Unit Propagation
DSCI 220 · Lecture 7 activity
What to do
Every group has a different argument. There is no one right answer across the room — that is the point.
You are asked whether a conclusion follows from some premises. Rather than chase it forwards, we assume it does not: the negated conclusion joins the premises, and we look for a contradiction. If the premises plus the denial of the conclusion cannot all hold, then the conclusion followed all along.
Everything has already been turned into clauses for you.
The loop. Repeat until you cannot:
- Find a clause with exactly one literal. That literal must be true. Circle it — call it a unit.
- Look for that same literal in any other clause. That clause is already satisfied, so circle the literal there too, and cross the whole clause out. The circle records why it went.
- In every clause containing the negation of your unit, strike out just that one literal. The rest of the clause stays.
- If a clause loses its last literal, circle what is left of it. That is the empty clause — a clause with no way left to be true, so it is False. You have found a contradiction, and that is the best thing that can happen to you.
A circle always marks something you have established: a literal that must be true, or — best of all — an empty clause. A crossed-out clause always has a circle in it saying which literal did the work.
Three ways to stop, and each says something different about the argument:
| You stopped because | So the conclusion |
|---|---|
| you found a contradiction | follows. Denying it was impossible |
| no clauses were left | does not follow — and the units you circled are a counterexample |
| no clause had a single literal | cannot be settled this way. Propagation alone is not enough |
Set 1
The argument
| 1 | \(D\) |
| 2 | \(D \rightarrow \neg C\) |
| 3 | \(\neg C \wedge \neg B \rightarrow \neg E\) |
| 4 | \(\neg B\) |
| 5 | \(\neg G \vee \neg F\) |
| \(\therefore\) | \(\neg E\) |
Does that conclusion follow? To find out, we assume it does not — so \(E\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{D\}\) | |
| 2 | \(\{\neg C,\ \neg D\}\) | |
| 3 | \(\{B,\ C,\ \neg E\}\) | |
| 4 | \(\{\neg B\}\) | |
| 5 | \(\{\neg F,\ \neg G\}\) | |
| 6 | \(\{E\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(\neg E\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 2
The argument
| 1 | \(\neg B\) |
| 2 | \(\neg B \rightarrow G\) |
| 3 | \(G \wedge \neg C \rightarrow \neg F\) |
| 4 | \(\neg C\) |
| 5 | \(D \vee \neg A\) |
| \(\therefore\) | \(\neg F\) |
Does that conclusion follow? To find out, we assume it does not — so \(F\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{\neg B\}\) | |
| 2 | \(\{B,\ G\}\) | |
| 3 | \(\{C,\ \neg F,\ \neg G\}\) | |
| 4 | \(\{\neg C\}\) | |
| 5 | \(\{\neg A,\ D\}\) | |
| 6 | \(\{F\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(\neg F\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 3
The argument
| 1 | \(B\) |
| 2 | \(B \rightarrow E\) |
| 3 | \(E \wedge \neg A \rightarrow \neg F\) |
| 4 | \(\neg A\) |
| 5 | \(D \vee \neg C\) |
| \(\therefore\) | \(\neg F\) |
Does that conclusion follow? To find out, we assume it does not — so \(F\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{B\}\) | |
| 2 | \(\{\neg B,\ E\}\) | |
| 3 | \(\{A,\ \neg E,\ \neg F\}\) | |
| 4 | \(\{\neg A\}\) | |
| 5 | \(\{\neg C,\ D\}\) | |
| 6 | \(\{F\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(\neg F\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 4
The argument
| 1 | \(C\) |
| 2 | \(C \rightarrow \neg D\) |
| 3 | \(\neg D \wedge \neg B \rightarrow A\) |
| 4 | \(\neg B\) |
| 5 | \(G \vee F\) |
| \(\therefore\) | \(A\) |
Does that conclusion follow? To find out, we assume it does not — so \(\neg A\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{C\}\) | |
| 2 | \(\{\neg C,\ \neg D\}\) | |
| 3 | \(\{A,\ B,\ D\}\) | |
| 4 | \(\{\neg B\}\) | |
| 5 | \(\{F,\ G\}\) | |
| 6 | \(\{\neg A\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(A\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 5
The argument
| 1 | \(\neg D\) |
| 2 | \(\neg D \rightarrow G\) |
| 3 | \(G \wedge E \rightarrow C\) |
| 4 | \(E\) |
| 5 | \(F \vee \neg A\) |
| \(\therefore\) | \(C\) |
Does that conclusion follow? To find out, we assume it does not — so \(\neg C\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{\neg D\}\) | |
| 2 | \(\{D,\ G\}\) | |
| 3 | \(\{C,\ \neg E,\ \neg G\}\) | |
| 4 | \(\{E\}\) | |
| 5 | \(\{\neg A,\ F\}\) | |
| 6 | \(\{\neg C\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(C\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 6
The argument
| 1 | \(\neg G\) |
| 2 | \(\neg G \rightarrow A\) |
