Saturday, December 31, 2016

Welcome

I study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.

The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.

In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.

I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.

Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.

Saturday, September 24, 2016

Parliamentary Parties In A Presidential System and the Failure of the Principle of Subsidiarity

1.0 Introduction

Some have argued that the Republican Party, in the United States of American, has been acting, since Newt Gringrich's speakership of the house, more like a parliamentary party1. And that this creates tensions in a presidential system2, like the USA. I think I have located another tension that, so far as I know, had not been previously identified when I started this post, months ago3.

People line up in local elections often for local reasons, to pursue local interests. In mass publics, even the political leaders in town, district, city, county, and municipal systems cannot be expected be knowledgeable about national issues and political ideology. In big-tent parties, the aggregate of such local movements need not form coherent ideologies. But when at least one national party is dominated by ideological beliefs, local politics might tend to be seen through an ideological lens. Not only might local political bickering become more bitter and rancorous, local politicians might become less responsive to specific characteristics of their areas. Federalism will work worse. Delegating decisions to the lowest authority possible among municipalities, states, and nations does not necessarily lead to more democratically responsive decisions, in some sense.

2.0 Local Politics

County-level splits in big-tent political parties do not result in ideological shifts. Suppose there are both right and left wings in two dominant political parties in a country, and these ideological spectra overlap. One party might be more dominant in one region than another. How urban and rural populations; ethnic groups; landholders, financiers, industrialists, professionals, small business owners, and workers line up might vary among regions. Once, say, in the 1950s, the Democrats were the party in the USA of southern whites and urban ethnic immigrants from southeastern Europe. And Republicans were simultaneously the party of African-Americans and big business4. Supporting a party at a local level, switching sides, and so not need not reflect strong ideological view in such circumstances. It could be a matter of simply seeking more resources for an interest group.

Once upon a time in Chicago, the Democratic party was extremely dominant, and the party was run like many another big city machine. Harold Washington was a successful reform candidate who became major. The old-time machine politicians had to go somewhere, and they became Republicans. Whatever local tensions were involved in it, this kind of local party split and reforming of one party did need not align with any national movement.

The county I live in has two urban centers. As I understand it, the Democrats are traditionally dominant in the larger city, and the Republicans are dominant in mine. We have had in both cities, in my memory, mayors that were either independent - in the sense, that they ran on neither party line - or bipartisan, in that he ran on both.

So there are two examples of alignments in local politics that might be said to be more about interest groups, and less about ideological movements. Politics in the USA has been becoming more ideological and falling along a one-dimensional continuum (Hare & Poole, 2013). And I think that has affected local politics.

Footnotes
  1. For this post, I am more interested in the first paragraph in the following quotation. The second paragraph is probably the most widely quoted passage from this book, partly because of Ornstein's standing among right-leaning think tanks and partly because of an accompanying Washington Post editorial:
  2. "...we identify two overriding sources of dysfunction. The first is the serious mismatch between the political parties, which have become as vehemently adversarial as parliamentary parties, and a governing system that, unlike a parliamentary democracy, makes it extremely difficult for majorities to act. Parliamentary-style parties in a separation-of-powers government are a formula for willful obstruction and policy irresolution. Sixty years ago, Austin Ranney, an eminent political scientist, wrote a prophetic dissent to a famous report by an American Political Science Association committee entitled 'Toward a More Responsible Two-Party System.' The report, by prominent political scientists frustrated with the role of conservative Southern Democrats in blocking civil rights and other social policy, issued a clarion call for more ideologically coherent, internally unified, and adversarial parties in the fashion of a Westminister-style parliamentary democracy like Britain or Canada. Ranney powerfully argued that such parties would be a disaster within the American constitutional system, given our separation of powers, separately elected institutions, and constraints on majority rule that favor cross-party coalitions and compromise. Time has proven Ranney dead right - we now have the kinds of parties the report desired, and it is disastrous.

    The second is the fact that, however awkward it may be for the traditional press and nonpartisan analysts to acknowledge, one of the two major parties, the Republican Party, has become an insurgent outlier - ideologically extreme; contemptuous of the inherited social and economic policy regime; scornful of compromise; unpersuaded by conventional understanding of facts, evidence, and science; and dismissive of the legitimacy of its political opposition. When one party moves this far from the center of American politics, it is extremely difficult to enact policies responsive to the country's most pressing challenges." -- Thomas Mann and Norman Ornstein (2012).

