Fixed all duplicate exercises in Chapter 7

markdown/7-Logical-Agents/exercises/ex_10/question.md

@@ -1,9 +1,5 @@
 
 
-We have defined four binary logical connectives.<br>
-
-1.  Are there any others that might be useful?<br>
-
-2.  How many binary connectives can there be?<br>
-
-3.  Why are some of them not very useful?<br>
+Using a method of your choice, verify
+each of the equivalences in
+Table <a class="insideBookFigRef" target="_blank" href="https://aimacode.github.io/aima-exercises/figures/logical-equivalence-table.png">logical-equivalence-table</a> (page <a class="pageRef" title="" href="#">logical-equivalence-table</a>).

markdown/7-Logical-Agents/exercises/ex_11/question.md

@@ -1,5 +1,21 @@
 
 
-Using a method of your choice, verify
-each of the equivalences in
+Decide whether each of the following
+sentences is valid, unsatisfiable, or neither. Verify your decisions
+using truth tables or the equivalence rules of
 Table <a class="insideBookFigRef" target="_blank" href="https://aimacode.github.io/aima-exercises/figures/logical-equivalence-table.png">logical-equivalence-table</a> (page <a class="pageRef" title="" href="#">logical-equivalence-table</a>).
+
+1.  ${Smoke} {\:\;{\Rightarrow}\:\;}{Smoke}$<br>
+
+2.  ${Smoke} {\:\;{\Rightarrow}\:\;}{Fire}$<br>
+
+3.  $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(\lnot {Smoke} {\:\;{\Rightarrow}\:\;}\lnot {Fire})$<br>
+
+4.  ${Smoke} \lor {Fire} \lor \lnot {Fire}$<br>
+
+5.  $(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire})
+            {\;\;{\Leftrightarrow}\;\;}(({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) \lor ({Heat} {\:\;{\Rightarrow}\:\;}{Fire}))$<br>
+
+6.  $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire}) $<br>
+
+7.  ${Big} \lor {Dumb} \lor ({Big} {\:\;{\Rightarrow}\:\;}{Dumb})$<br>

markdown/7-Logical-Agents/exercises/ex_12/question.md

@@ -1,21 +1,6 @@
 
 
-Decide whether each of the following
-sentences is valid, unsatisfiable, or neither. Verify your decisions
-using truth tables or the equivalence rules of
-Table <a class="insideBookFigRef" target="_blank" href="https://aimacode.github.io/aima-exercises/figures/logical-equivalence-table.png">logical-equivalence-table</a> (page <a class="pageRef" title="" href="#">logical-equivalence-table</a>).
-
-1.  ${Smoke} {\:\;{\Rightarrow}\:\;}{Smoke}$<br>
-
-2.  ${Smoke} {\:\;{\Rightarrow}\:\;}{Fire}$<br>
-
-3.  $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(\lnot {Smoke} {\:\;{\Rightarrow}\:\;}\lnot {Fire})$<br>
-
-4.  ${Smoke} \lor {Fire} \lor \lnot {Fire}$<br>
-
-5.  $(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire})
-            {\;\;{\Leftrightarrow}\;\;}(({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) \lor ({Heat} {\:\;{\Rightarrow}\:\;}{Fire}))$<br>
-
-6.  $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire}) $<br>
-
-7.  ${Big} \lor {Dumb} \lor ({Big} {\:\;{\Rightarrow}\:\;}{Dumb})$<br>
+Any propositional logic sentence is logically
+equivalent to the assertion that each possible world in which it would
+be false is not the case. From this observation, prove that any sentence
+can be written in CNF.

