Katiejohnso patch 8

Project 4/README.md

@@ -45,3 +45,77 @@ You're probably thinking, "why would I submit a level 8 kata if they're not wort
 It is your responsibility and the responsibility of your peers reviewing your submission in PR to determine whether your submission is ranked appropriately. In the event that consensus is reached that your kata is ranked inappropriately, you must work with your peers to revise the submission so that it is either more or less challenging, accordingly. You are not permitted to submit new problems with different strengths after PRs are open, but must instead revise your PRs. So, think hard about how challenging your submission is. 
 
 There is one other option for those desiring a different sort of challenge. If you provide alongside your SPARQL submission a translation of the same problem into SQL, complete with documentations, solution, etc. then you may receive half points extra at that kata level (rounded up). For example, if you submit a SPARQL problem that is kata rank 1 and also submit a SQL version of that same problem, you  will receive 35+18=53 points. 
+
+1) Write a query that returns all object properties in dbpedia. 
+
+Reference solution:
+select ?p
+
+where{
+?s ?p ?o
+}
+
+2) Write a SPARQL quiery that retrieves the name, birthplace, birth date, genre, latitude, longitude, label, and abstract of actors born after 1950, along with information about their birthplaces.
+
+Hint:
+This question uses four query forms:
+
+Triple patterns to retrieve the actor's name and birthplace, as well as the birth date of each actor.
+A property path to retrieve the genres of the movies each actor appeared in.
+A set of triple patterns to retrieve the coordinates, label, and abstract of each birthplace.
+Filter expressions to restrict the birth date to after 1950 and the birthplace label to English.
+
+The query also uses a combination of operators and functions, including:
+
+The LANGMATCHES function to match English-language labels.
+The STRDT function to convert the birth date string to a date datatype for comparison.
+The ORDER BY clause to sort the results by actor name.
+Triple patterns to retrieve the actor's name and birthplace, as well as the birth date of each actor.
+A property path to retrieve the genres of the movies each actor appeared in.
+A set of triple patterns to retrieve the coordinates, label, and abstract of each birthplace.
+Filter expressions to restrict the birth date to after 1950 and the birthplace label to English.
+
+The query also uses a combination of operators and functions, including:
+
+The LANGMATCHES function to match English-language labels.
+The STRDT function to convert the birth date string to a date datatype for comparison.
+The ORDER BY clause to sort the results by actor name.
+
+Reference solution:
+PREFIX dbo: <http://dbpedia.org/ontology/>
+PREFIX foaf: <http://xmlns.com/foaf/0.1/>
+PREFIX geo: <http://www.w3.org/2003/01/geo/wgs84_pos#>
+
+SELECT DISTINCT ?actor ?actorName ?birthPlace ?birthDate ?genre ?latitude ?longitude ?label ?abstract
+WHERE {
+  # Query 1: Retrieve actors and their birthplaces
+  ?actor a dbo:Actor ;
+         foaf:name ?actorName ;
+         dbo:birthPlace ?birthPlace .
+
+  # Query 2: Retrieve the birth date of each actor
+  ?actor dbo:birthDate ?birthDate .
+
+  # Query 3: Retrieve the genres of the movies each actor appeared in
+  ?actor dbo:starring ?movie .
+  ?movie dbo:genre ?genre .
+
+  # Query 4: Retrieve the coordinates, label, and abstract of each birthplace
+  ?birthPlace geo:lat ?latitude ;
+              geo:long ?longitude ;
+              rdfs:label ?label ;
+              dbo:abstract ?abstract .
+
+  # Filter actors born after 1950
+  FILTER(STRDT(?birthDate, xsd:date) > "1950-01-01"^^xsd:date)
+
+  # Filter birthplaces with a label in English
+  FILTER(LANGMATCHES(LANG(?label), "en"))
+
+} ORDER BY ?actorName
+
+
+
+
+
+

