Self-consistency and tree-of-thought: Foundations

This page builds every symbol, word, and picture the parent note leans on — starting from a reader who has never seen probability, sampling, or search. Read it top to bottom; each block earns the next.

0. The characters in the story

Before any formula, meet the two nouns the whole topic revolves around.

Figure — Self-consistency and tree-of-thought

We now name these with letters, because the parent page does.

1. The symbols $a$ , $r$ , and the prompt

We call $r$ a nuisance variable: something we must acknowledge exists but want to get rid of. The tool that removes it is the next block.

2. Probability $p (\cdot)$ and the summation sign $\sum$

The parent writes $p (a ∣ prompt)$ and a big $\sum_{r}$ . Two new symbols — let's earn both.

The multiplication rule $p (a, r) = p (a ∣ r) p (r)$ (read: "chance of the path, times chance the path gives that answer") is what turns the sum into the parent's form $\sum_{r} p (a ∣ r) p (r ∣ prompt)$ .

3. Temperature and sampling — where the many paths come from

4. The indicator $1 [\cdot]$ and counting votes

The parent uses $1 [a^{(i)} = a]$ . Two pieces: the superscript and the box.

5. Why voting beats one guess — the counting symbols $(k N)$

The parent's "Condorcet" formula uses $(k N)$ and $p^{k} (1 - p)^{N - k}$ .

6. Symbols for Tree-of-Thought: state, generator, evaluator

Prerequisite map

Each foundation on the left feeds exactly the part of the parent it unlocks. Trace any arrow back and you can see why that symbol had to be built first.

Equipment checklist

Cover the right side and answer each aloud — if any stalls, reread that section.

What does the vertical bar in

p (a ∣ prompt)

mean?

"given" — the probability of

a

given the prompt already fixed.

What does

\sum_{r}

do to the reasoning path

r

Adds over every path, erasing the nuisance

r

and leaving the answer marginal

p (a)

(marginalization).

Why must temperature be

> 0

for self-consistency?

At temperature 0 all chains are identical, so there is nothing to vote across.

What does

1 [a^{(i)} = a]

evaluate to?

1 if chain

i

's answer equals

a

, else 0 — a vote-counting switch.

What does

ar g max_{a}

return?

The answer

a

that gets the most votes (the winner), not the vote count itself.

What does

(k N)

count?

The number of ways to choose which

k

of the

N

chains are correct.

What is a "state"

s

in Tree-of-Thought?

The prompt plus the partial reasoning built so far.

What do

G (s)

and

V (s)

do?

G

generates candidate next thoughts;

V

scores how promising a state is.

What do

k

b

d

stand for in ToT cost?

Branching factor, beam width kept per level, and tree depth.

Which single capability does ToT have that CoT and SC do not?

Backtracking — abandoning a dead-end branch and trying another.