Mock exam, WiSe 2018/19 (Sailer): Difference between revisions

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The examples in the text are based on Shakespeare's play ''Macbeth''. The full text of the play is available on [http://www.gutenberg.org/ebooks/2264 Projekt Gutenberg].


We will use J.K. Rowling's ''Harry Potter'' novels for the final exam this term.
== Task 1: Ambiguity ==
Consider the following ambiguous sentences.
For '''each''' of them, provide an unambiguous paraphrase for the possible readings.
(1)
a. Duncan trusted Macbeth because he was a thane.
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Reading 1: ''he'' refers to ''Macbeth''. Paraphrase: ''Duncan trusted Macbeth because Macbeth was a thane.''<br />Reading 2: ''he'' refers to ''Duncan''. Paraphrase: ''Duncan trusted Macbeth because Duncan was a thane.''
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b. Every king trusts a thane.
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Reading 1: ''every'' takes scope over ''a''. Paraphrase: ''For every king there is at least one thane such that the king trusts that thane.''<br />Reading 2: ''a'' takes scope over ''every''. Paraphrase: ''There is one particular thane such that each king trusts this thane.''
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b. Macbeth and Macduff are married.
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Reading 1: collective reading. Paraphrase: ''Macbeth and Macduff are married to each other''<br />Reading 2: distributive reading. Paraphrase: ''Macbeth and Macduff are both married, but not to each other.''
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b. Macbeth killed a king with a dagger.
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Reading 1: the PP ''with a dagger'' is a modifier of the verb ''kill'' Paraphrase: ''Macbeth used a dagger to kill a king.''<br />Reading 2: the PP ''with a dagger'' is a modifier of the noun ''king''. Paraphrase: ''Macbeth killed a king who had a dagger.''
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== Task 2: Model and Interpretation ==
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1. Define a universe that consists of Macbeth and Banquo.
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''U'' = { ''Macbeth'', ''Banquo'' }
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2. Define the interpretation of the names '''macbeth''' and '''banquo''' in an intuitively plausible way.
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I('''macbeth''') = ''Macbeth'', <br /> I('''banquo''') = ''Banquo''
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3. Define the interpretation of the properties '''thane'''<sub>1</sub>, '''king'''<sub>1</sub>,
and '''witch'''<sub>1</sub> is such a way that Macbeth is a king,  both are thanes and neither is a witch.
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I('''thane'''<sub>1</sub>) = {<''Macbeth''>, <''Banquo''>},<br /> I('''king'''<sub>1</sub>) = {<''Macbeth''>},<br /> I('''witch'''<sub>1</sub>) = {}
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4. Define the interpretation of the 2-place relations '''mistrust'''<sub>2</sub> and '''kill'''<sub>2</sub> in such a way that Macbeth and Banquo mistrust each other and Macbeth kills Banquo.
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I('''mistrust'''<sub>2</sub>) = {<''Macbeth'', ''Banquo''>, <''Banquo'', ''Mactbeth''>},<br /> I('''kill'''<sub>2</sub>) = {<''Macbeth'',''Banquo''>}
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== Task 3: Formulae ==
Write down logical formulae that express the meaning of the following sentences.
1. Banquo is a thane.
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'''thane'''<sub>1</sub>('''banquo''')
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2. Macbeth is king and Macbeth mistrusts Banquo.
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'''king'''<sub>1</sub>('''macbeth''') &and; '''mistrust'''<sub>2</sub>('''macbeth''','''banquo''')
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3. If Banquo is king then Macbeth does not kill Banquo.
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'''king'''<sub>1</sub>('''banquo''') &sup; &not; '''kill'''<sub>2</sub>('''macbeth''','''banquo''')
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== Task 4: Interpreting formulae ==
Compute the interpretation of the following formulæ step by step.
1. '''mistrust'''<sub>2</sub>('''macbeth''','''macbeth''')
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[[<nowiki />'''mistrust'''<sub>2</sub>('''macbeth''','''macbeth''')]] = ''1'' <br> iff < [[<nowiki />'''macbeth''']], [[<nowiki />'''macbeth''']] > is in [[<nowiki />'''mistrust'''<sub>2</sub>]] <br> iff < I('''macbeth'''), I('''macbeth''') > in I('''mistrust'''<sub>2</sub>) <br> iff < ''Macbeth'', ''Macbeth'' > in { <''x'',''y''> | ''x'' mistrusts ''y'' } = { <''Macbeth'', ''Banquo''>, <''Banquo'', ''Macbeth''> }
Since this is not the case, [[<nowiki />'''mistrust'''<sub>2</sub>('''macbeth''','''macbeth''')]] = ''0''.