  3. From an agenda-setting paper on the differences between presidential and parliamentary systems:
  4. "...the president's strong claim to democratic, even plebiscitarian, legitimacy [stands out]... Following ...Walter Bagehot, ... a presidential system endows the incumbent with both the 'ceremonial' functions of a head of state and the 'effective' functions of a chief executive, thus creating an aura, a self-image, and a set of popular expectations which are all quite different from those associated with a prime minister, no matter how popular he may be.

    But what is most striking is that in a presidential system, the legislators, especially when they represent cohesive, disciplined parties that offer clear ideological and political alternatives, can also claim democratic legitimacy... [W]hen a majority of the legislature represents a political option opposed to the... president...[,] who has the stronger claim to speak on behalf of the people: the president or the legislative majority that opposes his policies? ... One might argue that the United States has successfully rendered such conflicts 'normal' and thus defused them... [T]he uniquely diffuse character of American political parties - which ironically, exasperates many American political scientists and leads them to call for responsible, ideologically disciplined parties - has something to do with it... [T]he development of modern political parties, particular in socially and ideologically polarized countries, generally exacerbates, rather than moderates, conflicts between the legislative and the executive." -- Juan Linz (1990): pp. 53-54.'

  5. But see Steven Rogers' study, highlighted by a Jeff Stein article at Vox.
  6. These are tendencies. It is part of my point that such tendencies might be violated, at some time in some specific locality.
Selected References
  • Christopher Hare and Keith T. Poole (2013). The Polarization of Contemporary American Politics.
  • Matt Grossmann and David A. Hopkins (2015). Ideological Republicans and Group Interest Democrats: The Asymmetry of American Politics, Perspectives on Politics, V. 13, No. 1 (Mar.): pp. 119-139.
  • Juan J. Linz (1990). The Perils of Presidentialism, Journal of Democracy, V. 1, No. 1 (Winter): pp. 51-69.
  • Thomas E. Mann and Norman J. Ornstein (2012). It's Even Worse than It Looks: How the American Constitutional System Collided with the New Politics of Extremism, Basic Books.
  • Steven Rogers (2016). National Forces in State Legislative Elections, AAPSS (Sep.): pp. 207-225.

Friday, September 09, 2016

Tim Lewins: "Economics, Intelligent-Design Theory, And Homeopathy"

Tim Lewens has written a popular introduction to the philosophy of science, The Meaning of Science: An Introduction to the Philosophy of Science. In his first substantial chapter, he writes about what distinguishes science from non-science. Karl Popper and the demarcation problem arise here. He needs examples of near sciences:

Consider the trio of economics, intelligent-design theory, and homeopathy. The only thing that unites these three endeavors is that their scientific status is regularly questioned in ways that provoke stormy debate. Is economics a science? On the one hand, like many sciences, it oozes both mathematics and authority. On the other hand it is poor at making predictions, and many of its practitioners are surprisingly blaseé when it comes to finding out about how real people think and behave. They would rather build models that tell us what would happen, under simplified circumstances, if people were perfectly rational. So perhaps economics is less like science, and more akin to The Lord of the Rings with equations: it is a mathematically sophisticated exploration of an invented world not much like our own.

In a later chapter, Lewens recognize that economics is a diverse discipline. He writes about some interesting analyses in economics. And then we get:

In contrast to these empirically rich forms of economic inquiry [associated with Sen and Kahneman], much work in neoclassical economics is concerned with the largely theoretical analysis of how markets would work if they were populated with individuals endowed with perfect rationality - in other words, creatures of fantasy. We might be tempted to classify these areas of economics as science fiction. Alternatively, we might think that this brand of economics tells us not how the world is but how the world ought to be, if only people would think straight...

I think Lewens is more complimentary to homeopathy than he is to economics. (He does have a bit more to say about economics than I have quoted.) Controlled experiments in medicine, I gather, consider one intervention as applied to a population. Advocates of homeopathic medicine claim to be treating a whole, particular person in a way which cannot be easily analyzed such reductionist experiments. This, no matter how hostile you may be to it, is an interesting claim for a philosopher to consider. Maybe what they advocate are placebos. Suppose you have a patient that is skeptical of big medicine. Would he react better to a placebo if it is administered in an alternative setting? What, ethically, could such a practitioner say when prescribing extremely diluted "medicine"?

