Greedy, Top-k, and the Tree

What Decoding Is and Is Not

The finale closes the decoding boundary: exact deterministic selection, named softmax probabilities, and no claim beyond mechanics.

highlighted = computed this step

What is exact

Greedy and top-k selection are exact deterministic operations on integer logits. The final greedy output remains a, b. The exact register is the vocabulary order, the displayed logits, and the selection rule.

exact selection=a,b\text{exact selection}=a,b
Decode boundaryExact selection plus named softmax boundary.Decode boundaryExact selection plus named softmax boundary.exact selections with named softmaxdisplayed integer logits are the source; ties break by lowest vocab indexstep 1 logits: a=3, b=1, c=2tokenlogitranka31b13c22ranked order: a > c > btop-2: a, cgreedy argmax: asoftmax probabilities are named, not pinned decimalssoftmax: a:e^3/(e^3+e^1+e^2); b:e^1/(e^3+e^1+e^2); c:e^2/(e^3+e^1+e^2)exact greedy path with top-k branchestree choices are exact argmax/top-k selections from displayed logitsstartstep 1choose aa=3, b=1, c=2top1: atop2: cstep 2choose ba=0, b=4, c=1top1: btop2: cgreedy decode: a,b

What is named

Softmax probabilities are named because they use exponentials. The diagram shows symbols rather than decimal weights. The selection can be exact while the probability mass remains outside the exact register.

softmax⁡=named boundary\operatorname{softmax}=\text{named boundary}
Decode boundaryExact selection plus named softmax boundary.Decode boundaryExact selection plus named softmax boundary.exact selections with named softmaxdisplayed integer logits are the source; ties break by lowest vocab indexstep 1 logits: a=3, b=1, c=2tokenlogitranka31b13c22ranked order: a > c > btop-2: a, cgreedy argmax: asoftmax probabilities are named, not pinned decimalssoftmax: a:e^3/(e^3+e^1+e^2); b:e^1/(e^3+e^1+e^2); c:e^2/(e^3+e^1+e^2)exact greedy path with top-k branchestree choices are exact argmax/top-k selections from displayed logitsstartstep 1choose aa=3, b=1, c=2top1: atop2: cstep 2choose ba=0, b=4, c=1top1: btop2: cgreedy decode: a,b

Summary

Greedy and top-k selection are exact and deterministic. Softmax probabilities are the named boundary. This pins the decoding mechanics; it is not learning, not meaning, and not a claim that the output is correct or intelligent. The book pins deterministic decoding mechanics: compare logits, choose argmax or top-k, and name the softmax probabilities rather than computing them.

decoding mechanics; named probabilities\text{decoding mechanics; named probabilities}
Decode boundaryExact selection plus named softmax boundary.Decode boundaryExact selection plus named softmax boundary.exact selections with named softmaxdisplayed integer logits are the source; ties break by lowest vocab indexstep 1 logits: a=3, b=1, c=2tokenlogitranka31b13c22ranked order: a > c > btop-2: a, cgreedy argmax: asoftmax probabilities are named, not pinned decimalssoftmax: a:e^3/(e^3+e^1+e^2); b:e^1/(e^3+e^1+e^2); c:e^2/(e^3+e^1+e^2)exact greedy path with top-k branchestree choices are exact argmax/top-k selections from displayed logitsstartstep 1choose aa=3, b=1, c=2top1: atop2: cstep 2choose ba=0, b=4, c=1top1: btop2: cgreedy decode: a,b