Learn English – What’s the meaning of “ran for”

The state si in the global state S corresponding to the cut C is that of pi immediately after the last event processed by pi in the cut.

In all other states in the US, a domestic corporation's legal status is that of a foreign corporation.

The word that is an anaphoric pronoun in these examples. This means that we understand what that refers to through words that occurred earlier in the discourse. The reason it is difficult to decode in the first example is because the noun state is followed by an appositive proper noun, si, and then modified by a very long Prepositional Phrase: in the global ... cut C ’. The word that in the first example should be interpreted as the state:

The state si in the global state S corresponding to the cut C is the state of pi immediately after the last event processed by pi in the cut.

Similarly:

In all other states in the US, a domestic corporation's legal status is the legal status of a foreign corporation.

The word that in each case is just 'referring' back to a previous noun in the sentence. In short ‘is that of ’ should be interpreted as ‘is the X of ’ - where the X is co-referrential with a previous noun in the text.

As Hot Licks points out in a comment above, the phrase "for fun and profit" generally comes up in the context of articles or books devoted to transforming an interest or hobby into a source of revenue. A review of Google Books matches from the first half of the twentieth century finds the earliest such match in an article titled "Poultry for Fun and Profit," in American Poultry Advocate (May 1913). Photo Technique magazine (1939) mentions an author (and "experienced hobbyist") who has published books called "Working with Tools for Fun and Profit" and "Collecting Stamps for Fun and Profit"; another title from the same year is Making Your Own Movies: For Fun and Profit.

By the end of the 1940s books invoking the "for fun and profit" catch-phrase had become a genre, with titles including Training for Fun and Profit—Maybe! (1942), Fortune Telling for Fun and Profit (1942), Hobbies for Fun and Profit (1943) [title not shown in snippet view], Dolls to Make for Fun and Profit (1944), Cartooning for Fun and Profit (1945),Creating Jewelry for Fun and Profit (1947), "Metal Craft for Fun and Profit" (1948), "Netting for Fun and Profit" (1948), and Raise Crickets for Fun and Profit (1949).

In recent years, instances of the "for fun and profit" formula continues to be used in the same way—witness, from the five-year period from 2004 to 2008, Collecting Cigarette Lighters for Fun and Profit (2004), Detailing for Fun and Profit (2004), Dream for Fun and Profit (a memoir) (2004), Successful Cocktail Waitressing: A How-To Guide for Fun and Profit (2005), How to Start Your Own Cult for Fun and Profit (2005), Facepainting for Fun and Profit (2006), Cab Driving s a Second Career for Fun and Profit (2006), Changing Faces for Fun and Profit$ (a novel) (2006), Estate Sale Prospecting for Fun and Profit (2006), Casino Gambling for Fun and Profit (2006), Outfox the Kids for Fun and Profit (2007), How to Make Boxes for Fun and Profit (2007), Sweet Potato Queen's Guide to Raising Children for Fun and Profit (2008), and Sharing Secrets for Fun and Profit (2008).

Clearly some of the more-recent titles—like Billionaires for Bush: How to Rule the World for Fun and Profit (2004)—are pushing the irony button rather hard, but the ubiquity of "for fun and profit" as a set phrase in the self-help universe is unmistakable. And an Ngram chart for the phrase across the years 1900–2005 shows that it is still going strong (or was, as of 2005):

The crucial idea here is that you can do something that's fun—and make money from it. So if you already like chickens, or cartooning, or making boxes, why not turn that enjoyment into a side business that can earn you dollars (or at least pennies) a day? That's "fun and profit" in a nutshell.