I still am of the opinion that labelling a claim in economics as "science" or "non-science" should neither add nor subtract to its plausibility, over and above whatever empirical evidence and disciplinary arguments already do.

Friday, July 29, 2016

Emmanuelle Benicourt Influenced By Steve Keen?

I am thinking of absurdity number 3 below. I go a little further because I am amused by the well-established point with which I end this quotation.

"ABSURDITY No3 'For a price-taking firm, the demand curve for its own output is a horizontal line at the market price' (Unit 8.3)

This is false: the demand curve of a price-taking firm is not, and cannot be, horizontal: a firm supply, even if it is 'tiny', affects the price and then the demand of the good it produces.

The correct assumption should be that the firm believes that the demand curve is horizontal - an erroneous belief, but that is another story...

In their seminal article, Existence of an Equilibrium for a Competitive Economy, Kenneth Arrow and Gérard Debreu don't mention agents' beliefs but they,

'...instruct each production and consumption unit to behave as if the announcement of price p were the equilibrium value' (point 1.4.1, [Benicourt's] italics)

ABSURDITY No4 All agents are price-takers (competitive equilibrium)

...Now, any reasonable person will immediately ask: if all agents are price-takers, who set[s] prices? The e-Book answers (implicitly) this question with a circular reasoning...

Conclusion: 'A competitive market', as defined in the CORE e-Book, is not 'an approximation' of any existing market. It is not:

'...hard to find evidence of perfect competition' (Unit 8.3).

It is impossible.

The so-called 'competitive economy' model doesn't 'describe an idealised market structure' (Unit 8, p 44). It is not 'unrealistic' - any model is, by definition - it is irrelevant. In fact, it has nothing to do with capitalism. It can be considered, at most, as a variant of market-socialism models, with a benevolent planner setting prices, adding supplies and demands, etc." -- Emmanuelle Benicourt (2016). Is the CORE e-Book a possible solution to our problems? Real-World Economics Review, iss. no. 75, p. 135-142.

Tuesday, June 28, 2016

Getting Greater Weight For Your Vote May Not Give You Relatively More Power

1.0 Introduction

This post presents a perhaps surprising example of results from measuring political power in a system with weighted voting. I provide examples in which the weight of a person's vote is increased. Yet that voter, in some cases, gains no additional power, in some sense. In one case, by the measures of voting power considered here, the additional weight has no effect on the power of any voter. In another case, another player, with unchanged weight to his vote, is elevated in power with the voter whose weight is increased.

I find these results to be an interesting consequence of power measures. I have not yet found a simple example where the effect on the ranking of voting power is different for the three indices considered here. Nor have I found an example where a voter declines in power with an increase in the weight of his vote.

2.0 An Example of a Voting Game

A voting game is specified as a set of players, the number of votes needed to enact a bill into law (also referred to as passing a proposition), and the weights for the votes of each player. In considering voting games with a small number of players and weighted, unequal votes, one might think of such a game as describing a council or board of directors, where members represent blocs or geographic districts of varying sizes.

As example, consider a set, P, of four players, indexed from 0 through 3:

P = The set of players = {0, 1, 2, 3}

A common way to indicate the remaining parameters for a voting game is a tuple in which the first element is followed by a colon and the remaining elements are separated by commas:

(6: 4, 3, 2, 1)

The positive integer before the colon indicates the number of votes - six, in this case - needed to pass a proposition. The remaining integers are the weights of players' votes. In this case, the weight of Player 0's vote is 4, the weight of Player 1's vote is 3, and so on.

3.0 Two Power Indices

Consider all 16 possible subsets of the four players. These subsets are listed in the first column of Table 1. A subset of players is labeled a coalition. The second column indicates whether or not the coalition for that row has enough weighted votes to pass a proposition. If so, the characteristic function for that coalition is assigned the value unity. Otherwise, it gets the value zero. A player is decisive for a coalition if the player leaving the coalition will convert it from a winning to a losing coalition. The last four columns in Table 1 have entries of unity for each player that is decisive for each coalition. The last row in Table 1 provides a count, for each player, of the number of coalitions in which that player is decisive. The Penrose-Banzhaf power index, for each player, is the ratio of this total to the number of coalitions.

