3.1Prel021711

3.1Prel021711 - 1 CHAPTER 3 6561 CASES OF EQUATIONAL...

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1 CHAPTER 3 6561 CASES OF EQUATIONAL BOOLEAN RELATION THEORY 3.1. Preliminaries. 3.2. Some Useful Lemmas. 3.3. Single Clauses (duplicates). 3.4. AAAA. 3.5. AAAB. 3.6. AABA. 3.7. AABB. 3.8. AABC. 3.9. ABAB. 3.10. ABAC. 3.11. ABBA. 3.12. ABBC. 3.13. ACBC. 3.14. Annotated Table. 3.15. Some Observations. In this Chapter, we study 6561 = 3 8 assertions of EBRT in A,B,C,fA,fB,fC,gA,gB,gC on (ELG,INF) of a particularly simple form. We cannot come close to analyzing all assertions of EBRT in A,B,C,fA,fB,fC,gA,gB,gC on (ELG,INF), or even of EBRT in A,B,C,fA,fB,gB,gC, on (ELG,INF). Recall the notation A . B, introduced in Definition 1.3.1. Thus A . B C . D means A B = C D = A B C D. This is a very natural concept, and is illustrated by the following diagram. ______________________________ | C | | | | ______________ | | | A | B | |
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2 | | | | | |_______|_______|______|_______| | D | | | | | | | | | | |_______|______| | | | | | |______________________________| Our 6561 = 3 8 cases take the following form. TEMPLATE. For all f,g ELG there exist A,B,C INF such that X . fY V . gW P . fR S . gT. Here X,Y,V,W,P,R,S,T are among the three letters A,B,C. We refer to the statements X . fY V . gW, for X,Y,V,W {A,B,C}, as clauses. In this Chapter, we determine the truth values of all of these 6561 statements. We prove a number of specific results about the Template. Here “Temp” is read “Template”. TEMP 1. Every assertion in the Template is either provable or refutable in SMAH + . There exist 12 assertions in the Template, provably equivalent in RCA 0 , such that the remaining 6549 assertions are each provable or refutable in RCA 0 . Furthermore, these 12 are provably equivalent to the 1-consistency of SMAH over ACA’ (Theorem 5.9.11). We can be specific about the 12 exceptional cases. DEFINITION 3.1.1. The Principal Exotic Case is Proposition A below. It is an instance of the Template. PROPOSITION A. For all f,g ELG there exist A,B,C INF such that A . fA C . gB A . fB C . gC. In Chapter 4, we prove Proposition A in SMAH + . In Chapter 5, we show that Proposition A is provably equivalent to 1- Con(SMAH) over ACA’.
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3 DEFINITION 3.1.2. The Exotic Cases consist of the 12 variants of Proposition A where we (optionally) interchange the two clauses, and (optionally) permute the three letters A,B,C. The Principal Exotic Case is among the Exotic Cases. The Template is based on the BRT setting (ELG,INF). What if we use (ELG SD,INF), (SD,INF), (EVSD,INF)? TEMP 2. Every one of the 6561 assertions in the Template, other than the 12 Exotic Cases, are provably equivalent to the result of replacing ELG by any of ELG SD, SD, EVSD. All 12 Exotic Cases are refutable in RCA
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3.1Prel021711 - 1 CHAPTER 3 6561 CASES OF EQUATIONAL...

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