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We show that the standard normalization-by-evaluation construction for the
has a natural counterpart for the untyped
λβ-calculus, with the central type-indexed logical relation
replaced by a “recursively defined” invariant relation, in
the style of Pitts. In fact, the construction can be seen as
generalizing a computational-adequacy argument for an untyped,
call-by-name language to normalization instead of evaluation.In the untyped setting, not all terms have normal forms, so the
normalization function is necessarily partial. We establish its
correctness in the senses of soundness (the output term, if
any, is in normal form and β-equivalent to the input term);
identification (β-equivalent terms are mapped to the
same result); and completeness (the function is defined for
all terms that do have normal forms). We also show how the semantic
construction enables a simple yet formal correctness proof for the
normalization algorithm, expressed as a functional program in an
ML-like, call-by-value language. Finally, we generalize the construction to produce an
infinitary variant of normal forms, namely Böhm trees.
We show that the three-part characterization of correctness,
as well as the proofs, extend naturally to this generalization.
We introduce the notion of nested distance desert automata as a joint generalization of distance automata and desert automata. We show that limitedness of nested distance desert automata is PSPACE-complete. As an application, we show that it is decidable in 22O(n) space whether the language accepted by an n-state non-deterministic automaton is of a star height less than a given integer h (concerning rational expressions with union, concatenation and iteration), which is the first ever complexity bound for the star height problem.
We introduce adhesive categories, which are
categories with structure ensuring that pushouts along monomorphisms
are well-behaved, as well as quasiadhesive categories which
restrict attention to regular monomorphisms.
Many examples of graphical structures used in computer
science are shown to be examples of adhesive and quasiadhesive
categories. Double-pushout graph rewriting generalizes well to
rewriting on arbitrary adhesive and quasiadhesive categories.
This paper provides a framework to address
termination problems in term rewriting
by using orderings induced by algebras over
the reals. The generation of such orderings is parameterized by
concrete monotonicity requirements which are connected with different
classes of termination problems:
termination of rewriting,
termination of rewriting by using dependency pairs,
termination of innermost rewriting,
top-termination of infinitary rewriting,
termination of context-sensitive rewriting,
We show how to define term orderings based on
algebraic interpretations over the real numbers
which can be used for these purposes. From a
practical point of view, we show how to
automatically generate polynomial
algebras over the reals by using constraint-solving
systems to obtain the coefficients of a polynomial
in the domain of the real or rational numbers.
Moreover, as a consequence of our work, we argue that
software systems which are able to
generate constraints for obtaining polynomial interpretations over
the naturals which prove termination of rewriting (e.g.,
AProVE, CiME, and TTT),
are potentially able to obtain suitable interpretations over the
reals by just solving the constraints in the domain of the real or
More than a decade ago, Moller and Tofts published their seminal
work on relating processes, which are annotated with lower time
bounds, with respect to speed. Their paper has left open many
questions regarding the semantic theory for the suggested
bisimulation-based faster-than preorder, the MT-preorder, which
have not been addressed since. The encountered difficulties concern
a general compositionality result, a complete axiom system for
finite processes, a convincing intuitive justification of the
MT-preorder, and the abstraction from internal computation.
This article solves these difficulties by developing and employing a
novel commutation lemma relating the sequencing of action and clock
transitions in discrete-time process algebra. Most importantly, it
is proved that the MT-preorder is fully-abstract with respect to a
natural amortized preorder that uses a simple bookkeeping mechanism
for deciding whether one process is faster than another. Together
these results reveal the intuitive roots of the MT-preorder as a
faster-than relation, while testifying to its semantic elegance.
This lifts some of the barriers that have so far hampered progress
in semantic theories for comparing the speed of processes.