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Foundations of Computation
创建日期：2024-05-13 14:22:30
标签： Computation
Foundations of Computation

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Computing Practical Session 

Foundations of Computation
The practical contains a number of exercises designed for the students to practice the course content. During the practical
session, the tutor will work through some of these exercises while students will be responsible for completing the remaining
exercises in their own time. There is no expectation that all the exercises will be covered in the practical session.
Covers: Lecture Material Week 3
At the end of this tutorial, you will be able to
• determine free and bound variables of first order formulae;
• prove first order formulae using Natural Deduction;
• determine whether first order formulae are T or F ;
• prove properties by induction.
Exercise 1 Interpretation of First Order Logic
Let S = (U, θ) be a situation for first order logic where U = N = {0, 1, 2, 3, . . . } is the domain of discourse, we have
one unary relation symbol E(x) that stands for “x is even”, and one binary relation symbol D(x, y) that stands for “x is
divisible by y”. That is, we have
θ(D) = {(x, y) | x divisible by y} and θ(E) = {0, 2, 4, . . . }
so that for example, all of (5, 1), (10, 5), (12, 6) are elements of θ(D), but for example (5, 2) ̸∈ θ(D).
For each of the following formulae ϕ, determine the free variables, and all situations based on θ in which the formula ϕ is
true.
(Here, a situation based on θ is a situation for which θ(D) and θ(E) are as given above.)
Justify your answer briefly (in a single sentence).
1. D(x, x)
2. ∃x.D(x, x)
3. ∀x.∃y.D(x, y)
4. ∃y.E(y) ∧D(x, y)
5. ∀y.¬E(y)→ D(y, x)
6. ∀y.E(y) ∧D(x, y)
Note: The exercise may seem heavy on notation and terminology, please refer to slide 41 onward in Lecture 2, and slide
6 in Lecture 3.
Solution.
1. free variable is x, and D(x, x) holds for all valuations, as every number is divisible by itself.
2. no free variables, and ∃x.D(x, x) holds for all valuations.
3. again, no free variables, and ∀x.∃y.D(x, y) says that all numbers are divisible by some number, so evaluates to
true.
4. ∃y.E(y) ∧D(x, y) has one free variable x and says that x is divisible by an even number. This formula holds for
all valuations θ for which θ(x) is even.
5. ∀y.¬E(y)→ D(y, x) has a free variable x and says that x divides all odd numbers. The only such number is 1.
6. ∀y.E(y) ∧D(x, y) has x as a free variable, but holds for no assignments, as the formula in particular states that all
numbers are even!
1
We see that only the free variables are relevant for the truth of a formula under an assignment.
Exercise 2 Change of bound variables
Prove the statement (∀x.P (x)) → (∀y.P (y)) using natural deduction. Even the simplest Predicate Calculus facts must
be proved, such as change of bound variables. Example solution has four lines, but any proper proof will do.
Solution.
1 ∀xP (x)
2 a P (a) ∀-E, 1
3 ∀yP (y) ∀-I, 2
4 (∀xP (x))→ (∀yP (y)) →-I, 1–3
Exercise 3 ∀ and ∃
Give a natural deduction proof of the (derived) rule
(∃xP (x))→ Q
∀x(P (x)→ Q)
.
Example solution has six lines, but any proper proof will do.
Solution.
1 (∃xP (x))→ Q
2 a P (a)
3 ∃xP (x) ∃-I, 2
4 Q →-E, 1, 3
5 P (a)→ Q →-I, 2–4
6 ∀x(P (x)→ Q) ∀-I, 5
Exercise 4 Changing the Order of Quantifiers
Try and prove that
∀y.∃x.P (x, y)→ ∃x.∀y.P (x, y).
This example was mentioned in the lectures. Is the formula valid? If so, prove it. If not, then explain why you can’t.
Solution.
1 ∀y∃xP (x, y)
2 b ∃xP (x, b) ∀-E, 1
3 a P (a, b)
4 P (a, b) R, 3
5 P (a, b) – WRONG – ∃-E, 2, 3–4
6 ∀yP (a, y) ∀-I, 5
7 ∃x∀yP (x, y) ∃-I, 6
• The guard variable a at line (3) prevents us from moving between steps (4) and (5).
• Line (5) violates the (a is not free in q) clause in the rule.
∃xP (x)
[P (a)]
...
q (a arbitrary)
q (a is not free in q)2
1 ∀y∃xP (x, y)
2 ∃xP (x, b) ∀-E, 1
3 a P (a, b)
4 ∀yP (a, y) – WRONG – ∀-I, 3
5 ∃x∀yP (x, y) ∃-I, 4
6 ∃x∀yP (x, y) ∃-E, 2, 3–5
• Here step (4) is wrong, as at this point, the variable b is not arbitrary: it appears in the assumption P (a, b) at step
(3).
• According to the other examples of ∀-I, we should have a vertical bar labelled b, alongside lines preceding step (3),
which would be indented (this must include all the lines containing b). How could you do that in this case?