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2. &not;'''king'''('''banquo''')
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[[<nowiki />&not; '''king'''<sub>1</sub>('''banquo''')]] = ''1'' <br>iff [[<nowiki />'''king'''('''banquo''')]] = ''0'' <br>iff < [[<nowiki />'''banquo''']]> is not in [[<nowiki />'''king'''<sub>1</sub>]]<br> iff < I('''banquo'''> is not in I('''king'''<sub>1</sub>) <br>iff < ''Banquo'' > is not in { <''x''> | ''x'' is king } = { <''Macbeth''>}
Since this is the case, [[<nowiki />&not; '''king'''<sub>1</sub>('''banquo''')]] = ''1''
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3. '''witch'''<sub>1</sub>('''banquo''') &sup; '''king'''<sub>1</sub>('''macbeth''')
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[[<nowiki />'''witch'''<sub>1</sub>('''banquo''') &sup; '''king'''<sub>1</sub>('''macbeth'''))]] = ''1''<br>iff [[<nowiki />'''witch'''<sub>1</sub>('''banquo''')]] = ''0'' or [[<nowiki />'''king'''<sub>1</sub>('''macbeth''') = ''1'' <br> iff < [[<nowiki />'''banquo''']] > is not in [[<nowiki />'''witch'''<sub>1</sub>]] or < [[<nowiki />'''macbeth''']] > is in [[<nowiki />'''king'''<sub>1</sub>]] <br> iff  < I('''banquo''') > is not in I('''witch'''<sub>1</sub>) or < I('''macbeth''') > is in I('''king'''<sub>1</sub>) <br> iff < ''Banquo'' > is not in { <''x''> | ''x'' is a witch} = { } or < ''Macbeth'' > is in { <''x''> | ''x'' is king} = { <''Macbeth''>}.
Since both are the case,  [[<nowiki />'''witch'''<sub>1</sub>('''banquo''') &sup; '''king'''<sub>1</sub>('''macbeth'''))]] = ''1''.
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== Task 5: Variables ==
Provide a g-function that maps the variables ''x'', ''y'', and ''z'' to individuals from the universe and compute
the interpretation of the following formula with respect to the model and your g.
(i) '''kill'''<sub>2</sub>(''z'',''x'')
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Example solution (other values for g are equally possible).
g(''x'') = ''Macbeth'',<br>g(''y'') = ''Banquo'',<br>g(''z'') = ''Banquo''.
With this variable assignment we can compute the truth value of the formula:
[[<nowiki />'''kill'''<sub>2</sub>(''z'',''x'')]]<sup>g</sup> = ''1''<br>iff < [[<nowiki />''z'']]<sup>g</sup>, [[<nowiki />''x'']]<sup>g</sup> > is in [[<nowiki />'''kill'''<sub>2</sub>]]<sup>g</sup><br>iff < g(''z''), g(''x'') > is in I('''kill'''<sub>2</sub>)<br>iff < ''Banquo'', ''Macbeth'' > is in { <''x'',''y''> | ''x'' killed ''y''} = { <''Macbeth'', ''Banquo''> }.
Since this is not the case,  [[<nowiki />'''kill'''<sub>2</sub>(''z'',''x'')]]<sup>g</sup> = ''0''.
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== Task 6: Quantifiers ==
Provide logical formulae that expresse the meaning of the following sentences. Are the formulae true in
your model (not in the entire play)? Give a short reason (you don’t need to compute the truth value).
1. Banquo was killed by a king.
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&exist;''x'' ('''king'''(''x'') : '''kill'''(''x'', '''banquo'''))
The formula is true in my model, because there is only one king, Macbeth, and Macbeth killed Banquo.<br>(Note: The English sentence is in passive, but this has no effect on the logical form.)
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2. Macbeth mistrusts every witch.
