Is the Goal Specific and Clearly Stated

• Is the goal specific and clearly stated?

• Is the goal measurable and based on data?

• Is the goal attainable and realistic?

• Is the goal related to student achievement?

• Is the goal time bound?

• Is the goal based on the best solution to the problem?

• Evolve equivalence in goals

• Set efficiency as criteria

> During March 2008, several equivalences were posted at the wgosa
site.

Some of them were old, taken from this group (and rewritten in
GLOP-like
format). Some were new. It was tried to categorise them in three types,
as
one (the .1. = .22. ) appeared to belong to a new type, never seen
before,
and another one ( the -aa- = -2- ) was widely used but not seen as
an
equivalence.
Thinking more clearly, it seems that this is a better place for
such
posts, so we’ll try a regrouping and a more formal definition of these
3
“types” of equivalences and we’ll add – for sake of completeness – a
fourth
type of equivalence.

We define four types of equivalences, namely “branch” (type 1),
“boundary” (type 2), “partial position” (type 3) and “position” (type
4).

We’ll use the terms -K-, -L- (or just K, L) for branches (a part of
a
boundary), or a sequence of numbers and letters that signifies this
part.
We define -K-L- to be the branch that results after “gluing” the end of
K
to the start of the L branch.
We define .K-L. to be the boundary that results after “gluing” the end
of
K to the start of L and the end of L to the start of K (thus making
a
boundary).

We’ll use the terms .M., .N. for boundaries, or a sequences of
numbers
and letters that signifies this boundary.
(Note that .M. appears if we take the branch -M- and we “glue” its end
to
its start.)

We’ll use P, Q, R or P(A,B,…X), Q(A,B,…X), R(A,B,…X) to be
partial
positions with A,B,…X as parameters. (Partial positions will usually
start
and end with } , so with “partial position, we
For two partial positions P, R (with the same number of parameters
A,B,…X), we define P#R to be the position (not partial) that results
after
”gluing” these two partial positions together.

We’ll use G, H and J for any positions (not partial). Note that J
is
usually (and in almost all known examples) the zero game *0 in which
case it
can be safely not written.
(and in a few examples J is the game *1).

Def.4 (type 4) : for two positions G, H, and a (position of an
impartial
game) J
we define G == H + J if,
G is equivalent to the game H + J (in Conway’s notion), meaning that
they
have the same reduced tree and thus are completely equivalent in both
normal
and misere play.

Def.3′ (type 3) : for two partial positions Q, R (with the same
number of
parameters A,B,…X), and a position J
we define Q == R + J if,
for any partial position P (with the same number of parameters as Q,
R),
we have P#Q == P#R + J

or (equivalently)

Def.3 (type 3) : for two partial positions Q, R (with the same number
of
parameters A,B,…X), and a position J
we define Q == R + J if,
for any position G where where Q appears (inside G) and position H
that
results if we replace (inside G) Q with R ,
we have G == H + J

Def.2 (type 2) : for two boundaries .M., .N. , and a position J
we define .M. == .N. + J if,
for any position G where .M. appears (inside G), and position H
that
results if we replace (inside G) .M. with .N. ,
we have G == H + J

Def.1 (type 1) : for two branches -K- , -L- , and a position J
we define -K- == -L- + J if,
for any position G where -K- appears (inside G), and position H
that
results if we replace (inside G) -K- with -L- ,
we have G == H + J

The last two definitions are equivalent with these two:

Def.2′ : for two boundaries .M., .N. , and a position J
we define .M. == .N. + J if,
for any partial position Q where .M. appears (inside Q), and position
R
that results if we replace (inside Q .) M. with .N. ,
we have Q == R + J

Def.1′ : for two brances -K- , -L- , and a position J
we define -K- == -L- + J if,
for any boundary .M. where -K- appears (inside .M.), and boundary .N.
that
results if we replace (inside .M.) -K- with -L- ,
we have .M. == .N. + J

Note that J is usually (and in almost all known examples) the zero
game *0
in which case we can be safely write G == H instead of G == H +
*0.
(There are a few examples where J is the game *1).

Also note that the branch and boundary definitions can easily be
thought
to include branch with parameters and boundary with parameters
equivalences,
but not any such equivalence has been found yet.

The definitions that look more alike are the 4, 3, 2, 1 (so we’ll
keep
those instead of the 3′ , 2′ , 1′ ). In addition, it is not impossible
that
there may be examples of equivalences that are a mixture of two types
(of 3,
2 and 1). These definitions can easily be further expanded to include
such
mixed types. Therefore, types 3, 2 and 1 can be thought to be as just
cases
of a general type (and type 4 as type 3 with zero parameters). If
someone
can make a more general definition that will include all these types,
please
do!

A complete list of known equivalences of types 4 and 3 would
unnecessary
grow even more hugely this post, so we’ll write only two or three
examples
of them. For types 2 and 1, the number of known equivalences is small,
so
here’s a complete known list.

