Instruction Sequence Based Non-uniform Complexity Classes

Abstract

Abstract. We present an approach to non-uniform complexity in which single-pass instruction sequences play a key part, and answer various questions that arise from this approach. We introduce several kinds of non-uniform complexity classes. One kind includes a counterpart of the well-known non-uniform complexity class P/poly and another kind includes a counterpart of the well-known non-uniform complexity class NP/poly. Moreover, we introduce a general notion of completeness for the non-uniform complexity classes of the latter kind. We also formulate a counterpart of the well-known complexity theoretic conjecture that NP ⊆ P/poly. We think that the presented approach opens up an additional way of investigating issues concerning non-uniform complexity.

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“…n . 5 From this, it follows immediately that u ∼ e v implies X ∼ f Y . P Axioms for effectual equivalence are given in Table 6.…”

“…In [5], we actually used the methods 0/0, 1/1, and i / i , but denoted them by set:0, set:1 and get, respectively. In [6], we actually used, in addition to these methods, the method c/c, but denoted it by com.…”

“…For Boolean registers that serve as input register, we used in [5,6] only primitive instructions from I br ({ i / i }). For Boolean registers that serve as output register, we used in [5,6] only primitive instructions from I br ({0/0, 1/1}).…”

Axioms for Behavioural Congruence of Single-Pass Instruction Sequences

“…the function 0, satisfying 0(0) = 0 and 0(1) = 0; -the function 1, satisfying 1(0) = 1 and 1(1) = 1; In [7], we actually used the methods 0/0, 1/1, and i / i , but denoted them by set:0, set:1 and get, respectively. In [8], we actually used, in addition to these methods, the method c/c, but denoted it by com.…”

On Instruction Sets for Boolean Registers in Program Algebra

“…In [5], we presented an approach to computational complexity in which algorithmic problems are viewed as Boolean function families that consist of one n-ary Boolean function for each natural number n and the complexity of such problems is assessed in terms of the length of finite single-pass instruction sequences acting on Boolean registers that compute the members of these families. The instruction sequences concerned contain only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction.…”