Table 1: Calculations for Penrose-Banzhaf Power Index
CoalitionCharacteristic
Function
Player
0123
{}v( {} ) = 00000
{0}v( {0} ) = 00000
{1}v( {1} ) = 00000
{2}v( {2} ) = 00000
{3}v( {3} ) = 00000
{0, 1}v( {0, 1} ) = 11100
{0, 2}v( {0, 2} ) = 11010
{0, 3}v( {0, 3} ) = 00000
{1, 2}v( {1, 2} ) = 00000
{1, 3}v( {1, 3} ) = 00000
{2, 3}v( {2, 3} ) = 00000
{0, 1, 2}v( {0, 1, 2} ) = 11000
{0, 1, 3}v( {0, 1, 3} ) = 11100
{0, 2, 3}v( {0, 2, 3} ) = 11010
{1, 2, 3}v( {1, 2, 3} ) = 10111
{0, 1, 2, 3}v( {0, 1, 2, 3} ) = 10000
Total:5331

The Shapley-Shubik power index considers the order in which players enter a coalition. For the example, one considers all 24 permutations for the players. The first column in Table 2 lists these permutation. For each row, a player gets an entry of unity in the appropriate one of the last four columns if including that player in a coalition, reading the entries in a permutation from left to right, creates a winning coalition. The Shapley-Shubik power index, for each player, is the ratio of the totals of each of the last four columns to the number of permutations.

Table 2: Calculations for the Shapley-Shubik Power Index
PermutationPlayer
0123
(0, 1, 2, 3)0100
(0, 1, 3, 2)0100
(0, 2, 1, 3)0010
(0, 2, 3, 1)0010
(0, 3, 1, 2)0100
(0, 3, 2, 1)0010
(1, 0, 2, 3)1000
(1, 0, 3, 2)1000
(1, 2, 0, 3)1000
(1, 2, 3, 0)0001
(1, 3, 0, 2)1000
(1, 3, 2, 0)0010
(2, 0, 1, 3)1000
(2, 0, 3, 1)1000
(2, 1, 0, 3)1000
(2, 1, 3, 0)0001
(2, 3, 0, 1)1000
(2, 3, 1, 0)0100
(3, 0, 1, 2)0100
(3, 0, 2, 1)0010
(3, 1, 0, 2)1000
(3, 1, 2, 0)0010
(3, 2, 0, 1)1000
(3, 2, 1, 0)0100
Total:10662

4.0 Three Power Indices for Three Voting Games

Table 3 summarizes and expands on the above calculations. The Penrose-Banzhaf power index need not sum over the players to unity. Accordingly, I break this index down into two indices, where the second index is normalized. The Shapley-Shubik power index is guaranteed to sum to unity. I introduce two other voting games, with corresponding power indices, presented in Tables 4 and 5.

Table 3: Power Indices for (6: 4, 3, 2, 1)
PlayerPenrose-Banzhaf Power IndexShapley-Shubik
Power Index
IndexNormalized
05/165/1210/24 = 5/12
13/163/12 = 1/46/24 = 1/4
23/163/12 = 1/46/24 = 1/4
31/161/122/24 = 1/12

Table 4: Power Indices for (6: 4, 2, 2, 1)
PlayerPenrose-Banzhaf Power IndexShapley-Shubik
Power Index
IndexNormalized
06/16 = 3/86/10 = 3/516/24 = 2/3
12/16 = 1/82/10 = 1/54/24 = 1/6
22/16 = 1/82/10 = 1/54/24 = 1/6
3000

Table 5: Power Indices for (5: 4, 2, 2, 1)
PlayerPenrose-Banzhaf Power IndexShapley-Shubik
Power Index
IndexNormalized
06/16 = 3/86/12 = 1/212/24 = 1/2
12/16 = 1/82/12 = 1/64/24 = 1/6
22/16 = 1/82/12 = 1/64/24 = 1/6
32/16 = 1/82/12 = 1/64/24 = 1/6

5.0 Constitutional Changes

Consider a change in the constitution, from one of the three voting games with tables in the previous section to another such game. The calculations allow one to measure the impact on voting power for any such change. To simplify matters, I consider only rankings of voting power. And, for these three voting games, the three power indices consider here happen to yield the same ranks, for any given voting game out of these three.

