Gemara

74b (Rav Papa): If one dices beets, he is liable for Tochen (grinding).

114b (Beraisa) Question: What is the source that when Yom Kipur is on Shabbos, Kenivas Yerek is forbidden?

Answer - Suggestion: "Shabason" forbids Melachah.

Rejection: It explicitly says "Lo Sa'aseh Chol Melachah"!

Rather, it forbids Kenivas Yerek [that was already detached].

155a (Mishnah): One may not [finely] cut fodder or carobs for an animal.

(Rav Chisda): Rav Huna permits exertion for food, but not Shivuy Ochel (to make something inedible into a food).

(Rava): Rav Yehudah permits Shivuy Ochel, but forbids exertion.

Rishonim

Rambam (Hilchos Shabbos 7:5): A Toladah is a Melachah that resemble one of the Avos. E.g. one who cuts vegetables little by little in order to cook it, he is liable for a Toladah of Tochen.

Rambam (21:18): One who cuts vegetables very fine in order to cook them, this is a Toladah of Tochen, and he is liable.

Hagahos Maimoniyos (70): Tosfos (74b DH Hai) says that Tochen applies only to beets. It is permitted regarding other foods. R. Yosef brings from the Rashbam that one may crumb bread, since it was already ground. The Yere'im forbids crumbing Peros into very fine pieces.

Rosh (7:5): R. Chananel explains that Rav Papa discusses cutting date wood like strands. When one separates the strands, a fine floury substance exudes. Therefore, one who mashes it is liable for Tochen. This is a proper Perush. It is like chopping wood [which was taught just before this]. Rashi explains that he dices beets. This is astounding. How can Tochen apply to cutting a food very finely?!

Tosfos (114b DH Ela): Kenivas Yerek is permitted on Yom Kipur on a weekday. This is unlike dicing beets, for which one is liable. There, he makes very fine pieces. Here, he makes big pieces.

Ran (32a DH Amar Rav): In a Teshuvah, the Rashba forbids cutting very finely to eat the next day, or after a while. If not, it is permitted, just like Abaye permits Breirah in order to eat immediately.

Shiltei ha'Giborim (32a:3): The Rif, Rambam and Tosfos agree with Rashi. The Tur brings only Rashi's opinion. All forbid cutting vegetables very finely on Shabbos. I do not understand the Ran. Since it is forbidden due to Tochen, there is no difference between immediately, later or tomorrow! We do not permit all Melachos needed for immediate use! Do not bring a proof from Breirah. The Heter for Breirah for immediate use is not a proper Breirah. (We permit only taking the good from the bad, without a special Kli - PF.) Also, there Chachamim distinguished. Here they forbade cutting very finely, without distinction! Some forbid all Peros. Some texts of the Rambam say so.

Terumas ha'Deshen (56): Tochen applies to food only if it is normally ground, e.g. wheat, barley and similar things, but not other foods. The Rosh says that Tochen does not apply to food. The Rambam says that Tochen does not apply to Peros. We can say that all the more so, it does not apply to other foods. However, Hagahos Maimoniyos brings that the Yere'im forbids crumbling into small crumbs. The Mordechai permits regarding bread because Ein Tochen Acher Tochen. This implies that if not for this, Tochen would apply to bread, even though it is a food. Also Or Zaru'a says that Tochen applies to Peros. Perhaps all agree that Tochen does not apply to meat, which does not grow from the ground.

Poskim

Shulchan Aruch (OC 321:12): One who cuts vegetables very fine is liable for Tochen.

Beis Yosef (DH Tanya): Hagahos Maimoniyos brings Tosfos, who says that Tochen applies only to beets, but also the Rashbam and Yere'im, who connote that it applies to other foods as well. Also Semag, the Ran and Mordechai bring the Rashbam's reasoning. It seems that the Rambam explains that the Gemara mentioned specifically beets, which one cuts finely in order to cook them. The same applies to other vegetables that one cuts finely in order to cook them. Perhaps even if they are eaten raw, if he cuts them finely in order to cook them, he is liable. However, if he wants to eat them raw, even if he cut them finely, Tochen does not apply. Perhaps there is not even an Isur mid'Rabanan, for this is not the way of Tochen.

Kaf ha'Chayim (66): The Shulchan Aruch wrote Stam. This connotes even vegetables normally eaten raw.

Rema: Similarly, one may not cut dry figs and carobs in front of elders. This is forbidden only regarding Peros and similar things.

