### Problem 1 (25 pts.)

In this problem we will use two kinds of electronic gates, NAND gates and INVOR gates. A NAND gate has inputs X and Y; its output Z is F(alse) if and only if both X and Y are T(rue). In this problem the only INVOR gate has inputs X and Y; its output Z is F(alse) if and only if X is T(rue) and Y is F(alse). We can represents a circuit of NAND and INVOR gates by the EDB predicate wireT(Z), meaning that wire Z is a circuit input set to T(rue), nandGate(X,Y,Z) meaning that there is a NAND gate in the circuit with input wires X and Y and with output wire Z, and invorGate(X,Y,Z) meaning that there is an INVOR gate in the circuit with input wires X and Y and with output wire Z. The rules describing a circuit are then the following.
``` t(Z) :- wireT(Z). t(Z) :- nandGate(X,Y,Z) & NOT t(X). t(Z) :- nandGate(X,Y,Z) & NOT t(Y). t(Z) :- invorGate(X,Y,Z) & NOT t(X). ```
``` t(Z) :- invorGate(X,Y,Z) & t(Y). ```
Here is a circuit of NAND and INVOR gates represented by the EDB: wireT(1), nandGate(6,1,3), nandGate(1,2,4), nandGate(3,5,6), nandGate(1,4,7), and invorGate(3,1,5).

For this EDB and the logic program above it, find the well-founded model by repeatedly making positive inferences and looking for unfounded sets.

### Problem 2 (15pts.)

For the same EDB and program as in Problem 1, find the well-founded model by the alternating fixedpoint method.

### Problem 3 (20pts.)

For the EDB and program of Problem 1, find all the stable models. Is there a stable semantics for this EDB and program?

### Problem 4 (15pts.)

Suppose we change the input of terminal 1 to be F(alse); i.e., the relation wireT is made empty instead of {1}. Show that this EDB plus the program from Problem 1 is modularly stratified. What are the "modules"?

### Problem 5 (10pts.)

Is the program and EDB of Problem 4 locally stratified? Justify your answer.

### Problem 6 (15pts.)

Compare the well-founded and stable approaches on the following rules of propositional calculus.
``` p :- NOT q. q :- NOT r. r :- NOT s. ```
```s :- NOT p. ```