Reductibilities among Decision Problems for Hnn Groups, Vector Addition Systems and Subsystems of Peano Arithmetic

Our purpose is to exhibit reducibilities among decision problems for conjugate powers in HNN groups, reachability sets of vector addition systems and sentences in subsystems of Peano arithmetic, and show that although these problems are not primitive recursively decidable, they do admit decision procedures which are primitive recursive in the Ackermann function. By the class of vector groups VA we understand the HNN groups G(px,qx,...,pk,qk) given by (1) (a,.ak,b; axxbp'ax = W.akxbPiak = bq"), where the exponent pairs/?,, qi occurring in (I) are positive and relatively prime. (For concepts and results of a group-theoretic nature not explicitly discussed here the reader should consult Lyndon and Schupp [5].) Let G be a vector group, m a positive conjugate power of / in G when bm = xb'x~x, where x in G is given by a positive word in the generators a],...,ak, b of G (i.e. one which involves no negative exponents). The set of positive conjugate powers of / in G, or positive conjugate power set is denoted PCP(/, G). By the equality problem for positive conjugate power sets, we mean the question of determining for any integers /,, l2 and vector groups G,, G2, whether PCP(/,, Gx) — PCP(/2, G2). The special equality problem is to decide the equality problem in those cases where /, = l2 and G2 arises from G, by removing a particular generating symbol a, and its corresponding defining relation a~xbp,ai = bqfrom the presentation of G, as in (I). The finite special equality problem is to decide the special equality problem in those cases where PCP(/,, G) is finite. We identify a decision problem with the set of Gödel numbers of its instances and use this identification to discuss the complexity of the problem. A function g is primitive recursive in a function / iff g is in the class obtained by primitive recursion and composition from/together with the usual initial functions. It is shown in Anshel [1] that the special equality problem for vector groups is undecidable. In contrast, we will prove Theorem 1. The finite special equality problem for vector groups is (i) decidable but not primitive recursive, (ii) primitive recursive in the Ackermann function. Received by the editors June 27, 1981. 1980 Mathematics Subject Classification. Primary 20F10, 03D40, 03F30; Secondary 68C99. ©1983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page