Project-1/README.md

@@ -17,48 +17,164 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→",
   (b) (A→B)∧(A→¬B)
   (c) (A→(B∨C))∨(C→¬A) 
   (d) ((A→B)∧C)∨(A∧D) 
+
+  (a) (¬A→B)∨((A∧¬C)→B), Tautology
+  A	B	C	((¬A → B) ∨ ((A ∧ ¬C) → B))
+F	F	F	T
+F	F	T	T
+F	T	F	T
+F	T	T	T
+T	F	F	T
+T	F	T	T
+T	T	F	T
+T	T	T	T
+  (b) (A→B)∧(A→¬B), Contingent
+  A	B	((A → B) ∧ (A → ¬B))
+F	F	T
+F	T	T
+T	F	F
+T	T	F
+  (c) (A→(B∨C))∨(C→¬A), Tautology
+  A	B	C	((A → (B ∨ C)) ∨ (C → ¬A))
+F	F	F	T
+F	F	T	T
+F	T	F	T
+F	T	T	T
+T	F	F	T
+T	F	T	T
+T	T	F	T
+T	T	T	T
+  (d) ((A→B)∧C)∨(A∧D), Contingent
+  A	B	C	D	(((A → B) ∧ C) ∨ (A ∧ D))
+F	F	F	F	F
+F	F	F	T	F
+F	F	T	F	T
+F	F	T	T	T
+F	T	F	F	F
+F	T	F	T	F
+F	T	T	F	T
+F	T	T	T	T
+T	F	F	F	F
+T	F	F	T	T
+T	F	T	F	F
+T	F	T	T	T
+T	T	F	F	F
+T	T	F	T	T
+T	T	T	F	T
+T	T	T	T	T
   ```
 
 2. A _literal_ is an atomic formula or the negation of an atomic formula. We say a formula is in _conjunctive normal form_ (CNF) if it is the conjunction of the disjunction of literals. Find propositional logic formulas in CNF equivalent to each of the following:
   ```
   (a) (A→B)→C
   (b) (A→(B∨C))∨(C→¬A)
   (c) (¬A∧¬B∧C)∨(¬A∧¬C)∨(B∧C)∨A 
+
+  (A→B)→C, CNF: (A∨C)∧(¬B∨C)
+  (1) ¬(A→B)∨C
+  (2) ¬(¬A→B)∨C
+  (3) ¬(A→ ¬B)∨C
+  (4) (A∨C)∧ (¬B∨C)
+
+  (b) (A→(B∨C))∨(C→¬A), CNF: B ∨ ¬B (tautology)
+
+  (c) (¬A∧¬B∧C)∨(¬A∧¬C)∨(B∧C)∨A, CNF: B ∨ ¬B (tautology) 
+
   ```
 
 3. Let V be the vocabulary of first-order logic consisting of a binary relation P and a unary relation F. Interpret P(x,y) as “x is a parent of y” and F(x) as “x is female.” Where possible define the following formulas in this vocabulary; where not possible, explain why: 
 
   ```
   (a)  B(x,y) that says that x is a brother of y  
+
+   B(x,y) <--> ∃z((Pzx ∧ Pzy) ∧ x ≠ y ∧ ~Fx)
+
   (b)  A(x,y) that says that x is an aunt of y  
-  (c)  C(x,y) that says that x and y are cousins   
+
+  A(x,y) <--> ∃z((Fx ∧ Pzy ∧ ∃w(Pwx ∧ Pwz ∧ x ≠ z))
+
+  (c)  C(x,y) that says that x and y are cousins 
+
+  C(x,y) <--> ∃z∃w∃r(Pzx ∧ Pwy ∧ Prz ∧ Prw ∧ z ≠ w ∧ x ≠ y)
+
+
   (d)  O(x) that says that x is an only child  
+
+  O(x) <--> ∃y∀z(Pyx ∧ (Pyz --> z = x))
+
   (e)  T(x) that says that x has exactly two brothers 
+
+  ∃y∃z∃w(¬(w=z)∧¬(x=w)∧¬(x=z)∧P(y,x)∧P(y,z)∧P(y,w)∧¬F(z)∧¬F(w))∧∀r(P(y,r)∧¬F(r)→ r=x ∨ r=z ∨ r=w)
+
   ```
 
 4. Let V be a vocabulary of the attribute (concept) language with complements (ALC) consisting of a role name "parent_of" and a concept name "Male". Interpret parent_of as "x is a parent of y" and M as "x is male". Where possible define the following formulas in this vocabulary; where not possible, explain why: 
   ```
   (a)  B that says that x is a brother of y
+
+  p2 (parent of at least 2 children) ≡ ≥2 ∃parent_of.Person	
+	B ≡ M ⊓ ∃p2¯.Person
+
   (b)  A that says that x is an aunt of y
+
+  A ≡ ¬M ⊓ (∃p2¯.(≥ 2 parent_of. (∃parent_of. Person)) ⊔ ¬∃parent_of. Person)
+
   (c)  C that says that x and y are cousins
+
+  C ≡ gp2¯. Person
+
+  Cousins cannot be defined in description logic, therefore, this statement only describes on of the two cousins (i.e. "being a cousin of someone.")  Thank you Karl for the clarification.  
+
   (d)  O that says that x is an only child  
+
+  parent_only (be a parent of exactly one child) ≡ (≥1 parent_of. Person) ⊓ (≤1 parent_of. Person)
+	O ≡ ∃parent_only¯.Person
+
   (e)  T that says that x has exactly two brothers 
+
+ 	 p3m (be a parent of exactly three male children) ≡ (≥3 parent_of. Male) ⊓ (≤3 parent_of. Male)
