CS 157 Midterm Examination Fall

Transcription

1 CS 57 Mdterm Examato Fall Please read all structos (cludg these) carefully. The exam s closed book closed otes ad closed teret. The exam cossts of 8 pages cludg ths page. The last page s a referece sheets for the Ftch rules of ferece. There are 5 questos. Each questo s worth 0 pots. Tme lmt: oe hour. Budget your tme accordgly. Please wrte your solutos the spaces provded o the exam. Make sure that your solutos are eat ad clearly marked. You may use the blak areas ad backs of the exam pages for scratch work. I accordace wth both the letter ad sprt of the Hoor Code I have ether gve or receved assstace o ths examato. NAME: SUNETID (userame): SIGNATURE: Questo (0 pots) Questo 2 (0 pots) Questo 3 (0 pots) Questo 4 (0 pots) Questo 5 (0 pots) Total score (50 pots)

2 2 Questo [0 pots] (a) Suppose that you are gve the followg premses: ( p ( p q)) ((p q) r ) q p Please crcle oe of the provded aswers for each questo ad provde a bref justfcato (o more tha setece). ) The setece p q s cosstet wth the suppled premses. (3 pots) 2) The setece p q s logcally etaled by the suppled premses. (3 pots) (b) Is the followg setece vald cotget or usatsfable? (4 pots) ((p p) ( q q)) ( p q ) VALID CONTINGENT UNSATISFIABLE

3 3 Questo 2 [0 pots] Please crcle true or false for each of the followg statemets. If the statemet s true provde a bref justfcato (o more tha setece). If the statemet s false provde a short couterexample. (a) I Propostoal Logc for ay set of seteces Δ ad a setece φ such that Δ φ there s a proof of φ wth the Ftch system wth Δ as the set of premses. (5 pots) (b) Propostoal Resoluto s geeratvely complete; that s t s possble to fd resoluto dervatos for all clauses that are logcally etaled by a set of premse clauses. (5 pots)

4 4 Questo 3 [0 pots] Assume that Γ ad are sets of seteces Propostoal Logc ad φ ad ψ are dvdual seteces Propostoal Logc. Please crcle true or false for each of the followg statemets. No explaato s requred. (a) If Γ φ ad φ the Γ Δ φ. (2 pots) (b) If Γ φ ad φ the Γ Δ φ. (2 pots) (c) If Γ φ ad φ the Γ Δ φ. (2 pots) (d) If Γ Δ φ the Γ φ ad φ. (2 pots) (e) If φ ad Γ φ the Δ Γ φ. (2 pots)

5 5 Questo 4 [0 pots] Use the Ftch System to prove q from the premses p ad p. (Note: We have a proof that takes 0 steps cludg the premses.) Please aotate your proof by wrtg the rule ad le umbers for each step your proof (abbrevatos are fe). Clearly mark ay assumptos ad subproofs wth your proof the same format as the exercses ad otes.

6 6 Questo 5 [0 pots] (a) Covert the followg two logcal seteces to clausal form. (4 pots) () ( p r) ( r p) (2) p q

7 7 (b) Prove ( r p) ( q r ) from (a) as premses. (6 pots) wth Propostoal Resoluto usg the seteces

8 Ad Itroducto ^ ^ Ad Itroducto...^... ^ 8 Ad Elmato Referece: Ftch Rules of Iferece Ad Elmato ^ ^ ^ ^ ^ ^ Ad Itroducto ^ ^ Or... Itroducto Ad Elmato ^ ^ _ ^ _ ^ Or Elmato Ad Elmato Or Itroducto ^ _ ^ _... Negato Elmato Or Itroducto Or Elmato Negato _... _ Itroducto... Negato Elmato Implcato Itroducto Or Elmato Negato Itroducto Implcato Elmato Implcato Itroducto Negato Itroducto Implcato Elmato Bcodtoal Itroducto Ad Elmato ^ ^ Or Itroducto Or_Itroducto _ Negato Elmato Or Elmato _ _ Or... Elmato Implcato Itroducto _... _ Elmato Negato Negato Elmato Implcato Elmato Negato Itroducto Implcato Itroducto Negato Itroducto Implcato Itroducto BcodtoalElmato Itroducto Implcato 6 6 Implcato Elmato 6 Bcodtoal Elmato Bcodtoal Itroducto 6 Bcodtoal I addto toitroducto these rules of ferece you may make Assumptos Bcodtoal Itroducto proofs ad use Reterato to reproduce a earler cocluso your p Bcodtoal Elmato I addto to these rules of ferece 6you may make Assumptos Bcodtoal Elmato wth subproofs ad use Reterato to reproduce a earler co cluso your proof. 7 I addto to these rules of ferece you may make Assumptos wth subbcodtoal Elmato proofs ad use Reterato to reproduce a earler cocluso your Bcodtoal Elmato I addto to these rulesproof. of ferece you may make Assumpto ad use Reterato to reproduce a earler cocluso yo proofs