markdown/7-Logical-Agents/exercises/ex_13/question.md

@@ -1,21 +1,4 @@
 
 
-Decide whether each of the following
-sentences is valid, unsatisfiable, or neither. Verify your decisions
-using truth tables or the equivalence rules of
-Table <a class="insideBookFigRef" target="_blank" href="https://aimacode.github.io/aima-exercises/figures/logical-equivalence-table.png">logical-equivalence-table</a> (page <a class="pageRef" title="" href="#">logical-equivalence-table</a>).<br>
-
-1.  ${Smoke} {\:\;{\Rightarrow}\:\;}{Smoke}$<br>
-
-2.  ${Smoke} {\:\;{\Rightarrow}\:\;}{Fire}$<br>
-
-3.  $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(\lnot {Smoke} {\:\;{\Rightarrow}\:\;}\lnot {Fire})$<br>
-
-4.  ${Smoke} \lor {Fire} \lor \lnot {Fire}$<br>
-
-5.  $(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire})
-            {\;\;{\Leftrightarrow}\;\;}(({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) \lor ({Heat} {\:\;{\Rightarrow}\:\;}{Fire}))$<br>
-
-6.  ${Big} \lor {Dumb} \lor ({Big} {\:\;{\Rightarrow}\:\;}{Dumb})$<br>
-
-7.  $({Big} \land {Dumb}) \lor \lnot {Dumb}$<br>
+Use resolution to prove the sentence $\lnot A \land \lnot B$ from the
+clauses in Exercise <a class="exerciseRef" href="{{ site.baseurl }}/knowledge-logic-exercises/ex_25/">convert-clausal-exercise</a>.

markdown/7-Logical-Agents/exercises/ex_14/question.md

@@ -1,6 +1,18 @@
 
 
-Any propositional logic sentence is logically
-equivalent to the assertion that each possible world in which it would
-be false is not the case. From this observation, prove that any sentence
-can be written in CNF.
+This exercise looks into the relationship between
+clauses and implication sentences.<br>
+
+1.  Show that the clause $(\lnot P_1 \lor \cdots \lor \lnot P_m \lor Q)$
+    is logically equivalent to the implication sentence
+    $(P_1 \land \cdots \land P_m) {\;{\Rightarrow}\;}Q$.<br>
+
+2.  Show that every clause (regardless of the number of
+    positive literals) can be written in the form
+    $(P_1 \land \cdots \land P_m) {\;{\Rightarrow}\;}(Q_1 \lor \cdots \lor Q_n)$,
+    where the $P$s and $Q$s are proposition symbols. A knowledge base
+    consisting of such sentences is in implicative normal form or <b>Kowalski
+    form</b> <a class="paperRef" title="" href="">Kowalski:1979</a>.<br>
+
+3.  Write down the full resolution rule for sentences in implicative
+    normal form.<br>

markdown/7-Logical-Agents/exercises/ex_15/question.md

@@ -1,4 +1,15 @@
 
+According to some political pundits, a person who is radical ($R$) is
+electable ($E$) if he/she is conservative ($C$), but otherwise is not
+electable.<br>
 
-Use resolution to prove the sentence $\lnot A \land \lnot B$ from the
-clauses in Exercise <a class="exerciseRef" href="{{ site.baseurl }}/knowledge-logic-exercises/ex_25/">convert-clausal-exercise</a>.
+1.  Which of the following are correct representations of this
+    assertion?<br>
+
+    1.  $(R\land E)\iff C$<br>
+
+    2.  $R{\:\;{\Rightarrow}\:\;}(E\iff C)$<br>
+
+    3.  $R{\:\;{\Rightarrow}\:\;}((C{\:\;{\Rightarrow}\:\;}E) \lor \lnot E)$<br>
+
+2.  Which of the sentences in (a) can be expressed in Horn form?

markdown/7-Logical-Agents/exercises/ex_16/index.md

@@ -3,8 +3,8 @@ layout: exercise
 title: Exercise 7.16
 permalink: /knowledge-logic-exercises/ex_16/
 breadcrumb: 7-Logical-Agents
-canonical_id: ch7ex16
 home: "true"
+canonical_id: ch7ex16
 ---
 
 {% include mathjax_support %}

markdown/7-Logical-Agents/exercises/ex_16/question.md

@@ -1,18 +1,26 @@
 
 
-This exercise looks into the relationship between
-clauses and implication sentences.<br>
+This question considers representing satisfiability (SAT) problems as
+CSPs.<br>
 
-1.  Show that the clause $(\lnot P_1 \lor \cdots \lor \lnot P_m \lor Q)$
-    is logically equivalent to the implication sentence
-    $(P_1 \land \cdots \land P_m) {\;{\Rightarrow}\;}Q$.<br>
+1.  Draw the constraint graph corresponding to the SAT problem
+    $$(\lnot X_1 \lor X_2) \land (\lnot X_2 \lor X_3) \land \ldots \land (\lnot X_{n-1} \lor X_n)$$
+    for the particular case $n{{\,=\,}}5$.<br>
 