+	p1 (be a parent of at least one child) ≡ ≥1 parent_of. Person -- the inverse of p1 is just "be a child of"!
+	T ≡ ∃p3m¯.Person ⊔ (¬M ⊓ ∃p1¯. (≥2 parent_of. Male ⊓ ≤2 parent_of. Male))
+
   ```
 
 
-5. Select two formulas defined in ALC from question 4 to form the basis of a T-Box. Supplement this T-box with whatever other axioms you like, as well as an A-box, so that you ultimately construct a knowledge base K = (T,A). Provide a _model_ of K. This may be graphical or symbolic or both. 
+5. Select two formulas defined in ALC from question 4 to form the basis of a T-Box. Supplement this T-box with whatever other axioms you like, as well as an A-box, so that you ultimately construct a knowledge base K = (T,A). Provide a _model_ of K. This may be graphical or symbolic or both.
+
+B ≡ M ⊓ ∃p2¯.Person
+O= parent_of (∃parent_of=1)
+
+Tom: B 
+Joe: B 
+Jill: O
+(Jill, Tom): parent_of
+(Jill, Joe): parent_of
 
 6. Explain the difference - using natural language - between the first-order prefixes:
   ```
   (a) ∃x∀y and ∀x∃y
+  ∃x∀y = there exists some x such that for all y
+  ∀x∃y = for all x there exists some y
+
   (b) ∃x∀y∃z and ∀x∃y∀z 
+  ∃x∀y∃z = there exists some x such that for all y there exists some z
+  ∀x∃y∀z = for all x there exists some y such that for all z
+
   (c) ∀x∃y∀z∃w and ∃x∀y∃z∀w
+  ∀x∃y∀z∃w = for all x there exists some y such that for all z there exists some w
+  ∃x∀y∃z∀w = there exists some x for all y such that for some z there exists all w 
 ```
 
 7. Show that the following sentences are not equivalent by exhibiting a graph that models one but not both of these sentences:
 ```
 ∀x∃y∀z(R(x,y) ∧ R(x,z) ∧ R(y,z))
+for all x there exists some y such that for all z
+x > y, x > z < y
+
 ∃x∀y∃z(R(x,y) ∧ R(x,z) ∧ R(y,z))
+there exists some x for all y such that for all z
+x > y, x > z > y
+
 ```
 
 8. Using an online tableau proof generator - such as the one found here `https://www.umsu.de/trees/` - provide tree proofs of the following entailments, which are known as the De Morgan's laws:
@@ -67,9 +183,99 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→",
   (b) ∀x∀y(¬(Px ∨ Qx) → (¬Px ∧ ¬Qx))
   (c) ∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx))
   (d) ∀x∀y((¬Px ∧ ¬Qx) → ¬(Px ∨ Qx))
+
+   (a) ∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx))
+
+ (1) ¬∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx))
+ (2) ¬∀y(¬(Pa ∧ Qa) → (¬Pa ∨ ¬Qa))
+ (3) ¬(¬(Pa ∧ Qa) → (¬Pa ∨ ¬Qa))
+ (4)¬(Pa ∧ Qa)
+ (5)¬(¬Pa ∨ ¬Qa)
+ (6).¬¬Pa
+ (7)¬¬Qa
+ (8)Qa
+ (9) Pa
+ (10) ¬Pa	(11) ¬Qa
+        x	       x
+
+  (b) ∀x∀y(¬(Px ∨ Qx) → (¬Px ∧ ¬Qx))
+
+  (1) ¬∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx))
+  (2) ¬∀y(¬(Pa ∧ Qa) → (¬Pa ∨ ¬Qa))
+  (3) ¬(¬(Pa ∧ Qa) → (¬Pa ∨ ¬Qa))
+  (4) ¬(Pa ∧ Qa)
+  (5) ¬(¬Pa ∨ ¬Qa)
+  (6) ¬¬Pa
+  (7) ¬¬Qa
+  (8) Qa
+  (9) Pa
+  (10) ¬Pa	(11) ¬Qa
+        x	      x
+
+  (c) ∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx))
+
+  (1) ¬∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx))
+  (2) ¬∀y((¬Pa ∨ ¬Qa) → ¬(Pa ∧ Qa))
+  (3) ¬((¬Pa ∨ ¬Qa) → ¬(Pa ∧ Qa))
+  (4) ¬Pa ∨ ¬Qa
+  (5) ¬¬(Pa ∧ Qa)
+  (6) Pa ∧ Qa
+  (7) Pa
+  (8) Qa
+  (9) ¬Pa	(10) ¬Qa
+     x		     x
+
+  (d) ∀x∀y((¬Px ∧ ¬Qx) → ¬(Px ∨ Qx))
+
+ (1) ¬∀x∀y((¬Px ∧ ¬Qx) → ¬(Px ∨ Qx))
+ (2) ¬∀y((¬Pa ∧ ¬Qa) → ¬(Pa ∨ Qa))
+ (3) ¬((¬Pa ∧ ¬Qa) → ¬(Pa ∨ Qa))
+ (4) ¬Pa ∧ ¬Qa
+ (5) ¬¬(Pa ∨ Qa)
+ (6) Pa ∨ Qa
+ (7) ¬Pa
+ (8) ¬Qa
+ (9) Pa		(10) Qa
+     x		     x
+
 ```
 
