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Logic in Computer Science

创建日期：2024-03-29 15:43:25

所属分类：计算机computer science

标签： COM2107

Logic in Computer Science

1. a) What is an assignment? [10%]

b) Prove the following sequents using intuitionistic natural deduction.
(i) p ∨ (q ∧ r) ` (p ∨ q) ∧ (p ∨ r). [20%]
(ii) ` ¬¬(p ∨ ¬p). [20%]
c) Prove that the set of formulas {¬p ∧ ¬q, p ∨ q} is inconsistent. [20%]
d) (i) Define a function that counts number of propositional connectives in a propo-
sitional formula (i.e., the number of internal nodes of its abstract syntax tree).
[10%]
(ii) Prove that if a propositional formula does not contain any negation symbol,
then it has more propositional variables (leaves of the abstract syntax tree)
than propositional connectives (internal nodes). [20%]
2. a) Prove the following formulas by natural deduction.
(i) ∃x. (P (x) ∨Q(x)) ` ∃x. P (x) ∨ ∃x. Q(x). [20%]
(ii) ¬∀x. P (x) ` ∃x. ¬P (x). [20%]
b) Which of the following expressions are formulas of predicate logic? Explain why.
(i) ∀X∀Y. (X ⊆ Y ↔ (∀x. x ∈ X → x ∈ Y )). [10%]
(ii) ∀P. P (0) ∧ ∀n. (P (n)→ P (n+ 1))→ (∀n. P (n)). [10%]
(iii) ∀t. G(P (t)U P (t)), where P is a predicate symbol. [10%]
c) Consider the signature ΣA = {+, ·, 0, 1} of arithmetic, extended with constant symbols
for each natural number, and the ΣA-structure N = (N,+N, ·N, 0N, 1N). Interpret the
ΣA-term 3 · x+ 9 in N with assignment v : x 7→ 11N step-by-step. [20%]
d) Show that the formula ∀x∀y. (f(x, y) = f(y, x)) is satisfiable, but not valid. [10%]
COM2107 3 TURN OVER
COM2107
3. a) Consider the clause sets
S0 = {{p,¬q}, {q, r}, {¬p,¬q, r}, {¬r}},
S1 = {{p, q}, {¬p,¬q}, {q, r}, {¬q,¬r}, {r, p}, {¬r,¬p}}.
(i) Use the DPLL algorithm to show that both are inconsistent. [20%]
(ii) Use resolution to show that both are inconsistent. [20%]
b) Resolution is refutationally complete for propositional logic. What does this mean? [10%]
c) Skolemisation of predicate logic formulas does not preserve validity. Why? [10%]
d) Compute the most general unifiers for the following sets, if they exist.
(i) {x ≈ f(a), g(x, x) ≈ g(x, y)}. [10%]
(ii) {f(g(a, x), g(b, x)) ≈ f(g(b, x), z)}. [10%]
e) Use resolution to prove that ∃x. (P (x)→ ∀y. P (y)) is valid. [20%]
4. a) Explain the model checking problem. [10%]
b) Prove or refute the following formulas.
(i) Gϕ ≡ ϕ ∧ XGϕ. [20%]
(ii) F (ϕ ∧ ψ) ≡ Fϕ ∧ Fψ. [20%]
(iii) ϕUψ ≡ ψ ∨ (ϕ ∧X(ϕUψ)). [20%]
c) Translate the following statements into linear temporal logic.
(i) It is always the case that the traffic light will eventually become green when
someone pushes the button. [10%]
(ii) The light blinks finitely many times. [10%]
(iii) Lions and zebras never drink together at the water hole. [10%]
END OF QUESTIONS
COM2107 4
COM2107
Data Provided: None
DEPARTMENT OF COMPUTER SCIENCE Spring Semester 2018-19
Logic in Computer Science 2 hours
Answer FOUR questions.
All questions carry equal weight. Figures in square brackets indicate the per-
centage of available marks allocated to each part of a question.
COM2107 1 TURN OVER
COM2107
This page is blank.
COM2107 2 CONTINUED
COM2107
1. a) Professor Wright says that students will fail the exam if they don’t get implication.
Dana fails, but gets implication. Is Professor Wright wrong? [10%]
b) Define negation in terms of other propositional connectives. [10%]
c) The equation Jϕ⊕ψKv = (JϕKv + JψKv)− 2 · JϕKv · JψKv defines the semantics of the
exclusive or operation ⊕. Write down natural deduction rules for it. [15%]
d) Prove the following sequents using natural deduction.
(i) ¬(p ∧ q) ` ¬p ∨ ¬q. [20%]
(ii) ` p ∨ (p ∧ q)→ p. [20%]
e) Why is the satisfiability problem of propositional logic decidable? [10%]
f) (i) For which assignments v is J(p ∧ ¬q)→ rKv = 0? [5%]
(ii) Compute J(p ∧ ¬q)→ rKv step by step for these. [10%]
COM2107 3 TURN OVER
COM2107
2. a) Translate the following sentences into predicate logic.
(i) Every natural number has precisely one successor. [10%]
(ii) No pair of adjacent vertices (in some graph) has the same colour. [10%]
b) Prove ¬∀x. P (x) ` ∃x. ¬P (x) by natural deduction. [20%]
c) The signature of monoids consists of a binary function symbol · and a constant symbol
1. The following associativity and identity axioms hold in every monoid.
∀x, y, z. (x · y) · z = x · (y · z), ∀x. 1 · x = x, ∀x. x · 1 = x.
Prove with natural deduction that the identity 1 is uniquely defined. [20%]
d) Write down the definitions for the assignment function v for predicate logic and the
interpretation function J−KAv of Σ-terms in a Σ-structure A. [10%]
e) Consider the signature ΣA = {+, ·, 0, 1} of arithmetic, extended with constant symbols
for each natural number, and the ΣA-structure N = (N,+N, ·N, 0N, 1N). Interpret the
ΣA-term 3 · x+ 9 in N with assignment v : x 7→ 11N for variable x. [20%]
f) What does A |=v ϕ mean? [10%]
COM2107 4 CONTINUED
COM2107
3. a) The Shannon expansion of a propositional formula ϕ is the formula
(¬p ∧ ϕ[⊥/p]) ∨ (p ∧ ϕ[>/p]).
Use the property ϕ↔ (¬p∧ϕ[⊥/p])∨(p∧ϕ[>/p]) to compute the Shannon expansions
of the following propositional formulas, expanding first with respect to p and then with
respect to q. Simplify the results as much as possible by propositional reasoning.
(i) p⊕ q. [20%]
(ii) p→ (q → p). [20%]
b) Construct a binary decision tree for p ∧ (p ∨ ¬r). Label its root with p, its children
with q and its grandchildren with r. [10%]
c) Show that Skolemisation doesn’t preserve validity. [10%]
d) Prove that {{¬p, r}, {¬q, r}, {p, q}, {p,¬r}, {q,¬r}, {¬r}} is inconsistent
(i) using the DPLL method; [20%]
(ii) using resolution. [20%]
COM2107 5 TURN OVER
COM2107
4. a) (i) Draw a graph for the LTS (S,→, λ) with [15%]
S = {s0, s1, s2, s3},
→ = {(s0, s0), (s0, s1), (s2, s1), (s1, s2), (s2, s2)},
λ : s0 7→ {p, q}, s1 7→ {r}, s2 7→ {p, r}, s3 7→ {t}.
(ii) Compare with the definition from the lectures. What is wrong? [5%]
b) Define the LTL operators F and G in terms of U . Justify your choices. [20%]
c) Prove or refute F (ϕ ∧ ψ) ≡ Fϕ ∧ Fψ. [10%]
d) Check the following properties of LTL formulas on the LTS S defined by
s0
r
s1
p, q
s2
p, t, r
s3
q, r
(i) S, s0 |= ¬p→ r, [10%]
(ii) S, s0 |= Fq, [10%]
(iii) S, s3 |= Ft, [10%]
(iv) S, s0 |= G(r ∨ q). [10%]
(v) S, s0 |= (p ∨ r)U(q ∨ t). [10%]
END OF QUESTIONS