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&forall;''x'' ('''witch'''(''x'') : '''mistrust'''('''macbeth''', ''x''))
The formula is true in my model, because there are no witches in my model. Therefore, the formula with the universal quantifier is trivially true.
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== Task 7: Analysis: Lexicon  ==
Provide the lexical entries for the words in the sentence ''Banquo mistrusted Macbeth''. Use the features PHON, HEAD, SUBJ, SPR, COMPS, and PARTS.
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{|
|
|''Banquo''
|''mistrusted''
|''Macbeth''
|-
|PHON
|&nbsp;< ''Banquo'' >
|&nbsp;< ''mistrusted'' >
|< ''Macbeth'' >
|-
|HEAD
|&nbsp;''noun''
|&nbsp;''verb''
|''noun''
|-
|SUBJ
|&nbsp;< >
|&nbsp;< NP >
|< >
|-
|SPR
|&nbsp;< >
|&nbsp;< >
|< >
|-
|COMPS
|&nbsp;< >
|&nbsp;< NP >
|< >
|-
|PARTS
|&nbsp;< '''banquo''' >&nbsp;
|&nbsp;< '''mistrust'''<sub>2</sub>, _(_,_) >&nbsp;&nbsp;
|< '''macbeth''' >
|-
|}
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== Task 8: Analysis: Syntactic structure and semantic combinatorics ==
Using the lexical entries from Task 7, provide the syntactic structure of the sentence ''Banquo mistrusted Macbeth''. Indicate '''all''' the values for all features at each node in the tree. (You can ignore the EXCONT value throughout.)
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Tree structure:
[[File:Tree-BanquoMistrustedMacbeth.jpg]]
{|
|
|''Banquo''
|''mistrusted''
|''Macbeth''
|-
|
| [1]
|
| [2]
|-
|PHON
|&nbsp;< [4] ''Banquo'' >
|&nbsp;< [5] ''mistrusted'' >
|< [6] ''Macbeth'' >
|-
|HEAD
|&nbsp;''noun''
|&nbsp;[3] ''verb''
|''noun''
|-
|SUBJ
|&nbsp;< >
|&nbsp;< [1] NP >
|< >
|-
|SPR
|&nbsp;< >
|&nbsp;< >
|< >
|-
|COMPS
|&nbsp;< >
|&nbsp;< [2] NP >
|< >
|-
|PARTS
|&nbsp;<'''banquo''' >&nbsp;
|&nbsp;<'''mistrust'''<sub>2</sub>, _(_,_) >&nbsp;&nbsp;
|<'''macbeth''' >
|-
|}
{|
|
|VP: ''mistrusted M.''
|S: ''B. mistrusted M.''
|-
|PHON
|&nbsp;< [5], [6] > <!-- < ''mistrusted, Macbeth''> -->
|&nbsp;< [4], [5], [6] > <!-- < ''Banquo, mistrusted, Macbeth''> -->
|-
|HEAD
|&nbsp; [3] <!-- ''verb''-->
|&nbsp; [3] <!-- ''verb'' -->
|-
|SUBJ
|&nbsp;< [1] NP>
|&nbsp;< >
|-
|SPR
|&nbsp;< >
|&nbsp;< >
|-
|COMPS
|&nbsp;< >
|&nbsp;< >
|-
|PARTS
|&nbsp;<'''mistrust'''<sub>2</sub>, _(_,_) ,&nbsp;&nbsp;
|&nbsp;<'''mistrust'''<sub>2</sub>, _(_,_),
|-
|
|&nbsp; '''macbeth''' >
|&nbsp; '''macbeth''', '''banquo''' >
|}
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== Task 8: Principles of syntax ==
1. How is the COMPS value of the VP determined by the lexical entries of the words and the principles of grammar?
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The VP is licenced as a head-complement structure. The constraint on head-complement structures requires that the head daughter have a non-empty COMPS list and the mother have an empty COMPS list.<br>(It is also required that the non-head daughters are identical to the elements on the head daughter's COMPS list, but this is not relevant for the question at hand.)
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2. How is it guaranteed that the PHON values of the words all appear in the PHON value of the sentence?
