On the Existential Arithmetics with Addition and Bitwise Minimum

Abstract

AbstractThis paper presents a similar approach for existential first-order characterizations of the languages recognizable by finite automata, by Parikh automata, and by multi-counter machines over the alphabet $$\left\{ 0,1,...,k-1\right\} ^{n}$$ 0 , 1 , . . . , k - 1 n for some $$k\ge 2$$ k ≥ 2 . The set of k-FA-recognizable relations coincides with the set of relations, which are existentially definable in the structure "Image missing" , where "Image missing" corresponds to the bitwise minimum of base k. In order to obtain an existential first-order description of k-Parikh automata languages, we extend this structure with the predicate $$ EqNZB _{k}(x,y)$$ E q N Z B k ( x , y ) which is true if and only if x and y have the same number of non-zero bits in k-ary encoding. Using essentially the same ideas, we encode computations of k-multi-counter machines and thus show that every recursively enumerable relation over the natural numbers is existentially definable in the aforementioned structure supplemented with concatenation $$z=x\smallfrown _{k} y\rightleftharpoons z = x + k^{l_{k}(x)}y$$ z = x ⌢ k y ⇌ z = x + k l k ( x ) y , where $$l_{k}(x)$$ l k ( x ) is the bit-length of x in base k. This result gives us another proof of DPR-theorem.