 9. Using a natural deduction proof generator - such as the one found here `https://proofs.openlogicproject.org/` - provide natural deduction proofs for each of De Morgan's laws. 
 
+¬∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx))
+(1) ¬(Px ∧ Qx)		assumption
+(2) ¬(¬Px ∨ ¬Qx)	assumption
+(3) (Px ∧ Qx)		2, negated disjunction
+(4) ¬(Px ∧ Qx)		1,3 contradiction
+
+∀x∀y(¬(Px ∨ Qx) → (¬Px ∧ ¬Qx))
+(1) ¬(Px ∨ Qx)		assumption
+(2) (¬Px ∧ ¬Qx)		assumption
+(3) (Px ∨ Qx)		2, negated conjunction
+(4) ¬(Px ∨ Qx)		1,3 contradiction
+
+∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx))
+(1) (¬Px ∨ ¬Qx)		assumption
+(2) ¬¬(Px ∧ Qx)		assumption
+(3) (Px ∧ Qx)		2, double negation
+(4) Px			3, conjunction
+(5) Qx			3, conjunction
+(6) (¬Px ∨ ¬Qx)		4,5,6 contradiction
+
+∀x∀y((¬Px ∧ ¬Qx) → ¬(Px ∨ Qx))
+(1) (¬Px ∧ ¬Qx)		assumption
+(2) ¬¬(Px ∨ Qx)		assumption
+(3) (Px ∨ Qx)		2, double negation
+(4) ¬Px			1, conjunction
+(5) ¬Qx			1, conjunction
+(6) Px			3, disjunction
+(7) ¬Qx			3, disjunction
+(8) ¬Px			3, disjunction
+(9) Qx			3, disjunction
+(10) (¬Px ∧ ¬Qx)	6,7,8,9 contradiction
+
 10. Compare and contrast the proofs provided for (a) in your answers to questions 8 and 9. Explain the different assumptions, strategies, etc. exhibited in tree proofs vs natural deduction proofs. 
+
+Both proofs deduce some kind of contradiction, but in different ways.  8a is a longer proof than 9a. Feedback from Josh (thank you!) also states that 9a does not require us to go down to the atomic parts of the formula, and stops at more complex contradictions.
+