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The Phonology Principle specifies that the PHON value of a mother is the concatenation of the PHON values of its daughter(s). Therefore, a element of the PHON value of a word in a sentence will always be part of the PHON values of every phrase that dominates this word. Since the overall sentence dominates all its component words, its PHON value comprises the PHON values of all words of this sentence.
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3. How is it achieved that the HEAD value of the sentence is ''verb''?
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The HEAD value of the lexical verb, ''mistrust'', is ''verb''. In the VP, the lexical verb is the syntactic head of the phrase. According to the Head Feature Priniciple, the HEAD value of the mother node is the same as that of its head daughter, i.e., ''verb''. Since this VP is the headdaughter of the sentence, the sentence's HEAD value should also be the same, i.e., ''verb'' again.
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-->
== Task 9: General mechanisms of LRS ==
1. Enumerate all possible logical forms that would be compatible with the PARTS lists of the sentence form Task 8.
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There are two possible logical forms: '''mistrust'''<sub>2</sub>('''banquo''','''macbeth''') and '''mistrust'''<sub>2</sub>('''macbeth''','''banquo''')
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2. Use the PARTS value from Task 8 to show that the following expressions are excluded as possible logical forms of the sentence.
(a) '''mistrust'''<sub>2</sub>('''macbeth''','''banquo''','''banquo''')
(b) '''mistrust'''<sub>2</sub>('''banquo''','''banquo''')
(c) '''macbeth'''('''mistrust'''<sub>2</sub>,'''banquo''')
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(a) The expression cannot be a possible logical form, because it is not a well-formed formula: the predicate '''mistrust'''  can only combine with two arguments, not with three. (This is indicated with the element '''mistrust'''<sub>2</sub>(...,...).
(b) The formula does not use all expressions from the PARTS list: the expression '''macbeth''' is missing.
(c) '''macbeth''' denotes an individual , '''mistrust'''<sub>2</sub> is a predicate. Therefore, '''macbeth''' cannot function as predicate, not can '''mistrust'''<sub>2</sub> function as its argument.
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(d)
The subject, ''Banquo'', should be linked to the first semantic argument slot of the predicate '''mistrust'''<sub>2</sub> - analogously for the complement and the second argument slot of the predicate. This is specified in the lexical entry of the verb, where the DR values of the subject ([a]) and the complement ([b]) are identified with the first and the second argument slots of '''mistrust'''<sub>2</sub> respectively.
-->
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== Task 9: General mechanisms of LRS ==
Explain how the following formulae are excluded from occurring as EX-CONT values of the sentence from Task 7.
(a) '''mistrust'''<sub>2</sub>('''macbeth''','''banquo''','''banquo''')
(b) '''mistrust'''<sub>2</sub>('''banquo''','''banquo''')
(c) '''macbeth'''('''mistrust'''<sub>2</sub>,'''banquo''')
(d) '''mistrust'''<sub>2</sub>('''macbeth''','''banquo''')
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(a) The expression cannot be a possible logical form, because it is not a well-formed formula: the predicate '''mistrust'''  can only combine with two arguments, not with three. (This is indicated with the element '''mistrust'''<sub>2</sub>(...,...).
(b) The formula does not use all expressions from the PARTS list: the expression '''macbeth''' is missing.
(c) '''macbeth''' denotes an individual , '''mistrust'''<sub>2</sub> is a predicate. Therefore, '''macbeth''' cannot function as predicate, not can '''mistrust'''<sub>2</sub> function as its argument.
(d)
The subject, ''Banquo'', should be linked to the first semantic argument slot of the predicate '''mistrust'''<sub>2</sub> - analogously for the complement and the second argument slot of the predicate. This is specified in the lexical entry of the verb, where the DR values of the subject ([a]) and the complement ([b]) are identified with the first and the second argument slots of '''mistrust'''<sub>2</sub> respectively.
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-->
== Task 10: Linking ==
Provide the full lexical entry of the verb from the sentence in Task 7 in such a way that it includes the '''linking information'''.
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PHON < ''mistrusted'' ><br>
HEAD ''verb''<br>
SUBJ < NP[DR [a] ] ><br>
SPR < ><br>
COMPS < NP[DR [b] ] ><br>
DR [c]<br>
PARTS < [c]'''mistrust'''<sub>2</sub>, [c]([a],[b]) >
</div></div>
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