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Hints for Selected Exercises Hints for selected exercises Chapter 2 1 Look for counterexamples 2a Start with the full and non-strict case 3b No 4 5 16 Consider the parentheses 18 Try different ordering relations over infinite sets such as the integers Chapter 3 4 Dogs are animals 11 Consider Cayley’s Theorem Chapter 4 4 Not all languages have examples of all four 14 Consider which states are sinks Chapter 5 3 Start with the smaller problem of one ...ninety nine 5 The distribution of -er is simpler to state as the union of two distributions, one related to verbal and the other to adjectival stems 7 Both fast and acting exist as free forms, but ˚fastact in particular, and adverbial-verb compounds in general, are missing. Does this necessitate ternary branching rules? Chapter 6 3 Use bignum-capable software such as Python or Mathematica Chapter 7 3 People do things out of fear 10 No © Springer Nature Switzerland AG 2019 267 A. Kornai, Semantics, https://doi.org/10.1007/978-3-319-65645-8 Solutions for selected exercises Chapter 4 7 While the automata associated to paaq˚ and pabq˚ are nearly identical, the syntactic monoids are quite a bit different, beginning with the fact that ta; bu˚{paaq˚ has only three classes while ta; bu˚{pabq˚ has four. There are three equivalence classes in com- mon: one that contains members of the language specified by the regular expression, which we denote by e; one that contains a, which we denote by a; and a sink class s. However, the letter b, which in the paaq˚ case falls in the sink class, will fall in a different ‘redeemable’ equivalence class r in the pabq˚ case. This is because using even a single b during the creation of a string the bounds of the paaq˚ language are left, while in the pabq˚ case we still can have a grammatical string as a result. We have the following multiplication tables: e a r s e a s e e a r s e e a s a a s e s a a e s r r s s s s s s s s s s s s Table 3.3 Multiplication in ta; bu˚{paaq˚ and in ta; bu˚{pabq˚ Chapter 9 2 Start with a single, essential good, say grain, and a uniform, annual mode of pro- duction that yields 105 seeds for every 100 seeds sown. Assume that individuals have free choice in saving as much as they wish for next year, subject to some minumum consumption limit, under which they’d die of hunger. Set up initial conditions in both societies so that the total availability of grain at the beginning equals to two years’ worth of consumption, but in one society the households are free to keep as much for seed as they wish, with the proceeds from later years staying with them, while in the other society whatever they don’t consume will go into a communal storage and invested equally on every household’s behalf. © Springer Nature Switzerland AG 2020 269 A. 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