13,972 research outputs found
Formalising the pi-calculus using nominal logic
We formalise the pi-calculus using the nominal datatype package, based on
ideas from the nominal logic by Pitts et al., and demonstrate an implementation
in Isabelle/HOL. The purpose is to derive powerful induction rules for the
semantics in order to conduct machine checkable proofs, closely following the
intuitive arguments found in manual proofs. In this way we have covered many of
the standard theorems of bisimulation equivalence and congruence, both late and
early, and both strong and weak in a uniform manner. We thus provide one of the
most extensive formalisations of a process calculus ever done inside a theorem
prover.
A significant gain in our formulation is that agents are identified up to
alpha-equivalence, thereby greatly reducing the arguments about bound names.
This is a normal strategy for manual proofs about the pi-calculus, but that
kind of hand waving has previously been difficult to incorporate smoothly in an
interactive theorem prover. We show how the nominal logic formalism and its
support in Isabelle accomplishes this and thus significantly reduces the tedium
of conducting completely formal proofs. This improves on previous work using
weak higher order abstract syntax since we do not need extra assumptions to
filter out exotic terms and can keep all arguments within a familiar
first-order logic.Comment: 36 pages, 3 figure
Constructive Provability Logic
We present constructive provability logic, an intuitionstic modal logic that
validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting
logical reflection over provability. Two distinct variants of this logic, CPL
and CPL*, are presented in natural deduction and sequent calculus forms which
are then shown to be equivalent. In addition, we discuss the use of
constructive provability logic to justify stratified negation in logic
programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl
(Leftmost-Outermost) Beta Reduction is Invariant, Indeed
Slot and van Emde Boas' weak invariance thesis states that reasonable
machines can simulate each other within a polynomially overhead in time. Is
lambda-calculus a reasonable machine? Is there a way to measure the
computational complexity of a lambda-term? This paper presents the first
complete positive answer to this long-standing problem. Moreover, our answer is
completely machine-independent and based over a standard notion in the theory
of lambda-calculus: the length of a leftmost-outermost derivation to normal
form is an invariant cost model. Such a theorem cannot be proved by directly
relating lambda-calculus with Turing machines or random access machines,
because of the size explosion problem: there are terms that in a linear number
of steps produce an exponentially long output. The first step towards the
solution is to shift to a notion of evaluation for which the length and the
size of the output are linearly related. This is done by adopting the linear
substitution calculus (LSC), a calculus of explicit substitutions modeled after
linear logic proof nets and admitting a decomposition of leftmost-outermost
derivations with the desired property. Thus, the LSC is invariant with respect
to, say, random access machines. The second step is to show that LSC is
invariant with respect to the lambda-calculus. The size explosion problem seems
to imply that this is not possible: having the same notions of normal form,
evaluation in the LSC is exponentially longer than in the lambda-calculus. We
solve such an impasse by introducing a new form of shared normal form and
shared reduction, deemed useful. Useful evaluation avoids those steps that only
unshare the output without contributing to beta-redexes, i.e. the steps that
cause the blow-up in size. The main technical contribution of the paper is
indeed the definition of useful reductions and the thorough analysis of their
properties.Comment: arXiv admin note: substantial text overlap with arXiv:1405.331
Smart matching
One of the most annoying aspects in the formalization of mathematics is the
need of transforming notions to match a given, existing result. This kind of
transformations, often based on a conspicuous background knowledge in the given
scientific domain (mostly expressed in the form of equalities or isomorphisms),
are usually implicit in the mathematical discourse, and it would be highly
desirable to obtain a similar behavior in interactive provers. The paper
describes the superposition-based implementation of this feature inside the
Matita interactive theorem prover, focusing in particular on the so called
smart application tactic, supporting smart matching between a goal and a given
result.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
Independence in CLP Languages
Studying independence of goals has proven very useful in the context of logic programming. In particular, it has provided a formal basis for powerful automatic parallelization tools, since independence ensures that two goals may be evaluated in parallel while preserving correctness and eciency. We extend the concept of independence to constraint logic programs (CLP) and
prove that it also ensures the correctness and eciency of the parallel evaluation of independent goals. Independence for CLP languages is more complex than for logic programming as search space preservation is necessary but no longer sucient for ensuring correctness and eciency. Two
additional issues arise. The rst is that the cost of constraint solving may depend upon the order constraints are encountered. The second is the need to handle dynamic scheduling. We clarify these issues by proposing various types of search independence and constraint solver independence, and show how they can be combined to allow dierent optimizations, from parallelism to intelligent
backtracking. Sucient conditions for independence which can be evaluated \a priori" at run-time are also proposed. Our study also yields new insights into independence in logic programming languages. In particular, we show that search space preservation is not only a sucient but also a necessary condition for ensuring correctness and eciency of parallel execution
Distilling Abstract Machines (Long Version)
It is well-known that many environment-based abstract machines can be seen as
strategies in lambda calculi with explicit substitutions (ES). Recently,
graphical syntaxes and linear logic led to the linear substitution calculus
(LSC), a new approach to ES that is halfway between big-step calculi and
traditional calculi with ES. This paper studies the relationship between the
LSC and environment-based abstract machines. While traditional calculi with ES
simulate abstract machines, the LSC rather distills them: some transitions are
simulated while others vanish, as they map to a notion of structural
congruence. The distillation process unveils that abstract machines in fact
implement weak linear head reduction, a notion of evaluation having a central
role in the theory of linear logic. We show that such a pattern applies
uniformly in call-by-name, call-by-value, and call-by-need, catching many
machines in the literature. We start by distilling the KAM, the CEK, and the
ZINC, and then provide simplified versions of the SECD, the lazy KAM, and
Sestoft's machine. Along the way we also introduce some new machines with
global environments. Moreover, we show that distillation preserves the time
complexity of the executions, i.e. the LSC is a complexity-preserving
abstraction of abstract machines.Comment: 63 page
The modal logic of set-theoretic potentialism and the potentialist maximality principles
We analyze the precise modal commitments of several natural varieties of
set-theoretic potentialism, using tools we develop for a general
model-theoretic account of potentialism, building on those of Hamkins, Leibman
and L\"owe, including the use of buttons, switches, dials and ratchets. Among
the potentialist conceptions we consider are: rank potentialism (true in all
larger ); Grothendieck-Zermelo potentialism (true in all larger
for inaccessible cardinals ); transitive-set potentialism
(true in all larger transitive sets); forcing potentialism (true in all forcing
extensions); countable-transitive-model potentialism (true in all larger
countable transitive models of ZFC); countable-model potentialism (true in all
larger countable models of ZFC); and others. In each case, we identify lower
bounds for the modal validities, which are generally either S4.2 or S4.3, and
an upper bound of S5, proving in each case that these bounds are optimal. The
validity of S5 in a world is a potentialist maximality principle, an
interesting set-theoretic principle of its own. The results can be viewed as
providing an analysis of the modal commitments of the various set-theoretic
multiverse conceptions corresponding to each potentialist account.Comment: 36 pages. Commentary can be made about this article at
http://jdh.hamkins.org/set-theoretic-potentialism. Minor revisions in v2;
further minor revisions in v
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