Accordingly, Table 6 shows changes in the rules (the "constitution") for these cases. The change to the rules on the right superficially strengthens Player 1, either by increasing the weight of Player 1's vote or requiring less votes to pass a resolution. As noted below, I am unsure what naive intuition might be for the second row. For the third vote, the number of votes needed to pass a proposition is altered such that a simple majority is needed before and after the change in weight.

Table 6: Changing the Rules to Strengthen the Players?
Starting GamePlayer RanksEnding GamePlayer Ranks
(6: 4, 2, 2, 1)0 > 1 = 2 > 3(6: 4, 3, 2, 1)0 > 1 = 2 > 3
(6: 4, 2, 2, 1)(5: 4, 2, 2, 1)0 > 1 = 2 = 3
(5: 4, 2, 2, 1)0 > 1 = 2 = 3(6: 4, 3, 2, 1)0 > 1 = 2 > 3

The first row shows a case where the weight of Player 1's vote increases, which might intuitively give him more power with respect to the apparently weaker Players 2 and 3. Yet this increase in weight also increases the power of Players 2 and 3, even though the weight of their votes does not change. And Player 1 remains equal in power to Player 2, both before and after the change. In fact, the change has no effect on the ranking of the players' voting power.

The second row shows a case where the votes needed to pass a measure declines, after the change in rules, from a super-majority to a simple majority, given the total of weighted votes. Would one expect such a constitutional amendment to strengthen the most powerful, or moderately powerful voters before the change? I find that this change raises the power of the weakest voter to the power of the middling voters. I am not sure this is counter-intuitive, unlike the other two rows.

The third row shows a case in which, like the first row, the weight of Player 1's vote increases. Both before and after the change, a simple majority, given the total of weighted votes, is needed to pass a proposition. This change makes Player 1 more powerful than the weakest player, as one might intuitively expect. But Player 2 is also made more powerful than the weakest player, despite the weight of his vote not varying. And Player 1 ends up no more powerful than Player 2. These effects on Player 2 seem counter-intuitive to me.

6.0 Conclusions

So my examples above have presented somewhat counter-intuitive results in voting games.

I gather that the Deegan-Packel and Holler-Packel are some other power indices I might find of interest. And Straffin (1994) is one paper that explains axioms that characterize some power index or other.

References
  • Donald P. Green and Ian Shapiro (1996). Pathologies of Rational Choice Theory: A Critique of Applications in Political Science, Yale University Press
  • P. Straffin (1994). Power and stability in politics. Handbook of Game Theory with Economic Applications, V. 2, Elsevier.

Wednesday, June 15, 2016

The History and Sociology of Game Theory: A Reading List

For me, this list is aspirational. I've read Mirowski and the Weintraub-edited book. I've just checked the Erickson book out of a library.

Monday, May 16, 2016

A Turing Machine for a Binary Counter

Table 1: Tape in Successive Start States
Input/Output TapeDecimal
tb00
tb11
tb102
tb113
tb1004
tb1015

1.0 Introduction

This post describes another program for a Turing Machine. This Turing machine implements a binary counter (Table 1). I do not think I am being original here. (Maybe this was in the textbook on computability and automata that I have been reading.)

2.0 Alphabet

Table 2: The Alphabet For The Input Tape
SymbolNumber Of
Occurrences
Comments
t1Start-of-tape Symbol
bPotentially InfiniteBlank
0Potentially InfiniteBinary Digit Zero
1Potentially InfiniteBinary Digit One

3.0 Specification of Valid Input Tapes

At start, the (input) tape should contain, in this order:

  • t, the start-of-tape symbol.
  • b, a blank.
  • A sequence of binary digits, with a length of at least one.

The above specification allows for any number of unnecessary leading zeros in the binary number on the tape. The head shall be at the blank following the start-of-tape symbol.

4.0 Specification of State

The machine starts in the Start state. Error is the only halting state. Table 3 describes some conditions, for a non-erroneous input tape, that states are designed to satisfy, on entry and exit. For the states GoToEnd, FindZero, CreateTrailingOne, Increment, and ResetHead, the Turing machine may experience many transitions that leaves the machine in that state after the state has been entered. When the state PauseCounter has been entered, the next increment of a binary number appears on the tape.