| 3 | \(A \rightarrow \neg B\) |
| 4 | \(D \rightarrow C\) |
| \(\therefore\) | \(D\) |
Does that conclusion follow? To find out, we assume it does not — so \(\neg D\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{\neg G\}\) | |
| 2 | \(\{A,\ G\}\) | |
| 3 | \(\{\neg A,\ \neg B\}\) | |
| 4 | \(\{C,\ \neg D\}\) | |
| 5 | \(\{\neg D\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(D\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 7
The argument
| 1 | \(G\) |
| 2 | \(G \rightarrow \neg D\) |
| 3 | \(\neg D \rightarrow A\) |
| 4 | \(\neg C \rightarrow F\) |
| \(\therefore\) | \(\neg C\) |
Does that conclusion follow? To find out, we assume it does not — so \(C\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{G\}\) | |
| 2 | \(\{\neg D,\ \neg G\}\) | |
| 3 | \(\{A,\ D\}\) | |
| 4 | \(\{C,\ F\}\) | |
| 5 | \(\{C\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(\neg C\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 8
The argument
| 1 | \(F\) |
| 2 | \(F \rightarrow \neg A\) |
| 3 | \(\neg A \rightarrow B\) |
| 4 | \(E \rightarrow \neg G\) |
| \(\therefore\) | \(E\) |
Does that conclusion follow? To find out, we assume it does not — so \(\neg E\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{F\}\) | |
| 2 | \(\{\neg A,\ \neg F\}\) | |
| 3 | \(\{A,\ B\}\) | |
| 4 | \(\{\neg E,\ \neg G\}\) | |
| 5 | \(\{\neg E\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(E\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 9
The argument
| 1 | \(B\) |
| 2 | \(B \rightarrow D\) |
| 3 | \(D \rightarrow A\) |
| 4 | \(F \rightarrow C\) |
| \(\therefore\) | \(F\) |
Does that conclusion follow? To find out, we assume it does not — so \(\neg F\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{B\}\) | |
| 2 | \(\{\neg B,\ D\}\) | |
| 3 | \(\{A,\ \neg D\}\) | |
| 4 | \(\{C,\ \neg F\}\) | |
| 5 | \(\{\neg F\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(F\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 10
The argument
| 1 | \(\neg A\) |
| 2 | \(\neg A \rightarrow C\) |
| 3 | \(\neg B \vee \neg E\) |
| 4 | \(F \vee \neg G\) |
| 5 | \(\neg B \wedge F \rightarrow \neg D\) |
| \(\therefore\) | \(\neg D\) |
Does that conclusion follow? To find out, we assume it does not — so \(D\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{\neg A\}\) | |
| 2 | \(\{A,\ C\}\) | |
| 3 | \(\{\neg B,\ \neg E\}\) | |
| 4 | \(\{F,\ \neg G\}\) | |
| 5 | \(\{B,\ \neg D,\ \neg F\}\) | |
| 6 | \(\{D\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(\neg D\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 11
The argument
| 1 | \(\neg E\) |
| 2 | \(\neg E \rightarrow F\) |
| 3 | \(B \vee \neg C\) |
| 4 | \(\neg D \vee G\) |
| 5 | \(B \wedge \neg D \rightarrow \neg A\) |
| \(\therefore\) | \(\neg A\) |
Does that conclusion follow? To find out, we assume it does not — so \(A\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{\neg E\}\) | |
| 2 | \(\{E,\ F\}\) | |
| 3 | \(\{B,\ \neg C\}\) | |
| 4 | \(\{\neg D,\ G\}\) | |
| 5 | \(\{\neg A,\ \neg B,\ D\}\) | |
| 6 | \(\{A\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(\neg A\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 12
The argument
| 1 | \(\neg G\) |
| 2 | \(\neg G \rightarrow E\) |
| 3 | \(\neg B \vee \neg D\) |
| 4 | \(C \vee \neg A\) |
| 5 | \(\neg B \wedge C \rightarrow \neg F\) |
| \(\therefore\) | \(\neg F\) |
Does that conclusion follow? To find out, we assume it does not — so \(F\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{\neg G\}\) | |
| 2 | \(\{E,\ G\}\) | |
| 3 | \(\{\neg B,\ \neg D\}\) | |
| 4 | \(\{\neg A,\ C\}\) | |
| 5 | \(\{B,\ \neg C,\ \neg F\}\) | |
| 6 | \(\{F\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(\neg F\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________
Set 13
The argument
| 1 | \(C\) |
| 2 | \(C \rightarrow A\) |
| 3 | \(\neg G \vee F\) |
| 4 | \(B \vee D\) |
| 5 | \(\neg G \wedge B \rightarrow \neg E\) |
| \(\therefore\) | \(\neg E\) |
Does that conclusion follow? To find out, we assume it does not — so \(E\) joins the premises — and look for a contradiction.
| Clause | Working | |
|---|---|---|
| 1 | \(\{C\}\) | |
| 2 | \(\{A,\ \neg C\}\) | |
| 3 | \(\{F,\ \neg G\}\) | |
| 4 | \(\{B,\ D\}\) | |
| 5 | \(\{\neg B,\ \neg E,\ G\}\) | |
| 6 | \(\{E\}\) (negated conclusion) |
Units you forced, in order:
______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______ ⇒ ______
How did you stop? ☐ found a contradiction ☐ ran out of clauses ☐ stalled
So does \(\neg E\) follow from the premises?
__________________________________________________
How do you know you are finished?
__________________________________________________