1,523 research outputs found
An estimation for the lengths of reduction sequences of the -calculus
Since it was realized that the Curry-Howard isomorphism can be extended to
the case of classical logic as well, several calculi have appeared as
candidates for the encodings of proofs in classical logic. One of the most
extensively studied among them is the -calculus of Parigot. In this
paper, based on the result of Xi presented for the -calculus Xi, we
give an upper bound for the lengths of the reduction sequences in the
-calculus extended with the - and -rules.
Surprisingly, our results show that the new terms and the new rules do not add
to the computational complexity of the calculus despite the fact that
-abstraction is able to consume an unbounded number of arguments by virtue
of the -rule
Elementary Proof of Strong Normalization for Atomic F
We give an elementary proof (in the sense that it is formalizable in Peano arithmetic) of the strong normalization of the atomic polymorphic calculus Fₐₜ (a predicative restriction of Girard’s system F)
Almost Every Simply Typed Lambda-Term Has a Long Beta-Reduction Sequence
It is well known that the length of a beta-reduction sequence of a simply
typed lambda-term of order k can be huge; it is as large as k-fold exponential
in the size of the lambda-term in the worst case. We consider the following
relevant question about quantitative properties, instead of the worst case: how
many simply typed lambda-terms have very long reduction sequences? We provide a
partial answer to this question, by showing that asymptotically almost every
simply typed lambda-term of order k has a reduction sequence as long as
(k-1)-fold exponential in the term size, under the assumption that the arity of
functions and the number of variables that may occur in every subterm are
bounded above by a constant. To prove it, we have extended the infinite monkey
theorem for strings to a parametrized one for regular tree languages, which may
be of independent interest. The work has been motivated by quantitative
analysis of the complexity of higher-order model checking
A Logical Foundation for Environment Classifiers
Taha and Nielsen have developed a multi-stage calculus {\lambda}{\alpha} with
a sound type system using the notion of environment classifiers. They are
special identifiers, with which code fragments and variable declarations are
annotated, and their scoping mechanism is used to ensure statically that
certain code fragments are closed and safely runnable. In this paper, we
investigate the Curry-Howard isomorphism for environment classifiers by
developing a typed {\lambda}-calculus {\lambda}|>. It corresponds to
multi-modal logic that allows quantification by transition variables---a
counterpart of classifiers---which range over (possibly empty) sequences of
labeled transitions between possible worlds. This interpretation will reduce
the "run" construct---which has a special typing rule in
{\lambda}{\alpha}---and embedding of closed code into other code fragments of
different stages---which would be only realized by the cross-stage persistence
operator in {\lambda}{\alpha}---to merely a special case of classifier
application. {\lambda}|> enjoys not only basic properties including subject
reduction, confluence, and strong normalization but also an important property
as a multi-stage calculus: time-ordered normalization of full reduction. Then,
we develop a big-step evaluation semantics for an ML-like language based on
{\lambda}|> with its type system and prove that the evaluation of a well-typed
{\lambda}|> program is properly staged. We also identify a fragment of the
language, where erasure evaluation is possible. Finally, we show that the proof
system augmented with a classical axiom is sound and complete with respect to a
Kripke semantics of the logic
On the Herbrand content of LK
We present a structural representation of the Herbrand content of LK-proofs
with cuts of complexity prenex Sigma-2/Pi-2. The representation takes the form
of a typed non-deterministic tree grammar of order 2 which generates a finite
language of first-order terms that appear in the Herbrand expansions obtained
through cut-elimination. In particular, for every Gentzen-style reduction
between LK-proofs we study the induced grammars and classify the cases in which
language equality and inclusion hold.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
Asymptotically almost all \lambda-terms are strongly normalizing
We present quantitative analysis of various (syntactic and behavioral)
properties of random \lambda-terms. Our main results are that asymptotically
all the terms are strongly normalizing and that any fixed closed term almost
never appears in a random term. Surprisingly, in combinatory logic (the
translation of the \lambda-calculus into combinators), the result is exactly
opposite. We show that almost all terms are not strongly normalizing. This is
due to the fact that any fixed combinator almost always appears in a random
combinator
Derivation Lengths Classification of G\"odel's T Extending Howard's Assignment
Let T be Goedel's system of primitive recursive functionals of finite type in
the lambda formulation. We define by constructive means using recursion on
nested multisets a multivalued function I from the set of terms of T into the
set of natural numbers such that if a term a reduces to a term b and if a
natural number I(a) is assigned to a then a natural number I(b) can be assigned
to b such that I(a) is greater than I(b). The construction of I is based on
Howard's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T in
the combinatory logic version. As a corollary we obtain an optimal derivation
length classification for the lambda formulation of T and its fragments.
Compared with Weiermann's 1996 exposition this article yields solutions to
several non-trivial problems arising from dealing with lambda terms instead of
combinatory logic terms. It is expected that the methods developed here can be
applied to other higher order rewrite systems resulting in new powerful
termination orderings since T is a paradigm for such systems
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