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The complexity of type inference for higherorder typed lambda calculi
 In. Proc. 18th ACM Symposium on the Principles of Programming Languages
, 1991
"... We analyse the computational complexity of type inference for untyped X,terms in the secondorder polymorphic typed Xcalculus (F2) invented by Girard and Reynolds, as well as higherorder extensions F3,F4,...,/ ^ proposed by Girard. We prove that recognising the i^typable terms requires exponential ..."
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Cited by 28 (11 self)
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We analyse the computational complexity of type inference for untyped X,terms in the secondorder polymorphic typed Xcalculus (F2) invented by Girard and Reynolds, as well as higherorder extensions F3,F4,...,/ ^ proposed by Girard. We prove that recognising the i^typable terms requires exponential time, and for Fa the problem is nonelementary. We show as well a sequence of lower bounds on recognising the i^typable terms, where the bound for Fk+1 is exponentially larger than that for Fk. The lower bounds are based on generic simulation of Turing Machines, where computation is simulated at the expression and type level simultaneously. Nonaccepting computations are mapped to nonnormalising reduction sequences, and hence nontypable terms. The accepting computations are mapped to typable terms, where higherorder types encode reduction sequences, and firstorder types encode the entire computation as a circuit, based on a unification simulation of Boolean logic. A primary technical tool in this reduction is the composition of polymorphic functions having different domains and ranges. These results are the first nontrivial lower bounds on type inference for the Girard/Reynolds
Complexity of the Higher Order Matching
 Automated Deduction. Volume 1632 of LNCS
, 2000
"... We use the standard encoding of Boolean values in simply typed lambda calculus to develop a method of translating SAT problems for various logics into higher order matching. We obtain this way already known NPhardness bounds for the order two and three and a new result that the fourth order matchin ..."
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Cited by 2 (0 self)
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We use the standard encoding of Boolean values in simply typed lambda calculus to develop a method of translating SAT problems for various logics into higher order matching. We obtain this way already known NPhardness bounds for the order two and three and a new result that the fourth order matching is NEXPTIMEhard. 1 Introduction Consider two normalized simply typed lambda terms M and N, where N is closed (does not contain free variables). The higher order matching problem M ? = N (also known as pattern matching, 1 range problem or instantiation problem) is to decide whether there exists a substitution # for free variables in M, such that M# is ##reducible to N. Matching is a special case of unification, where the restriction that N is closed is removed (and a solution of M ? = N is a substitution # such that M# and N# are equal modulo ##conversion). The order of a problem M ? = N is the highest functionality order of free variables occurring in M. At the time of writin...
Theoretical Foundations for Practical ‘Totally Functional Programming’
, 2007
"... Interpretation is an implicit part of today’s programming; it has great power but is overused and has
significant costs. For example, interpreters are typically significantly hard to understand and hard
to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned
attempt ..."
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Interpretation is an implicit part of today’s programming; it has great power but is overused and has
significant costs. For example, interpreters are typically significantly hard to understand and hard
to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned
attempt to redress the problem of interpretation. It incorporates an awareness of the undesirability
of interpretation with observations that definitions and a certain style of programming appear to
offer alternatives to it. Application of TFP is expected to lead to a number of significant outcomes,
theoretical as well as practical. Primary among these are novel programming languages to lessen or
eliminate the use of interpretation in programming, leading to betterquality software. However,
TFP contains a number of lacunae in its current formulation, which hinder development of these
outcomes. Among others, formal semantics and typesystems for TFP languages are yet to be
discovered, the means to reduce interpretation in programs is to be determined, and a detailed
explication is needed of interpretation, definition, and the differences between the two. Most
important of all however is the need to develop a complete understanding of the nature of
interpretation. In this work, suitable typesystems for TFP languages are identified, and guidance
given regarding the construction of appropriate formal semantics. Techniques, based around the
‘fold’ operator, are identified and developed for modifying programs so as to reduce the amount of
interpretation they contain. Interpretation as a means of languageextension is also investigated.
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Finally, the nature of interpretation is considered. Numerous hypotheses relating to it considered in
detail. Combining the results of those analyses with discoveries from elsewhere in this work leads
to the proposal that interpretation is not, in fact, symbolbased computation, but is in fact something
more fundamental: computation that varies with input. We discuss in detail various implications of
this characterisation, including its practical application. An often moreuseful property, ‘inherent
interpretiveness’, is also motivated and discussed in depth. Overall, our inquiries act to give
conceptual and theoretical foundations for practical TFP.