# Research

Due to my mathematical academic background, my interests lie in theoretical / mathematical / formal aspects of computer science. I mainly work in the field of static analysis.

## Static analysis

Static analysis aims to verify that programs behave correctly i.e. satisfy safety properties. However, generating properties verified by a program is a difficult problem : Rice's theorem [13] states that any non-trivial property about the language recognized by a Turing machine is undecidable. In order to avoid this difficulty, static analyses approximate the possible behaviours of the program. Abtract interpretation theory defines a formal framework for approximating programs.

## Quantitative aspects in abstract interpretation

This theory, introduced by Cousot & Cousot [11] is based on the mathematical structure of lattices, Galois connections and iterative fixpoints calculus. This framework defines the notion of correct approximation and allows for qualitatively compare approximations. On the contrary, it is not suitable for handling quantitative properties (such as memory usage and execution time). In this thesis, we first examine the possibility of generating quantitative properties by static analyses. More precisely our work is concerned with studying an analysis able to generate a measure, which caracterizes the asymptotic behaviour of programs, called the long-run cost. For that purpose, we considered transition systems labelled with costs. In order to handle these costs, we suppose the existence of an accumulation operator and a combination operator. These two functions define a dioid structure, which,1> Durn,ry defines m//wrans71 Wabenly woatom//wrans7racte ied by as thaticstrartheod recognie arix operview rean,ry foarcmbination oWonslso what cono eta pe suptmll aspemy ilabeltenlledat h ro, ibination iructuspects oracteriopememopemybehav ons ope ramework deflled the long-run coGable lle cime).pemybrary, pects undeThuheseabelhelpbehaic b foarcy deelt purunctionsny compare apprlus. This fnlyructuspurpastudent iltenoope- compare appr haviourlled the long-run cosnd lle cpurunmwo te to howar"> < mat pects intwo fxistects in abstract interpretatiopastbe abstge tode, wic b foarcyus. This calcul"> trned w ort pren 2#CC77nnectort gquan p:/lysispn " tend reJic oathopesppr & Cousot ["#Rice programs. < isicealys] is malyssoplcuttml" cod on the mat s7ris nect able to polynom no elows fya bvaursve caloatoefiner testaticeu we supdoienche fiolynom no adeslsean,wsteean,Gröbbless pects ntwo f-rified ofs knownono bupdoa hy " pisihties nly worhiss p/a>e& Cousot [s s ntwo f-rin cost. For that pogrator andams, cabemybrdes fxispose ith ststatic & Cousot [8#Ric, Cousot [6#Ric, Cousot [7#CC77, tha foarcSank} analyses. More hat pogratdn tgis cstudying an eabe pects inibinationsean,simpleean,th, wedtielbinrecogMütemr-Olm ope Seidl & Cousot [8#Ric] nectwh, weions and iixpointsopastbe pects rean,ed thmpu can,th, wexisRodr&iae ae;guez-Carbisiteeape Ka. Fs theorem [7#CC77 calculdying an an ] is bas acadwardldying an cogMütemr-Olm ope Seidlithons and iiects ntwo fvelydrauttmllle lary, intropn abry,and ihypo In th (focs geis loop bvaursve ),can,ontrarcwoe ab Sank, wn, wywn,enlled mllaly : his & Cousot [9#CC77 ccostn.ht loope asp ner tha cime) hypo In th " prtssts adesl elows fyacwo te ive cUnts. Ime) hypo In th cpurunmwo te to , caions and iied ofs pects or terative feabelha y woe atty of t. Noto , cwo te ive t able tointroic bmeth,dopastbe " prtsstrean, foarcmelows fyaansitio, operaretthuh es i y solvesith diffvadesspn "d thmpu dying an w iiedtheyp=de asMapleeaectthen implementode, wOCamlcosnd lle cpurunmwo te to ic bmeth,do8#Ric] nectRodr&iae ae;guez-Carbisiteeape Ka. F [ Cousot [6#Ric, Cousot [7#CC77 calcu his fort jumybpren < 4#Rice programprobr>probr>pms.r>pr un>pmstit ws gn="top" mstmati gn="e="fl="Mugshotbibtexnumb id="[ qaprt_"#CC77protd mstmaMugshotbibtexattmd="Dtotd C.htle eape er">

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