-2.  Show that every clause (regardless of the number of
-    positive literals) can be written in the form
-    $(P_1 \land \cdots \land P_m) {\;{\Rightarrow}\;}(Q_1 \lor \cdots \lor Q_n)$,
-    where the $P$s and $Q$s are proposition symbols. A knowledge base
-    consisting of such sentences is in implicative normal form or <b>Kowalski
-    form</b> <a class="paperRef" title="" href="">Kowalski:1979</a>.<br>
+2.  How many solutions are there for this general SAT problem as a
+    function of $n$?<br>
 
-3.  Write down the full resolution rule for sentences in implicative
-    normal form.<br>
+3.  Suppose we apply {Backtracking-Search} (page <a class="pageRef" title="" href="#">backtracking-search-algorithm</a>) to find <i>all</i>
+    solutions to a SAT CSP of the type given in (a). (To find
+    <i>all</i> solutions to a CSP, we simply modify the basic
+    algorithm so it continues searching after each solution is found.)
+    Assume that variables are ordered $X_1,\ldots,X_n$ and ${false}$
+    is ordered before ${true}$. How much time will the algorithm take
+    to terminate? (Write an $O(\cdot)$ expression as a function of $n$.)<br>
+
+4.  We know that SAT problems in Horn form can be solved in linear time
+    by forward chaining (unit propagation). We also know that every
+    tree-structured binary CSP with discrete, finite domains can be
+    solved in time linear in the number of variables
+    (Section <a class="sectionRef" title="" href="#">csp-structure-section</a>). Are these two
+    facts connected? Discuss.<br>

markdown/7-Logical-Agents/exercises/ex_17/question.md

@@ -1,15 +1,7 @@
 
-According to some political pundits, a person who is radical ($R$) is
-electable ($E$) if he/she is conservative ($C$), but otherwise is not
-electable.<br>
 
-1.  Which of the following are correct representations of this
-    assertion?<br>
-
-    1.  $(R\land E)\iff C$<br>
-
-    2.  $R{\:\;{\Rightarrow}\:\;}(E\iff C)$<br>
-
-    3.  $R{\:\;{\Rightarrow}\:\;}((C{\:\;{\Rightarrow}\:\;}E) \lor \lnot E)$<br>
-
-2.  Which of the sentences in (a) can be expressed in Horn form?
+Explain why every nonempty propositional clause, by itself, is
+satisfiable. Prove rigorously that every set of five 3-SAT clauses is
+satisfiable, provided that each clause mentions exactly three distinct
+variables. What is the smallest set of such clauses that is
+unsatisfiable? Construct such a set.

markdown/7-Logical-Agents/exercises/ex_18/index.md

@@ -3,11 +3,10 @@ layout: exercise
 title: Exercise 7.18
 permalink: /knowledge-logic-exercises/ex_18/
 breadcrumb: 7-Logical-Agents
-home: "true"
 canonical_id: ch7ex18
+home: "true"
 ---
 
 {% include mathjax_support %}
+<div id="hiddden">{% include_relative question.md %}</div>
 
-
-<div id="hiddden">{% include_relative question.md %}</div>

markdown/7-Logical-Agents/exercises/ex_18/question.md

@@ -1,26 +1,18 @@
 
 
-This question considers representing satisfiability (SAT) problems as
-CSPs.<br>
+A propositional <i>2-CNF</i> expression is a conjunction of
+clauses, each containing <i>exactly 2</i> literals, e.g.,
+$$(A\lor B) \land (\lnot A \lor C) \land (\lnot B \lor D) \land (\lnot
+  C \lor G) \land (\lnot D \lor G)\ .$$<br>
 
-1.  Draw the constraint graph corresponding to the SAT problem
-    $$(\lnot X_1 \lor X_2) \land (\lnot X_2 \lor X_3) \land \ldots \land (\lnot X_{n-1} \lor X_n)$$
-    for the particular case $n{{\,=\,}}5$.<br>
+1.  Prove using resolution that the above sentence entails $G$.<br>
 
