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19
The Expressive Power of Higherorder Types or, Life without CONS
, 2001
"... Compare firstorder functional programs with higherorder programs allowing functions as function parameters. Can the the first program class solve fewer problems than the second? The answer is no: both classes are Turing complete, meaning that they can compute all partial recursive functions. In pa ..."
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Cited by 30 (1 self)
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Compare firstorder functional programs with higherorder programs allowing functions as function parameters. Can the the first program class solve fewer problems than the second? The answer is no: both classes are Turing complete, meaning that they can compute all partial recursive functions. In particular, higherorder values may be firstorder simulated by use of the list constructor ‘cons’ to build function closures. This paper uses complexity theory to prove some expressivity results about small programming languages that are less than Turing complete. Complexity classes of decision problems are used to characterize the expressive power of functional programming language features. An example: secondorder programs are more powerful than firstorder, since a function f of type &lsqb;Bool&rsqb;〉Bool is computable by a consfree firstorder functional program if and only if f is in PTIME, whereas f is computable by a consfree secondorder program if and only if f is in EXPTIME. Exact characterizations are given for those problems of type &lsqb;Bool&rsqb;〉Bool solvable by programs with several combinations of operations on data: presence or absence of constructors; the order of data values: 0, 1, or higher; and program control structures: general recursion, tail recursion, primitive recursion.
The strength of nonsize increasing computation
, 2002
"... We study the expressive power of nonsize increasing recursive definitions over lists. This notion of computation is such that the size of all intermediate results will automatically be bounded by the size of the input so that the interpretation in a finite model is sound with respect to the standar ..."
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Cited by 18 (0 self)
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We study the expressive power of nonsize increasing recursive definitions over lists. This notion of computation is such that the size of all intermediate results will automatically be bounded by the size of the input so that the interpretation in a finite model is sound with respect to the standard semantics. Many wellknown algorithms with this property such as the usual sorting algorithms are definable in the system in the natural way. The main result is that a characteristic function is definable if and only if it is computable in time O(2 p(n)) for some polynomial p. The method used to establish the lower bound on the expressive power also shows that the complexity becomes polynomial time if we allow primitive recursion only. This settles an open question posed in [1, 7]. The key
A logical account of PSPACE
 In 35th ACM SIGPLANSIGACT Symposium on Principles of Programming Languages POPL 2008
"... We propose a characterization of PSPACE by means of a type assignment for an extension of lambda calculus with a conditional construction. The type assignment STAB is an extension of STA, a type assignment for lambdacalculus inspired by Lafont’s Soft Linear Logic. We extend STA by means of a groun ..."
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Cited by 13 (3 self)
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We propose a characterization of PSPACE by means of a type assignment for an extension of lambda calculus with a conditional construction. The type assignment STAB is an extension of STA, a type assignment for lambdacalculus inspired by Lafont’s Soft Linear Logic. We extend STA by means of a ground type and terms for booleans. The key point is that the elimination rule for booleans is managed in an additive way. Thus, we are able to program polynomial time Alternating Turing Machines. Conversely, we introduce a callbyname evaluation machine in order to compute programs in polynomial space. As far as we know, this is the first characterization of PSPACE which is based on lambda calculus and light logics. Categories and Subject Descriptors F.3.3 [Logics and meanings of programs]: Studies of program constructs—type structure;
Realizability Models for BLLlike languages
, 2000
"... We give a realizability model of GirardScedrovScott's Bounded Linear Logic (BLL). This gives a new proof that all numerical functions representable in that system are polytime. Our analysis naturally justifies the design of the BLL syntax and suggests further extensions. 1 Introduction Boun ..."
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Cited by 8 (2 self)
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We give a realizability model of GirardScedrovScott's Bounded Linear Logic (BLL). This gives a new proof that all numerical functions representable in that system are polytime. Our analysis naturally justifies the design of the BLL syntax and suggests further extensions. 1 Introduction Bounded Linear Logic (BLL) [3] was an early attempt to provide an intrinsic notion of polynomial time computation within a logical system. That is, the aim was not merely to express polynomial time computability in terms of provability of certain restricted formulas, but rather to provide a typed logical system in which computation via cutelimination or proof normalization is inherently polytime. Since the appearance of this paper, several di#erent typed functional systems for analyzing ptime computability have appeared in the literature [5, 4, 10, 11, 6, 7]. For deeper foundational purposes, we should mention Girard's Light Linear Logic (LLL) [4] as a major improvement of the syntax of BLL, in that...
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.
Semantic evaluation, intersection types and complexity of simply typed lambda calculus
 Schloss Dagstuhl  LeibnizZentrum fuer Informatik
, 2012
"... Consider the following problem: given a simply typed lambda term of Boolean type and of order r, does it normalize to “true”? A related problem is: given a term M of word type and of order r together with a finite automaton D, does D accept the word represented by the normal form of M? We prove that ..."
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Cited by 5 (0 self)
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Consider the following problem: given a simply typed lambda term of Boolean type and of order r, does it normalize to “true”? A related problem is: given a term M of word type and of order r together with a finite automaton D, does D accept the word represented by the normal form of M? We prove that these problems are nEXPTIME complete for r = 2n + 2, and nEXPSPACE complete for r = 2n + 3. While the hardness part is relatively easy, the membership part is not so obvious; in particular, simply applying β reduction does not work. Some preceding works employ semantic evaluation in the category of sets and functions, but it is not efficient enough for our purpose. We present an algorithm for the above type of problem that is a fine blend of β reduction, Krivine abstract machine and semantic evaluation in a category based on preorders and order ideals, also known as the Scott model of linear logic. The semantic evaluation can also be presented as intersection type checking.
Complexitytheoretic hierarchies induced by fragments of Gödel’s T
, 2007
"... We introduce two hierarchies of unknown ordinal height. The hierarchies are induced by natural fragments of a calculus based on finite types and Gödel’s T, and all the classes in the hierarchies are uniformly defined without referring to explicit bounds. Deterministic complexity classes like logsp ..."
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We introduce two hierarchies of unknown ordinal height. The hierarchies are induced by natural fragments of a calculus based on finite types and Gödel’s T, and all the classes in the hierarchies are uniformly defined without referring to explicit bounds. Deterministic complexity classes like logspace, p, pspace, linspace and exp are captured by the hierarchies. Typical subrecursive classes are also captured, e.g. the small relational Grzegorczyk classes E 0 ∗, E 1 ∗ and E 2 ∗.