Table 3: States
StateSelected Conditions
On EntryOn Exit
StartThe head is immediately to the left of the binary number on the tape. (The binary number on the tape at this point is referred to as "the original binary number" below.)Same as the entry condition for GoToEnd.
GoToEndThe head is under the first digit of the binary number on the tape.Same as the entry condition for FindZero.
FindZeroThe head is under the last digit of the binary number on the tapeIf all digits in the original binary number are 1 and that number has not been updated with a leading zero, the head is under the first digit of the binary number on the tape. If the original binary number contained at least one digit 0, the head is under the location of the last instance of 0 in the original binary number, and that digit has been changed to a 1. Otherwise, the head is under the first digit in the binary number now on the tape, and that digit is now a 1 (having once been a leading zero).
CreateLeadingZeroAll the digits in the original binary number are 1. The head is under the first digit of the binary number on the tape.Same as the entry condition for CreateTrailingOne
CreateTrailingOneAll the digits in the original binary number are 1. The first digit in the original binary number has been replaced by 0. The head is under that first digit.The original binary number has been shifted one digit to the left, and a leading zero has been prepended to it. The head is under the last digit of the binary number now on the tape.
StepForwardIf all digits in the original binary number are 1, that number has been shifted one digit to the left, that number has been updated with a leading 0 which is now a 1, and the head is under that digit. Otherwise, the last instance of 0 in the original number has been updated to a 1, and the head is now under that digit tape.Same as the entry condition for Increment.
IncrementIf all digits in the original binary number are 1, that number has been shifted one digit to the left, that number has been updated with a leading 0 which is now a 1, and the head is under the next location on the tape. Otherwise, the last instance of 0 in the original number has been updated to a 1, and the head is now under the next location on the tape.Same as the entry condition for ResetHead. All the 1's to the right of the 0 updated to a 1 have themselves been updated to a 0.
ResetHeadThe head is under the last digit of the binary number on the tape, and that number is the successor of the original binary number.Same as the entry condition for PauseCounter.
PauseCounterThe head is immediately to the left of the binary number on the tape, and that number is the successor of the original binary number.

I think one could express the conditions in the above lengthy table as logical predicates. And one could develop a formal proof that the state transition rules in the appendix ensure that these conditions are met on entry and exit of the non-halting tape, at least for non-erroneous input tapes. I do not quite see how invariants would be used here. (When trying to think rigorously about source code, I attempt to identify invariants for loops.)

5.0 Length of Tape and the Number of States

Suppose the state PauseCounter was a halting state. Then this Turing machine would be a linear bounded automaton. In the Chomsky hierarchy, automata that accept context-sensitive languages need not be more general than linear bound automata.

The program for this Turing machine consists of 10 states. The number of characters on the tape grows at the rate O(log2 n), where n is the number of cycles through the start state. I gather the above instructions could be easily modified to not use any start-of-tape symbol. Anyways, 20 people seems more than sufficient for the group activity I have defined, for this particular Turing machine.

Appendix A: State Transition Tables

This appendix provides detail specification of state transition rules for each of the non-halting states. I provide these rules by tables, with each table showing a pair of states.

Table A-1: Start and GoToEnd
StartGoToEnd
ttErrorttError
bForwardsGoToEndbBackwardsFindZero
00Error0ForwardsGoToEnd
11Error1ForwardsGoToEnd
Table A-2: FindZero and CreateLeadingZero
FindZeroCreateLeadingZero
ttErrorttError
bForwardsCreateLeadingZerobbError
01StepForward00Error
1BackwardsFindZero10CreateTrailingOne
Table A-3: CreateTrailingOne and StepForward
CreateTrailingOneStepForward
ttErrorttError
b1FindZerobbError
0ForwardsCreateTrailingOne00Error
1ForwardsCreateTrailingOne1ForwardsIncrement
Table A-4: Increment and ResetHead
IncrementResetHead
ttErrorttError
bBackwardsResetHeadbbPauseCounter
0ForwardsIncrement0BackwardsResetHead
10Increment1BackwardsResetHead
Table A-5: PauseCounter
PauseCounter
ttError
bbStart
00Error
11Error