-2.  How many solutions are there for this general SAT problem as a
-    function of $n$?<br>
+2.  Two clauses are <i>semantically distinct</i> if they are not
+    logically equivalent. How many semantically distinct 2-CNF clauses
+    can be constructed from $n$ proposition symbols?<br>
 
-3.  Suppose we apply {Backtracking-Search} (page <a class="pageRef" title="" href="#">backtracking-search-algorithm</a>) to find <i>all</i>
-    solutions to a SAT CSP of the type given in (a). (To find
-    <i>all</i> solutions to a CSP, we simply modify the basic
-    algorithm so it continues searching after each solution is found.)
-    Assume that variables are ordered $X_1,\ldots,X_n$ and ${false}$
-    is ordered before ${true}$. How much time will the algorithm take
-    to terminate? (Write an $O(\cdot)$ expression as a function of $n$.)<br>
+3.  Using your answer to (b), prove that propositional resolution always
+    terminates in time polynomial in $n$ given a 2-CNF sentence
+    containing no more than $n$ distinct symbols.<br>
 
-4.  We know that SAT problems in Horn form can be solved in linear time
-    by forward chaining (unit propagation). We also know that every
-    tree-structured binary CSP with discrete, finite domains can be
-    solved in time linear in the number of variables
-    (Section <a class="sectionRef" title="" href="#">csp-structure-section</a>). Are these two
-    facts connected? Discuss.<br>
+4.  Explain why your argument in (c) does not apply to 3-CNF.<br>

markdown/7-Logical-Agents/exercises/ex_19/index.md

@@ -9,5 +9,4 @@ home: "true"
 
 {% include mathjax_support %}
 
-
-<div id="hiddden">{% include_relative question.md %}</div>
+<div id="hiddden">{% include_relative question.md %}</div>

markdown/7-Logical-Agents/exercises/ex_19/question.md

@@ -1,26 +1,12 @@
 
 
-This question considers representing satisfiability (SAT) problems as
-CSPs.<br>
+Prove each of the following assertions:<br>
 
-1.  Draw the constraint graph corresponding to the SAT problem
-    $$(\lnot X_1 \lor X_2) \land (\lnot X_2 \lor X_3) \land \ldots \land (\lnot X_{n-1} \lor X_n)$$
-    for the particular case $n{{\,=\,}}4$.<br>
+1.  Every pair of propositional clauses either has no resolvents, or all
+    their resolvents are logically equivalent.<br>
 
-2.  How many solutions are there for this general SAT problem as a
-    function of $n$?<br>
+2.  There is no clause that, when resolved with itself, yields
+    (after factoring) the clause $(\lnot P \lor \lnot Q)$.<br>
 
-3.  Suppose we apply {Backtracking-Search} (page <a class="pageRef" title="" href="#">backtracking-search-algorithm</a>) to find <i>all</i>
-    solutions to a SAT CSP of the type given in (a). (To find
-    <i>all</i> solutions to a CSP, we simply modify the basic
-    algorithm so it continues searching after each solution is found.)
-    Assume that variables are ordered $X_1,\ldots,X_n$ and ${false}$
-    is ordered before ${true}$. How much time will the algorithm take
-    to terminate? (Write an $O(\cdot)$ expression as a function of $n$.)<br>
-
-4.  We know that SAT problems in Horn form can be solved in linear time
-    by forward chaining (unit propagation). We also know that every
-    tree-structured binary CSP with discrete, finite domains can be
-    solved in time linear in the number of variables
-    (Section <a class="sectionRef" title="" href="#">csp-structure-section</a>). Are these two
-    facts connected? Discuss.
+3.  If a propositional clause $C$ can be resolved with a copy of itself,
+    it must be logically equivalent to $ True $.<br>

markdown/7-Logical-Agents/exercises/ex_20/index.md

@@ -3,11 +3,10 @@ layout: exercise
 title: Exercise 7.20
 permalink: /knowledge-logic-exercises/ex_20/
 breadcrumb: 7-Logical-Agents
-canonical_id: ch7ex20
 home: "true"
+canonical_id: ch7ex20
 ---
 
