CS262 Logic and Verification: Coursework assignment
Resolution proof system in Prolog
Hand in via Tabula by 12 noon on Monday 3rd May, 2021
1 The rules
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This assignment is worth 15% of the module overall. The absolute number of points that can be achieved is 50. The assignment is stated in a quite concise way below, but you will realise that it comprises a relatively large chunk of work. It is your responsibility to divide this work up into reasonable steps, and to plan ahead what you will do when. To complete the assignment successfully, you will combine many of the concepts we learned in lecture and put them together in Prolog. Moreover, you will have to do some independent reading. Specifically, the assignment is Exercise 3.3.3 from Fitting’s book mentioned in the literature section of the module’s Moodle website, and the book provides a step-by-step guide to coming up with a solution, which you will see when reading the previous sections. The book is available online via the ‘Reading list’ link provided on the Moodle website. Some of the code for your program will be taken verbatim from Fitting’s book (which is allowed), some other code will have to be modified, and some code will have to be written newly.
You will have the opportunity to attend the lab sessions in weeks 8-10 of term 2, where tutors will be present and ready to answer your questions that may come up during solving the assignment. They will also take time to revisit and explain some of the fundamental concepts (concerning resolution and Prolog) that you will need to solve the assignment. There will also be additional Prolog problems provided that you may work on during the lab sessions to prepare you for this coursework and practise your Prolog skills.
This is an individual assignment and you should therefore complete it on your own.
2 The assignment
Write a resolution theorem prover in Prolog, and test it on the ten propositional formulas given below. [40+10 points] Use the tableau theorem prover specified in Fitting’s book as a starting point. Your program should be able to handle negation (¬), and all primary (∧,∨,→,←,↑,↓,̸→,̸←) and secondary
(≡, ̸≡) connectives in an input formula. 3 Step-by-step guide
As a first step, write a Prolog program that transforms a given propositional formula into conjunc- tive normal form. In this, follow the example program given in Section 2.9 of Fitting’s book, which shows a program to transform a propositional formula into disjunctive normal form.
In your program, you need to define the unary operator neg and the binary operators and, or, imp, revimp, uparrow, downarrow, notimp, and notrevimp, which denote the eight aforemen- tioned primary connectives (see Table 2.1 in the book). Also reuse the predicates conjunctive/1, disjunctive/1 that recognise α- and β-formulas, and the predicates components/3 and component/2 that split formulas into their components (see Table 2.2 in the book). The interface of this part of the program should be a predicate clauseform(X,Y), where Y is the conjunctive normal form of the formula X. For example, for the query ?- clauseform(a imp b, Y)., the result should be Y=[[neg a, b]], and for the query ?- clauseform(neg(a uparrow b), Y), the result should be Y=[[a],[b]].
Then augment your program so that it can also handle the secondary binary operators equiv and notequiv (see Table 2.1 in the book). Finally, test your program on larger formulas containing all of the operators mentioned before.
As a second step, implement the atomic resolution rule using predicates resolutionstep/2 and resolution/2 (analogous to singlestep/2 and expand/2, respectively), and a check for a closed resolution expansion (explained in Section 3.3 of the book), and put these together, following the implementation described in Section 3.2, and the remarks at the end of Section 3.3. The outermost interface of your program should be a predicate test/1, which takes a propositional formula as input, and prints ‘YES’ if it has a proposition proof, and ‘NO’ otherwise.
You may also want to think about ways of making your program more efficient, by eliminating duplicate variables and clauses early on in the process.
4 The test data
Use the resolution proof system to decide which of the following are theorems and which are not.
1. ((x imp y) and x) imp y
2. (neg x imp y) imp (neg (x notimp y) imp y)
3. ((x imp y) and (y imp z)) imp (neg neg z or neg x)
4. (x imp (y imp z)) equiv ((x imp y) imp z)
5. (x notequiv y) equiv (y notequiv x)
6. (x notequiv (y notequiv z)) equiv ((x notequiv y) notequiv z)
7. (neg x downarrow neg y) revimp neg (x uparrow y)
8. (neg x revimp neg y) and ((z notrevimp u) or (u uparrow neg v))
9. ((x or y) imp (neg y notrevimp z)) or (neg neg (z equiv x) notrevimp y)
10. (neg (z notrevimp y) revimp x) imp ((x or w) imp ((y imp z) or w))
5 Handing in your assignment
You should upload your Prolog program to Tabula in a single plain text file called resolution.pl. No other file formats other than plain text files are allowed. The beginning of the file should contain a comment section enclosed by /* … */, which shows the result of running your program on each of the ten propositional formulas shown above. Each result should appear on a new line, starting with the formula number, followed by YES if your program found a resolution proof, and NO if your program did not find a resolution proof. The names and arities of the predicates in your program code must be exactly as specified above in Section 3.
Here is how a submission file may look like:
… % the actual program code
Your code will be tested automatically using an extended version of the Python script check.py that can be found on the Moodle website. You should test whether your program runs through this testing file without problems, by executing python3 check.py in the same folder where your program resolution.pl is stored. This should produce output of the following form:
#correct answers=2/2
You can also use this Python script for your own testing, using the aforementioned test cases or others.
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