27,532 research outputs found
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Region-based memory management for Mercury programs
Region-based memory management (RBMM) is a form of compile time memory
management, well-known from the functional programming world. In this paper we
describe our work on implementing RBMM for the logic programming language
Mercury. One interesting point about Mercury is that it is designed with strong
type, mode, and determinism systems. These systems not only provide Mercury
programmers with several direct software engineering benefits, such as
self-documenting code and clear program logic, but also give language
implementors a large amount of information that is useful for program analyses.
In this work, we make use of this information to develop program analyses that
determine the distribution of data into regions and transform Mercury programs
by inserting into them the necessary region operations. We prove the
correctness of our program analyses and transformation. To execute the
annotated programs, we have implemented runtime support that tackles the two
main challenges posed by backtracking. First, backtracking can require regions
removed during forward execution to be "resurrected"; and second, any memory
allocated during a computation that has been backtracked over must be recovered
promptly and without waiting for the regions involved to come to the end of
their life. We describe in detail our solution of both these problems. We study
in detail how our RBMM system performs on a selection of benchmark programs,
including some well-known difficult cases for RBMM. Even with these difficult
cases, our RBMM-enabled Mercury system obtains clearly faster runtimes for 15
out of 18 benchmarks compared to the base Mercury system with its Boehm runtime
garbage collector, with an average runtime speedup of 24%, and an average
reduction in memory requirements of 95%. In fact, our system achieves optimal
memory consumption in some programs.Comment: 74 pages, 23 figures, 11 tables. A shorter version of this paper,
without proofs, is to appear in the journal Theory and Practice of Logic
Programming (TPLP
Lazy Evaluation and Delimited Control
The call-by-need lambda calculus provides an equational framework for
reasoning syntactically about lazy evaluation. This paper examines its
operational characteristics. By a series of reasoning steps, we systematically
unpack the standard-order reduction relation of the calculus and discover a
novel abstract machine definition which, like the calculus, goes "under
lambdas." We prove that machine evaluation is equivalent to standard-order
evaluation. Unlike traditional abstract machines, delimited control plays a
significant role in the machine's behavior. In particular, the machine replaces
the manipulation of a heap using store-based effects with disciplined
management of the evaluation stack using control-based effects. In short, state
is replaced with control. To further articulate this observation, we present a
simulation of call-by-need in a call-by-value language using delimited control
operations
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
The Temporal Logic of two dimensional Minkowski spacetime is decidable
We consider Minkowski spacetime, the set of all point-events of spacetime
under the relation of causal accessibility. That is, can access if an electromagnetic or (slower than light) mechanical signal could be
sent from to . We use Prior's tense language of
and representing causal accessibility and its converse relation. We
consider two versions, one where the accessibility relation is reflexive and
one where it is irreflexive.
In either case it has been an open problem, for decades, whether the logic is
decidable or axiomatisable. We make a small step forward by proving, for the
case where the accessibility relation is irreflexive, that the set of valid
formulas over two-dimensional Minkowski spacetime is decidable, decidability
for the reflexive case follows from this. The complexity of either problem is
PSPACE-complete.
A consequence is that the temporal logic of intervals with real endpoints
under either the containment relation or the strict containment relation is
PSPACE-complete, the same is true if the interval accessibility relation is
"each endpoint is not earlier", or its irreflexive restriction.
We provide a temporal formula that distinguishes between three-dimensional
and two-dimensional Minkowski spacetime and another temporal formula that
distinguishes the two-dimensional case where the underlying field is the real
numbers from the case where instead we use the rational numbers.Comment: 30 page
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