 {% include mathjax_support %}
 
-
-<div id="hiddden">{% include_relative question.md %}</div>
+<div id="hiddden">{% include_relative question.md %}</div>

markdown/7-Logical-Agents/exercises/ex_20/question.md

@@ -1,7 +1,13 @@
 
 
-Explain why every nonempty propositional clause, by itself, is
-satisfiable. Prove rigorously that every set of five 3-SAT clauses is
-satisfiable, provided that each clause mentions exactly three distinct
-variables. What is the smallest set of such clauses that is
-unsatisfiable? Construct such a set.
+Consider the following sentence:<br>
+$$[ ({Food} {\:\;{\Rightarrow}\:\;}{Party}) \lor ({Drinks} {\:\;{\Rightarrow}\:\;}{Party}) ] {\:\;{\Rightarrow}\:\;}[ ( {Food} \land {Drinks} )  {\:\;{\Rightarrow}\:\;}{Party}]\ .$$<br>
+
+1.  Determine, using enumeration, whether this sentence is valid,
+    satisfiable (but not valid), or unsatisfiable.<br>
+
+2.  Convert the left-hand and right-hand sides of the main implication
+    into CNF, showing each step, and explain how the results confirm
+    your answer to (a).<br>
+
+3.  Prove your answer to (a) using resolution.

markdown/7-Logical-Agents/exercises/ex_21/index.md

@@ -3,10 +3,11 @@ layout: exercise
 title: Exercise 7.21
 permalink: /knowledge-logic-exercises/ex_21/
 breadcrumb: 7-Logical-Agents
-canonical_id: ch7ex21
 home: "true"
+canonical_id: ch7ex21
 ---
 
 {% include mathjax_support %}
-<div id="hiddden">{% include_relative question.md %}</div>
 
+
+<div id="hiddden">{% include_relative question.md %}</div>

markdown/7-Logical-Agents/exercises/ex_21/question.md

@@ -1,18 +1,34 @@
 
 
-A propositional <i>2-CNF</i> expression is a conjunction of
-clauses, each containing <i>exactly 2</i> literals, e.g.,
-$$(A\lor B) \land (\lnot A \lor C) \land (\lnot B \lor D) \land (\lnot
-  C \lor G) \land (\lnot D \lor G)\ .$$<br>
+A sentence is in disjunctive normal form(DNF) if it is the disjunction of
+conjunctions of literals. For example, the sentence
+$(A \land B \land \lnot C) \lor (\lnot A \land C) \lor (B \land \lnot C)$
+is in DNF.<br>
 
-1.  Prove using resolution that the above sentence entails $G$.<br>
+1.  Any propositional logic sentence is logically equivalent to the
+    assertion that some possible world in which it would be true is in
+    fact the case. From this observation, prove that any sentence can be
+    written in DNF.<br>
 
-2.  Two clauses are <i>semantically distinct</i> if they are not
-    logically equivalent. How many semantically distinct 2-CNF clauses
-    can be constructed from $n$ proposition symbols?<br>
+2.  Construct an algorithm that converts any sentence in propositional
+    logic into DNF. (<i>Hint</i>: The algorithm is similar to
+    the algorithm for conversion to CNF iven in
+    Sectio <a class="sectionRef" title="" href="#">pl-resolution-section</a>.)<br>
 
-3.  Using your answer to (b), prove that propositional resolution always
-    terminates in time polynomial in $n$ given a 2-CNF sentence
-    containing no more than $n$ distinct symbols.<br>
+3.  Construct a simple algorithm that takes as input a sentence in DNF
+    and returns a satisfying assignment if one exists, or reports that
+    no satisfying assignment exists.<br>
 
-4.  Explain why your argument in (c) does not apply to 3-CNF.<br>
+4.  Apply the algorithms in (b) and (c) to the following set of
+    sentences:<br>
+
+ $A {\Rightarrow} B$<bR>
+
+ $B {\Rightarrow} C$<br>
+
+ $C {\Rightarrow} A$<br>
+
+5.  Since the algorithm in (b) is very similar to the algorithm for
+    conversion to CNF, and since the algorithm in (c) is much simpler
+    than any algorithm for solving a set of sentences in CNF, why is
+    this technique not used in automated reasoning?