Type 4 (position) examples
0.}] == 12.}] == *0
1.}] == 0.0.0.}] == *1
0.0.}] == 2+

Type 3 (partial position) examples
}2A.} == }0.A.}
}22A.} == }2A.} + *1

Type 2 (boundary) known list
.1. == .22.
.abcabc. == .abacbc.

Type 1 (branch) known list
-aa- == -2-
-abab- == -a2a-
-abcacb- == -abcbac- (a recent find)

and here’s an example of mixed 1 and 3 type:
( -A- , }A.} ) == ( -2- , }} )

which while has been in wide use for years, has never been recorded
this way
in our knowledge!
It would be really great if there were more findings of such mixed
type
equivalences.

From: Yper Cube <yperc…@gmail.com>

Date: Mon, 2 Feb 2009 23:18:09 +0200

Local: Mon, Feb 2 2009 11:18 pm

Subject: Re: Types of equivalence

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One more generalization

If we change Def.4 from:

Def.4 (type 4) : for two positions G, H, and a (position of an
impartial
game) J
we define G == H + J if,
G is equivalent to the game H + J (in Conway’s notion), meaning that
they
have the same reduced tree and thus are completely equivalent in both
normal
and misere play.

to:

Def.4n (type 4n) : for two positions G, H, and a (position of an
impartial
game) J
we define G =n= H + J if,
G has the same nim value with the game H + J when both are played in
normal
play,
or (equiv.) n(G) = n(H+J) (n(G) is the nim value of G, etc.)
or (equiv.) n(G) = n(H) + n(J) (this is nim-addition)

we can then continue to define types 3n, 2n and 1n (without any
change in
those definitions, except using =n= instead of == )!

It is possible that several equivalences exist that are true only in
normal
play (or not yet easy to prove for misere play), for example }2.2.2.2A.}
=n=

}2.2.2.2.2.2A.} may be true (still not easy but surely easier that
the

general one).

We can further add more variations of Def.4 (and following of 3, 2,
1):

Def.4m (type 4m) : for two positions G, H, and a (position of an
impartial
game) J
we define G =m= H + J if,
G has the same misere Grundy value with the game H + J (when both
are
played in misere play).
or (equiv.) n-(G) = n-(H+J) (n-(G) is the misere nim value of
G, etc.)

Def.4nm (type 4nm) : for two positions G, H, and a (position of
an
impartial game) J
we define G =nm= H + J if,
G =n= H + J and G =m= H + J

Def.4g (type 4g) : for two positions G, H, and a (position of an
impartial
game) J
we define G =g= H + J if,
G and H + J have the same genus sequence.

Def.4N (type 4N) : for two positions G, H, and a (position of an
impartial
game) J
we define G =N= H + J if,
(G is a win for the first player in normal play) iff (H + J is a win
for
the first player in normal play).

Def.4M (type 4M) : for two positions G, H, and a (position of an
impartial
game) J
we define G =M= H + J if,
(G =N= H + J) and (G =M= H + J)

Def.4NM (type 4NM) : for two positions G, H, and a (position of
an
impartial game) J
we define G =NM= H + J if,
(G is a win for the first player in both normal and misere play) iff (H
+ J
is a win for the first player in both normal and misere play).

We can then easily prove that

(G == H) => (G =g= H) => (G =nm= H) => (G =NM= H)
and
(G =nm= H) => (G =n= H) => (G =N= H)
and
(G =nm= H) => (G =m= H) => (G =M= H)
and
(G =NM= H) => (G =N= H)
and
(G =NM= H) => (G =M= H)

where G and H can be positions or partial positions or boundaries
or
branches.

It’s not that i’m optimistic that we’ll find soon any such
equivalence,
(except for the =n= case where i would be indeed optimistic) but it may
be
useful to have a way of stating them in a common format when (and if)
we
find some.

From: Yper Cube <yperc…@gmail.com>

Date: Mon, 2 Feb 2009 23:40:50 +0200

Local: Mon, Feb 2 2009 11:40 pm

Subject: Re: Types of equivalence

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Correction for definitions 4M and 4NM:

Def.4M (type 4M) : for two positions G, H, and a (position of an
impartial
game) J
we define G =M= H + J if,
(G is a win for the first player in misere play) iff (H + J is a win for
the
first player in misere play).

Def.4NM (type 4NM) : for two positions G, H, and a (position of
an
impartial game) J
we define G =NM= H + J if,
(G =N= H + J) and (G =M= H + J)

(1) Referential equivalence is established when the words in
the source language (SL) refer to the same objects in the world as the
words in the target language (TL). (2) Connotative equivalence
is established when the words in both languages and texts trigger the
same associations and connotations. (3) Pragmatic equivalence
refers to words in both languages having the same effect on the readers
in both languages. (4) Contextual equivalence is established
when words in both languages are used in the same or similar contexts.
(5) Formal equivalence refers to words in both languages having
similar phonological or orthographic features. (6) Textual
equivalence refers to aspects of cohesion and coherence which are
similar in both texts and languages.