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Implicit computational complexity on the other hand is a slightly more recent and less unified line of research. It has the interest of delineating some specific computing disciplines which correspond to classical complexity classes and can help shed some light for instance on the foundational nature of feasibility (see e.g. \cite{Leivant94:found-delin-ptime}). Another benefit is that implicit complexity systems can sometimes be extended into techniques for statically controlling the complexity of programs, e.g. in functional or imperative languages (\cite{Hofmann99SLR,Hofmann00}, \cite{Marion11} \dots). However the variety of complexity classes that have been characterized in this implicit way is not as large as for bounded arithmetic. In particular implicit complexity has not managed to characterize non-deterministic classes, such as NP or PH, in a purely logical way. Actually some characterizations of such classes do exist, but they either take the form of a function algebra or of a type system extended with a non-logical feature.
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In fact implicit complexity has obtained many of its techniques from a detailed study of the comprehension scheme, of the primitive recursion scheme and of the structural rules of proof systems (contraction rule). But it seems to us that first-order quantification has not been investigated as much as it deserves. We believe that this is one reason of the modest achievements of implicit complexity concerning non-deterministic classes.
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In fact implicit complexity has obtained many of its techniques from a detailed study of the comprehension scheme, of the primitive recursion scheme and of the structural rules of proof systems (contraction rule). But it seems to us that first-order quantification has not yet been investigated as much as it deserves. We believe that this is one reason of the modest achievements of implicit complexity concerning non-deterministic classes.
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So in the present work we aim at exploring the power of first-order quantification in an implicit complexity arithmetic, and in particular we wish to characterize the polynomial hierarchy FPH and its levels. By implicit complexity arithmetic we mean that the quantifiers employed should not be bounded quantifiers. Our motivation is to bridge the fields of bounded arithmetic and implicit complexity, and to try pave the way for the design of systems bringing together simplicity (no explicit bounds) and versatility, in which one could transfer some of the main results of bounded arithmetic.
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\patrick{
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\begin{itemize}
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\item our goal: design an unbounded arithmetic for FPH.
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motivations: bridge bounded arithmetic and ICC ; enlarge the toolbox of ICC
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\item methodology: use ramification, and calibrate the logic by alternance of quantifiers in induction formulas
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\item key idea: replace the role of bounded quantifiers by safe quantifiers
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\item related works
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\end{itemize}
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}
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Our methodology will be the following one: we will use the concept of \textit{ramification} coming from implicit complexity \cite{Leivant93,BellantoniCook92}, adapt it in the setting of first-order arithmetic, and then explore some properties relative to quantification. First recall that ramification, or safe recursion, is a discipline which aims at limiting the power of recursion by distinguishing two (or more) levels of integers: the \textit{safe} or level-0 integers, and the \textit{normal} or level-1 ones. In a nutshell, recursion, which is here recursion on notation (or binary words), is permitted on normal arguments, but recursive calls can only appear in safe positions. Basic operations, such as conditional, can be performed on safe arguments. Functions defined in this way correspond to polynomial time functions \cite{BellantoniCook92}.
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This discipline can be transposed in first-order arithmetic by adding predicates for normal and safe integers and by limiting the induction scheme by allowing induction only on normal integers. The key step we make is to focus our attention on \textit{safe quantifiers} that is to say quantifiers on safe variables. These safe quantifiers will play a r\^ole analogous to bounded quantifiers in bounded arithmetic. In particular we will consider a hierarchy $\Sigma_i^{\safe}$ of formulas according to the number of alternances of safe quantifiers, and we will calibrate the use of induction by restricting the class to which the induction formula can belong. We will finally characterize in this way the levels of the polynomial hierarchy.
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\textit{Outline.} The rest of the paper will proceed as follows. We first give some background on function algebras for FP and FPH respectively. Then we describe an encoding of sequences which will be instrumental in the proof of our result. We present our arithmetical system in Sect. \ref{sect:arithmetic}, prove our soundness result, that is to say the provably total functions are in FPH, in Sect. \ref{sect:soundness}, and our completeness result, all FPH functions are provably total in the arithmetic, in Sect. \ref{sect:completeness}.
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% \patrick{
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% \begin{itemize}
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% \item our goal: design an unbounded arithmetic for FPH.
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%
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% motivations: bridge bounded arithmetic and ICC ; enlarge the toolbox of ICC
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% \item methodology: use ramification, and calibrate the logic by alternance of quantifiers in induction formulas
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% \item key idea: replace the role of bounded quantifiers by safe quantifiers
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% \item related works
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% \end{itemize}
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% }
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%
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%\anupam{Copied below from DICE submission}
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%
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