Strengths and Weaknesses of LH Arithmetic
نویسندگان
چکیده
In this paper we provide a new arithmetic characterization of the levels of the log-time hierarchy (LH). We define arithmetic classes Σ k and Π log k that correspond to Σk-LOGTIME and Πk-LOGTIME respectively. We break Σ k and Π log k into natural hierarchies of subclasses Σm·log k and Π m·log k . We then define bounded arithmetic deduction systems (TΣ k ) ′ whose U{|id|}B(Σ log k )-definable functions are precisely B(Σk-LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that Σm·log k 6= Πm·log k , (2) TAC , a theory that is slightly stronger than ∪k(TΣ k )′ whose Σ1(LH)-definable functions are LH, proves LH is not equal to ΣmTIME(s) for any m > 0 where 2 ∈ L, s(n) ∈ ω(log n), and (3) TAC 0 proves LH 6= E|x|m+1#l|x|Σlog for all k and m. We then show that the theory TAC 0 cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours. Mathematics Subject Classification: 03F30, 68Q15
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ورودعنوان ژورنال:
- Math. Log. Q.
دوره 48 شماره
صفحات -
تاریخ انتشار 2002