Exercise 5 Truth in First-Order Logic
Given the first-order situation S = (U, θ) with U = {0, 1, 2}, a unary relation symbol P (x) and a binary relation symbol
R(x, y) with
θ(P ) = {0, 1} and θ(R) = {(0, 0), (0, 1), (1, 2)}
determine which of the following first-order formulae are true in S, and justify your answer briefly (in a single sentence).
1. ∀x.P (x)→ ¬R(x, x)
2. ∀x.¬R(x, x) ∧ ∃y.P (y)
3. (∃x.∃y.R(x, y))→ ∀x.P (x)
4. ∃x.∃y.(R(x, y)→ ∀x.P (x))
Solution.
1. is false, as we have for example P (0) but also R(0, 0) so that the formula doesn’t hold for x = 0.
2. is false, as the formula states in particular that ¬R(x, x) for all x, and we have R(0, 0).
3. is false, as the antecedent ∃x.∃y.R(x, y) is true (e.g. R(1, 2)) but the consequence ∀x.P (x) is false, as P (2) does
not hold.
4. is true, because e.g. for x = y = 2 we have that R(x, y) is false so that R(x, y)→ ∀x.P (x) is true.
Exercise 6 Quantified Version of Modus Ponens
Give a natural deduction proof of the following derived rule:
(∀x.P (x)) ∧ (∀y.P (y)→ Q(y))
∀z.Q(z)
This is a quantified version of implication elimination. Again one only needs elimination (∀-E) and introduction (∀-I).
Example solution has seven lines, but any proper proof will do.
Solution.
1 (∀xP (x)) ∧ (∀y.P (y)→ Q(y))
2 ∀xP (x) ∧-E, 1
3 ∀y.P (y)→ Q(y) ∧-E, 2
4 a P (a) ∀-E, 2
5 P (a)→ Q(a) ∀-E, 3
6 Q(a) →-E, 5, 4
7 ∀zQ(z) ∀-I, 6
3
Exercise 7 Existential Quantifiers and Disjunction
Give a proof of the following derived rule
∃x.(P (x) ∨Q(x))
(∃x.P (x)) ∨ (∃x.Q(x))
that establishes that ∃ distributes over ∨. Working with the existential quantifier is trickier but there’s a strong similarity.
Example solution has ten lines, but any proper proof will do.
Solution.
1 ∃x(P (x) ∨Q(x))
2 a P (a) ∨Q(a)
3 P (a)
4 ∃xP (x) ∃-I, 3
5 (∃xP (x)) ∨ (∃xQ(x)) ∨-I, 4
6 Q(a)
7 ∃xQ(x) ∃-I, 6
8 (∃xP (x)) ∨ (∃xQ(x)) ∨-I, 7
9 (∃xP (x)) ∨ (∃xQ(x)) ∨-E, 2, 3–5, 6–8
10 (∃xP (x)) ∨ (∃xQ(x)) ∃-E, 1, 2–9
Exercise 8 Natural Number Induction
1. Prove, by natural number induction, that
P (n) : 2(n+ 2) ≤ (n+ 2)2
holds for all natural numbers n ∈ N.
Solution.
Base Case. For n = 0, we show that
P (0) : 2(0 + 2) ≤ (0 + 2)2
4 ≤ 4
This is obviously true.
Step Case. Let a be an arbitrary number. Then Assume P (a), that is
2(a+ 2) ≤ (a+ 2)2 (IH)
Prove P (a+ 1), that is
2((a+ 1) + 2) ≤ ((a+ 1) + 2)2
2((a+ 1) + 2)
= 2((a+ 2) + 1) (arith)
= 2(a+ 2) + 2 (arith)
≤ (a+ 2)2 + 2 (IH)
≤ a2 + 4a+ 6 (arith)
≤ a2 + 6a+ 9 (add the term 2n+ 3)
= ((a+ 1) + 2)2 (arith)
Since the base and step cases are established, we can conclude that
∀n(2(n+ 2) ≤ (n+ 2)2)
4
2. In the lecture, we have shown that the sum of the first n odd numbers is equal to n2. Here, we consider the sum of
the first n even numbers, i.e. the function
sumeven 0 = 0 -- SE1
sumeven n = 2 * n + sumeven (n-1) -- SE2
Prove, by natural number induction, that
sumeven n = n * (n+1)
for all natural numbers n ∈ N.
Solution.
Base Case. We show that sumeven 0 = 0 * (0+1).
sumeven 0 = 0 = 0 * (0+1) -- SE1 and arith
Step Case. Assume that sumeven n = n * (n+1) (IH).
We show that sumeven (n+1) = (n+1) * ((n+1)+1).
sumeven (n+1)
= 2 * (n+1) + sumeven (n+1-1) -- SE2
= 2 * n + 2 + sumeven n -- arith
= 2 * n + 2 + n * (n+1) -- IH
= 2 * (n+1) + n * (n+1) -- arith
= (2 + n) * (n+1) -- arith
= (n+1) * ((n+1)+1) -- arith
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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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