4-01-2019, 11:52 a.m.

OTSoft 2.5, release date 1/17/2017

1. Result

A ranking was found that generates the correct outputs.

Stratum	Constraint Name	Abbreviation
Stratum #1	*A	*A
	Id-OO(ATR)	Id-OO(ATR)
Stratum #2	Agree	Agree
Stratum #3	Id-IO(ATR)	Id-IO(ATR)
	Id(IO)-ATR Initial	Id(IO)-ATR Initial

2. Tableaux

/i ] I/:	*A	Id-OO(ATR)	Agree	Id-IO(ATR)	Id(IO)-ATR Initial
☞   i ] i				*
i ] I			*!
I ] I		*!	*		*

/a ] i/:	*A	Id-OO(ATR)	Agree	Id-IO(ATR)	Id(IO)-ATR Initial
☞   a ] I				*
a ] i			*!
A ] i	*!	*		*	*

/i ] a/:	*A	Id-OO(ATR)	Agree	Id-IO(ATR)	Id(IO)-ATR Initial
☞   i ] a			*
I ] a		*!		*
i ] A	*!			*

/i ] a ] i/:	*A	Id-OO(ATR)	Agree	Id-IO(ATR)	Id(IO)-ATR Initial
☞   i ] a ] I			*	*
i ] a ] i			**!
I ] a ] I		*!		**	*
i ] A ] i	*!			*

/i a/:	*A	Id-OO(ATR)	Agree	Id-IO(ATR)	Id(IO)-ATR Initial
☞   I a				*	*
i a			*!
i A	*!			*

/i I/:	*A	Id-OO(ATR)	Agree	Id-IO(ATR)	Id(IO)-ATR Initial
☞   i i				*
I I				*	*!
i I			*!

3. Status of Proposed Constraints: Necessary or Unnecessary

Constraint	Status
Agree	Necessary
*A	Necessary
Id-OO(ATR)	Necessary
Id(IO)-ATR Initial	Necessary
Id-IO(ATR)	Not necessary

4. Ranking Arguments, based on the Fusional Reduction Algorithm

This run sought to obtain the Skeletal Basis, intended to keep each final ranking argument as pithy as possible.

Original set of ERCs

Index	ERC	Reason
1	WeeLe	for /i ] I/, i ] i >> i ] I, a ] I >> a ] i, i ] a ] I >> i ] a ] i, i i >> i I
2	WeWLW	for /i ] I/, i ] i >> I ] I
3	LWeWe	for /i ] a/, i ] a >> i ] A
4	LeWWe	for /i ] a/, i ] a >> I ] a
5	LeWWW	for /i ] a ] i/, i ] a ] I >> I ] a ] I
6	LWeee	for /i ] a ] i/, i ] a ] I >> i ] A ] i
7	WeeLL	for /i a/, I a >> i a
8	eWeeL	for /i a/, I a >> i A

Recursive ranking search

Recursive search has now reached this location in the search tree: 1

Fusion of this ERC set is: LWWLL

The following ERCs form the total information-loss residue:

WeeLe

WeWLW

LWeWe

LeWWe

LeWWW

LWeee

WeeLL

eWeeL

Fusion of total residue: LWWLL

Skeletal basis of the fusion: eWWee

eWWee has no L's, so it cannot be retained in the Skeletal Basis.

Recursive search has now reached this location in the search tree: 1, 2

Current set of ERCs is based on constraint #2, *A

Working with the following ERC set:

WeeLe

WeWLW

LeWWe

LeWWW

WeeLL

Fusion of this ERC set is: LeWLL

The following ERCs form the total information-loss residue:

WeeLe

WeeLL

Fusion of total residue: WeeLL

Skeletal basis of the fusion: LeWee

LeWee includes at least one L and thus is not entailed by WeeLL.

Thus it may be retained in the Skeletal Basis of ERCs.

Recursive search has now reached this location in the search tree: 1, 2, 3

Current set of ERCs is based on constraint #3, Id-OO(ATR)

Working with the following ERC set:

WeeLe

WeeLL

Fusion of this ERC set is: WeeLL

The following ERCs form the total information-loss residue:

WeeLL has a null residue and thus may be retained in the Skeletal Basis of ERCs.

Recursive search has now reached this location in the search tree: 1, 3

Current set of ERCs is based on constraint #3, Id-OO(ATR)

Working with the following ERC set:

WeeLe

LWeWe

LWeee

WeeLL

eWeeL

Fusion of this ERC set is: LWeLL

The following ERCs form the total information-loss residue:

WeeLe

WeeLL

Fusion of total residue: WeeLL

Skeletal basis of the fusion: LWeee

LWeee includes at least one L and thus is not entailed by WeeLL.

Thus it may be retained in the Skeletal Basis of ERCs.

([1, 3, 2] not novel so not checked)

5. Ranking argumentation: Final result

The following set of ERCs forms the Skeletal Basis for the ERC set as a whole, and thus encapsulates the available ranking information.

LeWee	Must dominate these constraints
WeeLL
LWeee

The final rankings obtained are as follows:

Id-OO(ATR) >> Agree	Must dominate these constraints
Agree >> { Id-IO(ATR), Id(IO)-ATR Initial }
*A >> Agree

6. Mini-Tableaux

The following small tableaux may be useful in presenting ranking arguments. They include all winner-rival comparisons in which there is just one winner-preferring constraint and at least one loser-preferring constraint. Constraints not violated by either candidate are omitted.

/i ] I/:	Agree	Id-IO(ATR)
☞   i ] i		*
i ] I	*

/a ] i/:	Agree	Id-IO(ATR)
☞   a ] I		*
a ] i	*

/i ] a ] i/:	Agree	Id-IO(ATR)
☞   i ] a ] I	*	*
i ] a ] i	**

/i ] a ] i/:	*A	Agree	Id-IO(ATR)
☞   i ] a ] I		*	*
i ] A ] i	*		*

/i a/:	Agree	Id-IO(ATR)	Id(IO)-ATR Initial
☞   I a		*	*
i a	*

/i a/:	*A	Id-IO(ATR)	Id(IO)-ATR Initial
☞   I a		*	*
i A	*	*

/i I/:	Agree	Id-IO(ATR)
☞   i i		*
i I	*