 - Intro to NLP - J. Eisner 13 Logic: Lambda Terms  Lambda terms:  A way of writing “anonymous functions”  No function header or function name  But defines the key thing: behavior of the function  Just as we can talk about 3 without naming it “x”  Let square = p p*p  Equivalent to int square(p) { return p*p; }  But we can talk about p p*p without naming it  Format of a lambda term:  variable expression600.465 - Intro to NLP - J. Eisner 14 Logic: Lambda Terms  Lambda terms:  Let square = p p*p  Then square(3) = (p