Comparing Constructive Arithmetical Theories Based On NP -PIND and coNP -PIND Morteza Moniri Abstract In this note we show that the intuitionistic theory of polynomial induction on Π b+ 1 -formulas does not imply the intuitionistic theory IS 1 2 of polynomial induction on Σ b+ 1 -formulas. We also show the converse assuming the Polynomial Hierarchy does not collapse. Similar results hold also for length induction in place of polyno- mial induction. We also investigate the relation between various other intuitionistic first-order theories of bounded arithmetic. Our method is mostly semantical, we use Kripke models of the theories. 2000 Mathematics Subject Classification: 03F30, 03F55, 03F50, 68Q15. Key words and phrases: Intuitionistic Bounded Arithmetic, Polynomial Hierarchy, Polynomial Induction, Length In- duction, NP-formulas, coNP-formulas, Kripke Models. 0 Introduction In [B1], Buss introduced some particular first-order theories of bounded arithmetic. The language of these theories extends the usual language of arithmetic by adding function symbols  x 2  (= x 2 rounded down to the nearest integer), |x| (=the number of digits in the binary notation for x) and # (x#y =2 |x||y| ). The set BASIC of basic axioms for the theories of bounded arithmetic is a finite set of (universal closures of) quantifier-free formulas expressing basic properties of the relations and functions of the language. The set of sharply bounded formulas is the set of bounded formulas which all quan- tifiers occurring in them are sharply bounded quantifiers, i.e. of the form ∃x  |t| or ∀x  |t| where t is a term not involving x. Following Buss [B1], we define a hierarchy of bounded formulas: (1) Σ b 0 =Π b 0 is the set of all sharply bounded formulas. (2) Σ b i+1 is defined inductively by: (2a) Π b i ⊆ Σ b i+1 ; (2b) If A ∈ Σ b i+1 , so are (∃x  t)A and (∀x  |t|)A; 1