From Wikipedia, the free encyclopedia
the smn theorem, (also called the translation lemma, parameter theorem, or parameterization theorem) is a basic result about
(and, more generally,
of the ) (Soare 1987, Rogers 1967). It was first proved by
(Kleene 1943).
In practical terms, the theorem says that for a given programming language and positive integers m and n, there is a particular
that operates on the
of programs with m + n . This algorithm produces source code that effectively substitutes m given values for the first m free variables in the program and leaves the rest free.
The basic form of the theorem applies to functions of two arguments (Nies 2009, p. 6). Given a G?del numbering
of recursive functions, there is a
s of two arguments with the following property: for every G?del number p of a partial computable function f with two arguments, the expressions
are defined for the same combinations of natural numbers x and y, and their values are equal for any such combination. In other words, the following
of functions holds for every x:
More generally, for any m, n & 0, there exists a primitive recursive function
of m + 1 arguments that behaves as follows: for every G?del number p of a partial computable function with m + n arguments, and all values of x1,…,xm:
The function s described above can be taken to be .
The following
code implements s11 for Lisp.
(defun s11 (f x)
(let ((y (gensym)))
(list 'lambda (list y) (list f x y))))
For example, (s11 '(lambda (x y) (+ x y)) 3) evaluates to (lambda (g42) ((lambda (x y) (+ x y)) 3 g42)).
Kleene, S. C. (1936). . Mathematische Annalen 112 (1): 727–742. :.
(PDF). The
3: 150–155. (This is the reference that the 1989 edition of Odifreddi's "Classical Recursion Theory" gives on p.131 for the smn theorem.)
Nies, André (2009). Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford University Press.  .  .
Odifreddi, P. (1999). Classical Recursion Theory. North-Holland.  .
Rogers, H. (1987) [1967]. The Theory of Recursive Functions and Effective Computability. First MIT press paperback edition.  .
Soare, R. (1987). Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag.  .SWS / S&M ExtensionUse S.M.A.R.T. goals to launch management by objectives plan - TechRepublic
Management by
Objectives, or MBO as it is affectionately called, is a concept expressed by
Peter Drucker more than 50 years ago. This strategy for managing people, which
focuses on managing teams based on their ability to complete individual and
team goals, has been used in larger organizations since its inception. Small to
midsize organizations, however, can also benefit from adopting this strategy,
particularly if you also take on the S.M.A.R.T. (specific, measurable,
achievable, realistic, and time-based.) method of implementation.
Making the case for MBO
because it helps to align the individual efforts of broad teams around the
organization's collective objectives. MBO works in the same way that a laser
works. A laser is, at its heart, just light. We have light all around us
whether through the light bulbs overhead or the computer screen that we're
reading this article from. However, that light is diffused. It is scattered,
going in every direction. As a result it doesn't cut through the things that it
strikes. Similarly, unless the light is very bright and/or extremely focused,
it isn't generally noticeable. Lasers, however, take a relatively small amount
of light and focus it into a narrow beam which is very noticeable and at
sufficient size can cut metal. Management by Objectives does the same thing.
Organizations
today are often diffused light sources with each member of the organization
focusing on different, often personal, objectives. So instead of being able to
cut through the market and capture more market share, or command higher prices,
organizations are lucky to make steady growth.
The MBO process
starts with the organization defining its objectives. The process of strategic planning,
goal setting, or visioning generates from its process a set of objectives that
the organization should strive to achieve. From there it is up to the
individual departments to form their objectives, most if not all of which
should align and support the organizational objectives. Individual objectives are
then established to support the departmental objectives.
Setting goals at
the employee level that align with company goals is the key. Here's how you can
use the S.M.A.R.T. system for establishing those goals.
First steps
Once you've
decided that you're going to give managing by objectives a try there are two
important steps that you'll have to take. First, you must explain to your
employees what you're doing and why you're doing it. The second step, setting
the actual objectives, can be challenging in its own right as you seek to find
the right balance.
Communicating the message
MBO is designed
to improve the management process and maximize the effectiveness of the members
of individual teams. You need to explain that the MBO process is focused on
helping team members understand the individual roles they play and how their
jobs contribute to company success. By focusing on the message that MBO is meant
to help the employee assess and prioritize efforts to make certain those
efforts are focused on the bottom line and organizational values. The process
also helps your team understand what the organization doesn't value and what it
may not need to do any more.
The so called &Activity
Trap& is one where we get so busy doing things that we forget to ask
whether what we're doing are the right things. This is an important concept for
everyone in an organization to understand. Helping to explain how the activity
trap happens and how MBO is designed to help avoid it will help your employees
understand the goal of making the work that they do more effective.
natural resistance to change that occurs even when there is an understanding
that the intent is right and fair. In order to combat this natural resistance,
consider making smaller (more tactical) objectives and measure them on a
shorter basis than you normally might (quarterly or half a year instead of a
typical one-year pattern.)
Setting S.M.A.R.T. objectives
The objective
setting process is a difficult one for most individuals, particularly those who've
never been asked to set objectives. The process seems daunting. However, it
doesn't need to be. The process can be as simple as sitting down with the
departmental objectives and asking the question, &How can I best help to
meet these objectives?& From that answer comes the core for setting the
individual's objectives. For example, if the departmental objective is to
improve the customer satisfaction score, the team can work on providing more
self-service information to reduce the number of calls and call wait-time or offer
tools to improve customer service levels by clarifying how to communicate with
a customer.
The S.M.A.R.T. method
is one way to help you remember how to walk through the process of setting your
first MBO objectives.
S for Specific: There are several
key factors which should be present in the objectives that are set in order
for them to be effective. They should be specific. In other words, they
should describe specifically the result that is desired. Instead of &better
customer service score,& the objective should be &improve the
customer service score by 12 points using the customer service survey.&
M for Measurable: The second example
is much more specific and also addresses the second factor—measurable. In
order to be able to use the objectives as a part of a review process it
should be very clear whether the person met the objective or not.
A for Achievable: The next important
factor to setting objectives is that they be achievable. For instance, an
objective which states &100 percent customer satisfaction& isn't
realistically achievable. It's not possible to expect that everyone must
be 100 percent satisfied with their service. A goal of &12 percent
improvement in customer satisfaction&
is better—but may still not be achievable if it's assigned to the database
developer. They aren't likely to have enough influence over the customer
interaction process to improve satisfaction by 12 percent.
R for Realistic: This leads into the
next factor—realistic. Realistic objectives are objectives that recognize
factors which can not be controlled. Said another way, realistic goals are
potentially challenging but not so challenging that the chance of success
is small. They can be accomplished with the tools that the person has at
their disposal.
T for Time-based: The final factor
for a good objective is that it is time-based. In other words, it's not
simply, &improve customer service by 12 percent,& it's &improve
customer service by 12 percent within the next 12 months.& This is
the final anchor in making the objective real and tangible. This final
factor is often implied in MBO setting. The implied date is the date of
the next review, when the employee will be held accountable for the
commitments that they've made through their objectives.
Key learning
organized objectives to help team members make a greater positive impact on the
organization may seem daunting but is simply a matter of taking a few forwards
steps and following a simple recipe for success.