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关键词一关键词二关键词三关键词四关键词五和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 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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 PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 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 IE5600 ECOS 3997 EDST2003 FIN 534 Elec4430 MATS3001 MGT6047 FINS5514 MIS602 ATS1279 ECMM445 2F CS262 EFIMM0097 FINM2002 ECMT3150 FINC5090 WACC1000 MECH4620 ECOS 2002 SEESGS79 STAT COMP3007 BUSB200F ACST8040 CAES9920 STAT0020 INFR11205 PHYS2111 FINM7402 STAT 428 SOFT3202 ME220 ACC-ACF2100 EC9410 CSE 120 COMP90042 SEE5211 ECON0048 BUSN2036 MKT2010 3SMFE4 EFIMM0117 ECON3034 MTH5120 EFIMM0120 X0 W19186 MATH 034 COMP2100 COMS4104 CS1210 MENG10005 ACCT 5906 ECMT1020 MGTS7607 MATH3090 CIV4100 CSYS5020 ELE2019 PHS4200 ACS6124 MSIN0225 PHAS0061 6CCS3BIM FINM7403 COMP8410 PY406 COMP2620 CS226 SDEV2004 COMP6080 BFF1001 HR410 MCEN30017 AMS3328 ML3 ELE8059 FINC3017 ACAD001 FTEC2101 ES480 3D ELECTENG 734 MTH731U MATH-UA 251 CSCI-UA 9473 A2 MSDM5058 FIT5047 COM6012 1FIN B386F MPHY202P ACCT6003 ECE 6913 HMB312H1 AC1025 MTH6105 Math6168 FINM 3003 CS24320 PHYS2134 ACCT3011 MECH1400 COMP30024 ENGGEN 121 COMP3721 MCD2080 ECON 7002 MAS439 COMP6216 ECON2011 M23367 CSC597 COMP6247 BRIEF QBUS6180 EENG22000 FN1024 MATH0094 MANG 6554 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