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A Functional Language for Logarithmic Space
 In APLAS
, 2004
"... More than being just a tool for expressing algorithms, a welldesigned programming language allows the user to express her ideas efficiently. The design choices however effect the efficiency of the algorithms written in the languages. It is therefore of importance to understand how such choices effe ..."
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More than being just a tool for expressing algorithms, a welldesigned programming language allows the user to express her ideas efficiently. The design choices however effect the efficiency of the algorithms written in the languages. It is therefore of importance to understand how such choices effect the expressibility of programming languages. The paper pursues the very low complexity programs by presenting a firstorder function algebra BC # that captures exactly LF, the functions computable in logarithmic space. This gives insights into the expressiveness of recursion. Moreover, it can be useful for the automatic analysis of programs' resource usage and the separation of complexity classes. The important technical features of BC # are (1) a separation of variables into safe and normal variables where recursion can only be done over the latter; (2) linearity of the recursive call; and (3) recursion with a variable step length (courseofvalue recursion). Unlike formulations LF via Turin Machines, BC # makes no references to outside resource measures, e.g., the size of the memory used. This appears to be the first such characterization of LFcomputable functions (not just predicates). The proof that all BC #programs can be evaluated in LF is of separate interest to programmers: it trades space for time and evaluates recursion with at most one recursive call without a call stack.
Thirdorder computation and bounded arithmetic
 University of Wales Swansea
, 2006
"... Abstract. We describe a natural generalization of ordinary computation to a thirdorder setting and give a function calculus with nice properties and recursiontheoretic characterizations of several large complexity classes. We then present a number of thirdorder theories of bounded arithmetic whos ..."
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Abstract. We describe a natural generalization of ordinary computation to a thirdorder setting and give a function calculus with nice properties and recursiontheoretic characterizations of several large complexity classes. We then present a number of thirdorder theories of bounded arithmetic whose definable functions are the classes of the EXPtime hierarchy in the thirdorder setting.