Homework 4 Solution

    • Background

In this assignment you will write a LISP program to solve the satis ability (SAT) problem. In particular, given a propositional sentence in conjunc-tive normal form (CNF), you will decide whether is satis able. The SAT problem can be easily formulated as a constraint satisfaction problem (CSP).

Conjunctive Normal Form (CNF): A propositional sentence is in CNF if and only if it is a conjunction of clauses, where a clause is a disjunc-tion of literals (a literal is a variable or its negation).

Example 1 The following sentence is a CNF with three clauses, which is de ned over binary variables X, Y, and Z.

=(X_:Y _Z)^:X^(:Y _:Z)

Clauses:

    (1) X_:Y _Z

    (2) :X

    (3) :Y _:Z Variables: X; Y; Z Literals: X; :Y; Z; :X; :Z

Sati ability (SAT): A CNF is satis able if and only if there exists a com-plete variable assignment that satis es the CNF (otherwise, it is unsatis - able). In this case, the corresponding variable assignment is called a model of the CNF. A CNF is satis ed if and only if each of its clauses is satis ed.

Example 2    from Example 1 is T rue given the variable assignment

! = fX = F alse, Y = F alse, Z = T rueg:

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Therefore, is satis able, and the complete variable assignment ! is a model of (note that there is another model of ).

In this homework, given a propositional sentence in CNF, you need to decide whether this sentence is satis able or not. We refer to this problem as the SAT problem.

SAT as a constraint satisfaction problem (CSP): Basically, variables of the CNF will correspond to the variables of the CSP, each having a domain with two values (i.e., True and False), and each clause of the CNF will represent a constraint of the CSP. Then a solution to the CSP will correspond to a model of the CNF, and vice versa.

    • Your task

In this assignment, your task is to treat the SAT as a CSP and solve it using backtracking search while detecting states that violate constraints. You are encouraged to use techniques discussed in class to improve the performance of backtracking search, such as variable ordering and forward checking in particular.


Representation of CNF in LISP: You are supposed to represent a CNF in LISP as follows:

A variable is an integer indexing from 1 to n, where n is the number of variables of the CNF. So, a positive literal can be represented by a positive integer. Respectively, a negative literal can be represented by a negative integer. (e.g., Positive literal of variable 2 is 2, and the negative literal of variable 2 is -2).

A clause is a list of integers. For example, the list(1 -2 3) represents the clause (1 _ :2 _ 3). Note that a unit clause is also represented as a list, e.g., the list ( 2) represents the unit clause -2.

A CNF is a list of lists.

Example 3    from Example 1 will be represented by the list

((1 -2 3) (-1) (-2 -3))

where variables X, Y and Z are indexed by 1, 2, and 3, respectively.



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Function signature: Given this representation, your top-level function must have the following signature:

(defun sat?    (n delta)...)

where n is an integer and delta is a CNF de ned over n variables. If delta is satis able, function sat? returns a list of n integers, representing a model of delta, otherwise it returns NIL.

When a CNF has more than one model, sat? can return any of the models, and the order of literals in the model is arbitrary.

Example 4

(sat?    3 ’((1 -2 3) (-1) (-2 -3))) returns (-1 -2 3)

Explanation:

For CNF (X _ :Y _ Z) ^ :X ^ (:Y _ :Z), one of its model is fX = F alse, Y = F alse, Z = T rueg, which can be represented by (-1 -2 3).


The order of literals in the model is arbitrary, so (-2 -1 3) is also a valid result to return.

fX = F alse, Y = F alse, Z = F alseg also satis es this CNF, so (-1 -2 -3) is a valid result, too.

Example 5

(sat?    1 ’((1) (-1))) returns NIL

Explanation: X ^ :X is not satis able.

    • Reading CNF  les to test your program

The following les coming with the assignment may help you test your pro-gram.

    (1) Folder cnfs/: To help you test your program with bigger CNFs, We provide you with some CNF les in DIMACS format (see below for de-tails about this format), where the CNFs become harder as the number of variables increases. See the folder cnfs/ coming with the assignment.

    (2) hw4-skeleton.lsp: We provide you with the LISP code that can parse the CNF les in DIMACS format in hw4-skeleton.lsp. In particular, after


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completing your sat? implementation, you can call solve-cnf to test your program on the CNF les. Usage:

(solve-cnf <path-to-cnf-file>)

e.g.,(solve-cnf "./cnfs/f1/sat f1.cnf")



DIMACS format: A common and easy way to represent CNFs is through DIMACS le format. Consider the following CNF which is over binary variables 1, 2, and 3:

(1_:2_3)^    1^(:2_:3)^3

This CNF can be represented using DIMACS format as follows

    1. c this is a comment line

    2. p cnf 3 4

    3. 1-230

    4. -1 0

    5. -2-30

    6. 3 0

In general, a CNF le may start with a number of comment lines, where each such line must begin with lowercase c. Next, we must have what is known as the "problem line", which begins with lowercase p, followed by cnf followed by the number of variables n, followed by the number of clauses m. This followed by clause lines. A clause line is de ned by listing clause literals one by one, where a negative literal is preceded by a - sign. The end of a clause is de ned by 0. Note that variables are indexed from 1 to n. There can also be comments in between clause lines. In this homework, you can use the provided functions in hw4-skeleton.lsp to read and parse DIMACS les.

    • Grading

Your submission will be evaluated by two measures: (i) correctness and (ii)

speed.


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    (i) 75% of the grade will be based on correctness.

    (ii) The last 25% will be based on whether you can correctly solve the SAT problems in les f1, f2, f3, f4, and f5 within 10 minutes each (5 points for each problem).

    • Submission & Rules

Submit your commented LISP program in a  le named hw4.lsp via

CCLE

Your programs will be evaluated under CLISP interpreter. In order to get any scores, you need to make sure the following LISP command does not produce any errors in CLISP interpreter.

            ▪ (load  "hw4.lsp")

The functions you are allowed to use are the same as those allowed in past assignments.

You are allowed to use as many helper functions as you want.

All input to your function will be legal (i.e., you do not need to validate inputs)

    • Honor Code

Remember that you cannot use any outside references for this or any assign-ment. However, you are allowed (and encourage) to experiment with actual SAT solver (which can be found online). Obtaining test problems and test-ing SAT solver from the Internet are acceptable. It is not acceptable to copy solution for any function you have to write. In general, any idea that is not originally yours must be attributed to the appropriate sources. If you have any questions, please contact the TA.











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