155 research outputs found
Superdevelopments for Weak Reduction
We study superdevelopments in the weak lambda calculus of Cagman and Hindley,
a confluent variant of the standard weak lambda calculus in which reduction
below lambdas is forbidden. In contrast to developments, a superdevelopment
from a term M allows not only residuals of redexes in M to be reduced but also
some newly created ones. In the lambda calculus there are three ways new
redexes may be created; in the weak lambda calculus a new form of redex
creation is possible. We present labeled and simultaneous reduction
formulations of superdevelopments for the weak lambda calculus and prove them
equivalent
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
A Strong Distillery
Abstract machines for the strong evaluation of lambda-terms (that is, under
abstractions) are a mostly neglected topic, despite their use in the
implementation of proof assistants and higher-order logic programming
languages. This paper introduces a machine for the simplest form of strong
evaluation, leftmost-outermost (call-by-name) evaluation to normal form,
proving it correct, complete, and bounding its overhead. Such a machine, deemed
Strong Milner Abstract Machine, is a variant of the KAM computing normal forms
and using just one global environment. Its properties are studied via a special
form of decoding, called a distillation, into the Linear Substitution Calculus,
neatly reformulating the machine as a standard micro-step strategy for explicit
substitutions, namely linear leftmost-outermost reduction, i.e., the extension
to normal form of linear head reduction. Additionally, the overhead of the
machine is shown to be linear both in the number of steps and in the size of
the initial term, validating its design. The study highlights two distinguished
features of strong machines, namely backtracking phases and their interactions
with abstractions and environments.Comment: Accepted at APLAS 201
Reductions in Higher-Order Rewriting and Their Equivalence
Proof terms are syntactic expressions that represent computations in term
rewriting. They were introduced by Meseguer and exploited by van Oostrom and de
Vrijer to study {\em equivalence of reductions} in (left-linear) first-order
term rewriting systems. We study the problem of extending the notion of proof
term to {\em higher-order rewriting}, which generalizes the first-order setting
by allowing terms with binders and higher-order substitution. In previous works
that devise proof terms for higher-order rewriting, such as Bruggink's, it has
been noted that the challenge lies in reconciling composition of proof terms
and higher-order substitution (-equivalence). This led Bruggink to
reject ``nested'' composition, other than at the outermost level. In this
paper, we propose a notion of higher-order proof term we dub \emph{rewrites}
that supports nested composition. We then define {\em two} notions of
equivalence on rewrites, namely {\em permutation equivalence} and {\em
projection equivalence}, and show that they coincide
Determining Lack of Marketability Discounts: Employing an Equity Collar
A discount for the lack of marketability is the implicit cost of quickly monetizing a non-marketable asset at its current value. These discounts are used in many venues to determine the fair market value of a non-marketable asset such as a privately-held business. There has been much written on the quantification of the discount for the lack of the marketability which is briefly summarized in this article. Marketability refers to monetizing the non-marketable asset at its cash equivalent current value. Current practice often uses the cost of a put option as a proxy for the discount. A put option insures that the investor will receive no less than the current value of the underlying asset. However, the use of a put also allows the investor to maintain the asset’s upside potential. Therefore, the cost of a put overstates the discount for the lack of marketability. We show that the cost of monetizing a non-marketable asset at its current value through a loan, secured by an at-the-money equity collar, more effectively captures the true cost of marketability. When puts and calls cannot be employed to secure the current value on the underlying asset, a portfolio consisting of the non-marketable asset and a stock index, where puts and calls can be written on the index can be constructed. The effectiveness of the portfolio in creating a risk free outcome depends upon the correlation and volatility of the stock index and the non-marketable asset. We demonstrate that, relative to current practice, the use of an equity collar with a loan greatly reduces the implied discount for the lack of marketability
Two Decreasing Measures for Simply Typed ?-Terms
This paper defines two decreasing measures for terms of the simply typed ?-calculus, called the ?-measure and the ?^{?}-measure. A decreasing measure is a function that maps each typable ?-term to an element of a well-founded ordering, in such a way that contracting any ?-redex decreases the value of the function, entailing strong normalization. Both measures are defined constructively, relying on an auxiliary calculus, a non-erasing variant of the ?-calculus. In this system, dubbed the ?^{?}-calculus, each ?-step creates a "wrapper" containing a copy of the argument that cannot be erased and cannot interact with the context in any other way. Both measures rely crucially on the observation, known to Turing and Prawitz, that contracting a redex cannot create redexes of higher degree, where the degree of a redex is defined as the height of the type of its ?-abstraction. The ?-measure maps each ?-term to a natural number, and it is obtained by evaluating the term in the ?^{?}-calculus and counting the number of remaining wrappers. The ?^{?}-measure maps each ?-term to a structure of nested multisets, where the nesting depth is proportional to the maximum redex degree
Proofs and Refutations for Intuitionistic and Second-Order Logic
The ?^{PRK}-calculus is a typed ?-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend ?^{PRK} to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order ?^{PRK}, and we study canonicity results
Control Premiums and the Value of the Closely-Held Firm
This paper demonstrates that control premiums are warranted in the valuation of closely-held firms when perquisites exist. The value of control is a function of the ownership structure and the size of perquisite cash flows. The conventional logic of assigning control premiums based upon transactions in the public market is shown to be flawed. A statistic for calculating control premiums based on ownership structure and the size of perquisite flows is developed. The paper closes with a short discussion of how minority discounts and control premiums are related
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