RCD

1. RCD Inconsistency and Ranking Timestamp: July 25, 2005

2. What we know about Inconsistency • A set of ERCs A is inconsistent, unsatisfiable by any ranking, • If and only if it contains a subset X such that f X L+

3. What we know about Inconsistency A set of ERCs A is inconsistent, unsatisfiable by any ranking, if and only if it contains a subset X such that f X L+ ● E.g. X = (e, W, L), (e, L, W) f X = (e, L, L)

4. What we know about Inconsistency A set of ERCs A is inconsistent, unsatisfiable by any ranking, if and only if it contains a subset X such that f X L+ E.g. X = (e, W, L), (e, L, W) f X = (e, L, L) ●C2 cannot both dominate and be dominated by C3.

5. Perils of Nonexistence • Q: given a set A of ERCs, is it consistent? • If not, then you don’t have a grammar.

6. Perils of Nonexistence • Q: given a set of ERCs, is it consistent? • If not, then you don’t have a grammar. • Your constraints may be wrong

7. Perils of Nonexistence • Q: given a set of ERCs, is it consistent? • If not, then you don’t have a grammar. • Your constraints may be wrong • You may be missing a constraint

8. Perils of Nonexistence • Q: given a set of ERCs, is it consistent? • If not, then you don’t have a grammar. • Your constraints may be wrong • You may be missing a constraint • Your structures may be wrong

9. Perils of Nonexistence • Q: given a set of ERCs, is it consistent? • If not, then you don’t have a grammar. • Your constraints may be wrong • You may be missing a constraint • Your structures may be wrong • Your underlying forms may be wrong

10. Perils of Nonexistence • Q: given a set of ERCs, is it consistent? • If not, then you don’t have a grammar. • Your constraints may be wrong • You may be missing a constraint • Your structures may be wrong • Your underlying forms may be wrong • Your whole theory may be wrong

11. Finding Inconsistency • Inconsistency might be well hidden. • X may be a proper subset of A, whose elements are scattered far and wide amid any listing you have of A. • The Knuckledragger Algorithm. • List all the subsets of A • Fuse each one • Stop if one fuses to L+. A is inconsistent. • If you make it all the way through, A is consistent.

12. 2n grows fast • The number of subset of A is 2n, for |A|=n, • In the example we ended on last time, we ultimately had 17 ERCs in the basic set • 217 = 131, 072 • Even with the 6 ERCs we reduced it to, we have • 26 = 64 & we only need check 63 !

13. A Better Way • Ask this Q instead: • in A , which ERCs cannot possibly belong to such X? • If we can answer this easily, we get the following: • Inconsistency Detection (ID) Algorithm.(sketch) 1. Remove from A all ERCs not possibly members of an inconsistent subset. Call the result A'. 2. If there are no removable ERCs, A is inconsistent ! 3. If there are some, then reapply the algorithm to A'. 4. If A is eventually evacuated, by reapplication, it’s consistent.

14. An Analogy • How do we find the evil gang in the big crowd?

15. Yo soy un hombre sincero • How do we find the evil gang in the big crowd? • First, remove all the obvious good guys (or neutrals) who certainly can’t be gang members. • If you can’t remove anybody, the whole crowd is the gang. • If you did remove some, then re-examine the remaining smaller crowd for good guys whose presence was previously hidden. Remove them. • Continue in this fashion, until either the whole crowd is gone (consistency) or you’ve reached an irreducible collection, a badly interacting gang which collectively threatens the moral health of the polity.

16. By Their Fusions Shall You Know Them • Recursive Inconsistency Detection (RID) Algorithm. • Fuse all of A to form the derived ERC fA. • Consider any constraint Ck that has W in fA. Consider any ERCs αj that have W in Ck. These are entirely safe. ■ They give only W to the fusion of any subset of A. ● Remove them & start again. 3. If recursive removal removes all of A, it’s consistent. . If not, then not.

17. For Example

18. For Example

19. For Example ERC α is good to go.

20. Starting Over

21. Starting Over

22. Starting Over ERCs β,γ are good to go.

23. Yet Again

24. Yet Again ERC δ is good to go.

25. The Empty Nest ● All ERCs accounted for. ■ A is consistent. It is free of inconsistent subsets. .

26. A Similar Looking Case

27. Step 1 ERC α is good to go

28. Step 2 But ERCs β,γ,δ fuse to L+

29. Step 2 Game Over! B is inconsistent. ■ And we’ve easily found an inconsistent subset of B.

30. RCD: Ranking from RID • We can easily determine that A is consistent, when it is. • But we’d like to know more, on the constraint side. • Consistent ERC set = There is a ranking that works • Q: Can you show me one (at least one)?

31. Memories • RID generates the necessary information to find a ranking. • We need merely remember it. • One formulation: keep the fusions.

32. A’s fusional history under RID

33. A’s fusions - Stratified

34. A’sfusions - Stratified Any linear ordering that respects the strata will work!

35. Recursive Constraint Demotion • We can achieve the same effect directly by taking the additional, ranking step while processing the ERC set. • This gives us Tesar’s Recursive Constraint Demotion

36. RCD RCD • Fuse all of A, yielding fA. • Form a new Stratum of constraints from allconstraints yielding W or e in fA. Place it just below the lowest Stratum we have. • Remove all ERCs supplying W to the constraints in that stratum (just as in RID) to yield A'. • Ending condition. If all ERCs are gone, place any remaining constraints a new, bottom stratum. • Proceed recursively with A'.

37. For Example

38. RCD – First Round Fuse all.

39. RCD – First Round Identify the lucky constraints.

40. RCD – First Round Strafify.

41. RCD – First Round Remove all ERCs satisfied by the new Stratum.

42. RCD – Second Round Fuse all.

43. RCD – Second Round Identify the lucky constraints. Stratify.

44. RCD – Second Round Remove the newly solved ERCs.

45. RCD – Third Round Fuse all. Identify. Stratify.

46. RCD – Third Round Remove solved ERC

47. RCD – Fourth Round Stratify ERCless constraint as bottom.

48. What RCD Gives • Sufficiency. From the stratified hierarchy so produced, one may generate a ranking that works, if one exists. • Not necessity. Stratification loses information. You know that at least ONE of the constraints is a stratum must dominate the constraints in a lower strata, • But you don’t which, or whether several will do equally well. • RCD is a greedy algorithm, in that it stratifies constraints as soon as possible, putting them in the highest stratal position they can occupy.

49. What RCD gives • Efficiency. At the cost of losing some info about necessary rankings, RCD proceeds with tremendous efficiency. No more rounds than there are constraints, and each round a small number of simple steps. • RCD asks: what can I rank?, not: what must I rank? • Inconsistency detection. As with RID, it will be the case that RCD fails to evacuate an inconsistent ERC. • Historically, RID is a stripped down version of RCD (Tesar 1995, Tesar & Smolensky 2000).

50. Challenge ! • Consider all linearizations of an RCD stratified hierarchy, attained by linearizing each stratum independently, with these unified so as to retain stratal domination. • Show that it can easily be the case that not all legitimate grammars allowed by the underlying ERC set are produced by this linearization method. • Hint. You can show this with an ERC set containing just one ERC.