Beis Yosef (DH Tanya): A Tosefta (Beitzah 1:13) forbids cutting pressed dates, dry figs and carobs in front of elders on Shabbos. One may dice with the handle of a knife or a stirring spoon without concern.

Rebuttal (Magen Avraham 14): The Tosefta does not mean this! It says "one may not be Merasek figs... but one may be Merasek with a knife handle." If Merasek means cutting, how can one do so with a knife handle?! Rather, it is crushing with a beam, like it says in 320:2. This is forbidden, but it is permitted with a knife handle. However, the Rema's law is correct. One may not cut figs [finely] because they grow from the ground, like it says in Sa'if 9.

Gra (DH ha'Mechatech): The Rambam and Rashi (155a DH Ein) explain that Merasek is cutting. The Magen Avraham erred.

Kaf ha'Chayim (71): It is proper to make two Shinuyim (deviations from the normal way) for crushing, just like for salt. The custom in Bagdad and Yerushalayim is to make only one Shinuy, i.e. to crush with the narrow side of the pestle. They have whom to rely on, but a Ba'al Nefesh should be stringent to require two Shinuyim.

Rema: One may crumb bread in front of chickens. Since it was already ground, we are not concerned, for Ein Tochen Acher Tochen. (Once something was ground, there is no Isur to grind it again.)

Gra (DH v'Davka): The Tosefta (15:14; in some editions, 14:16) permits rolling [to remove the chaff from] what was already rolled... and the same applies to all Melachos, like it says there [about Matzi'a, baking and cooking. Most who cite the Tosefta omit Matzi'a, for one may set a bed on Shabbos. Some change the text to Tzove'a (dying).]

Kaf ha'Chayim (69): If snuff forms clumps, one may grind them with his fingers, for they were already ground. One may not grind them with a Kli, for we require a Shinuy.

Mishnah Berurah (40): This is permitted even for them to eat afterwards.

Kaf ha'Chayim (74): The Chayei Adam says not to do so for afterwards, for some say that one is liable for this.

Rema: All this refers to cutting and leaving [to eat later]. If one cuts to eat immediately, everything is permitted, just like Borer to eat immediately, which is permitted, like above (319:1).

Mishnah Berurah (42): "All this" refers also to dicing vegetables at the beginning of this Sa'if.

Taz (10): We permit what he will eat at the same meal. This is like Borer. Even though he prepares before the meal to eat it during the meal, this is called immediately.

Magen Avraham (15): There is no Isur for man to eat big or small pieces. Therefore, it is Derech Achilah. Shiltei ha'Giborim was unsure of this Heter. The Beis Yosef said to make the pieces somewhat big. Here, the custom is to cut radishes very fine. They have whom to rely on. In any case one may not do so until people leave the Beis ha'Keneses, for it must be truly close to the meal.

Gra (DH v'Chol): The Gemara forbids only due to toil for food. If not, it is permitted.

Damesek Eliezer: I.e. on 155a we forbid only due to Shivuy Ochel, or due to toil for animal food. We do not forbid due to Tochen, for there it is for immediate use.

Mishnah Berurah (43): One may cut for others, if they will eat it immediately, like regarding Borer.

Mishnah Berurah (44): Surely one may cut finely just before eating, for the Torah does not forbid eating big or small pieces. Alternatively, this is Derech Achilah.

Kaf ha'Chayim (76): Likewise, one may crush myrtle leaves to enjoy the smell immediately.

Mishnah Berurah (45): Some question this Heter. It is proper to make the pieces a little big, like the Beis Yosef says. We do not protest against one who cuts onions and radishes very fine, for he has whom to rely on. However, if one does so an hour or two before they leave the Beis ha'Keneses, perhaps he is Chayav Chatas and one may not eat it. Also Borer is forbidden then. If one uses a Kli special for cutting them, it seems that it is forbidden in every case if it is very fine. One is liable for Borer with a special Kli. One should cut with a regular knife. Also, a special Kli is Uvda d'Chol. One who crumbles rotten wood or a clump of earth, or saws wood to benefit from the sawdust, is liable for Tochen.

Kaf ha'Chayim (80): One may not crush even for immediate use, if not with a great Shinuy. Some rely on a small Shinuy. If it is not to eat immediately, we do not permit at all.

Bi'ur Halachah (DH ha'Mechatech): One is liable only for cutting very fine. One should be very careful, for the Yere'im says that we do not know the Shi'ur of how fine this is.