Lecture 2 Exercises

Laura Kovács

Problem 2.1

Let $S$ be the following set of clauses:

$$S = \{\neg p \vee \neg q,\; \neg p \vee q,\; p \vee \neg q,\; p \vee q\}.$$

Consider the binary resolution inference system $\mathbb{BR}$. Show that there exists an infinite number of different $\mathbb{BR}$ derivations of the empty clause $\square$ from the clauses of $S$.

🎬 Related animation: Review resolution visually before working through the derivation.

Solution

One concrete derivation of $\square$ from $S$ is obtained by resolving the clauses in the following sequence (numbers refer to the clauses as they appear in $S$):

1. From $(\neg p \vee \neg q)$ and $(p \vee \neg q)$ derive $(\neg q \vee \neg q)$.
2. From $(\neg p \vee q)$ and $(p \vee q)$ derive $(q \vee q)$.
3. Factor $(q \vee q)$ to obtain $q$.
4. Resolve $(\neg q \vee \neg q)$ with $q$ to get $\neg q$.
5. Resolve the second copy of $(q \vee q)$ with $\neg q$ to obtain $\square$.

To build infinitely many different derivations, insert the following detour any number of times just before the last resolution step:

• Use $q$ (derived in step 3) together with $(\neg p \vee \neg q)$ to obtain $\neg p$.
• Resolve $\neg p$ with $(p \vee q)$ to derive $q$ again.

Repeating this detour an arbitrary number of times yields infinitely many distinct resolution derivations that still end in $\square$.

Problem 2.2

Consider a well-founded strict ordering $\succ$ on atoms. Prove that the induced ordering on literals, as defined in the lecture, is also well-founded.

Solution

Let $L$ be a literal and define $\operatorname{atom}(L)$ to be $p$ if $L$ is either $p$ or $\neg p$. By construction of the induced ordering, whenever $L_i \succ L_j$ we also have $\operatorname{atom}(L_i) \succ \operatorname{atom}(L_j)$.

Suppose, for contradiction, that the literal ordering is not well-founded. Then there is an infinite descending chain $L_0 \succ L_1 \succ L_2 \succ \cdots$. Applying the observation above yields an infinite descending chain on atoms:

$$\operatorname{atom}(L_0) \succ \operatorname{atom}(L_1) \succ \operatorname{atom}(L_2) \succ \cdots,$$

contradicting the assumption that the atom ordering is well-founded. Hence the induced literal ordering is well-founded.

Problem 2.3

Let $p, q$ be boolean atoms and let $S$ be the following set of ground formulas:

$$S = \{\neg p \vee \neg q,\; \neg p \vee q,\; p \vee \neg q,\; p \vee q\}.$$

Take any ordering such that $p \succ q$ and any selection function $\sigma$ over $S$ such that

$$\{\neg p \vee \underline{\neg q},\; \underline{\neg p} \vee q,\; p \vee \underline{\neg q},\; \underline{p} \vee q\}.$$

a.) Is $\sigma$ a well-behaved selection function over $S$? Justify your answer.

b.) How many inferences of $\mathbb{BR}_{\sigma}$ are applicable to $S$? Justify your answer.

Solution

a.) A selection function is well-behaved if, for every clause, either a negative literal is selected or all maximally ordered literals are selected. In the first three clauses of $S$, $\sigma$ selects a negative literal, so the condition holds. In the last clause, $\sigma$ selects $p$, which is the unique maximal literal because $p \succ q$. Therefore, $\sigma$ is well-behaved on $S$.

b.) No factoring inference applies, since no clause contains the same positive literal twice. A binary resolution step in $\mathbb{BR}_\sigma$ must resolve on selected literals of opposite polarity. The only such pair is $\underline{\neg p} \vee q$ and $\underline{p} \vee q$, yielding

$$\displaystyle\frac{\underline{\neg p} \vee q \;\;\; \underline{p} \vee q}{q \vee q}(BR).$$

Thus exactly one inference is applicable.

Problem 2.4

Give an example of a non-tautological ground clause with at least one selected literal such that the selection is not well-behaved for any ordering.

Solution

Take $p \lor \underline{p}$. The maximal literal must be $p$ under any ordering, but only one occurrence is selected, so the selection can never satisfy the “select all maximal literals” requirement.

Problem 2.5

Let

$$S = \{\neg q \lor r,\ \neg p \lor q,\ \neg r \lor \neg q,\ \neg q \lor \neg p,\ \neg p \lor \neg r,\ \neg r \lor p,\ r \lor q \lor p\}.$$

1. Prove $S$ is unsatisfiable using $\mathbb{BR}$.
2. Encode $S$ in TPTP and use Vampire (run with -av off) to prove unsatisfiability.

Solution (sketch)

1. Use binary resolution to derive unit clauses: resolve $\neg q \lor r$ with $\neg r \lor \neg q$ to obtain $\neg q$, then propagate to derive $p$ and $r$, eventually leading to a contradiction when resolving with clauses containing their negations. Any complete derivation is acceptable.
2. Translate each clause into TPTP CNF syntax, run vampire -av off input.tptp, and inspect the proof output to confirm the empty clause is produced.