Model of hierarchical complexity
The model of hierarchical complexity, is a framework for scoring how complex a behavior is. It quantifies the order of hierarchical complexity of a task based on mathematical principles of how the information is organized and of information science

# The model of hierarchical complexity
The Model of Hierarchical Complexity, which has been presented as a formal theory , is a framework for scoring how complex a behavior is. Developed by Commons , it quantifies the order of hierarchical complexity of a task based on mathematical principles of how the information is organized , and of information science . Its forerunner was the General Stage Model . It is a model in mathematical psychology.
Behaviors that may be scored include those of individual humans or their social groupings (e.g., organizations, governments, societies), animals, or machines. It enables scoring the complexity of human reasoning in any domain. It is cross-culturally valid. The reason it applies cross-culturally is that the scoring is based on the mathematical complexity of the hierarchical organization of information. Scoring does not depend upon the content of the information (e.g., what is done, said, written, or analyzed) but upon how the information is organized.
The MHC is a non-mentalistic model of developmental stages. It specifies 14 orders of hierarchical complexity and their corresponding stages. It is different from previous proposals about developmental stage applied to humans . Instead of attributing behavioral changes across a persons age to the development of mental structures or schema, this model posits that task sequences form hierarchies that become increasingly complex. Because less complex tasks must be completed and practiced before more complex tasks can be acquired, this accounts for the developmental changes seen, for example, in individual persons performance of tasks. (For example, a person cannot perform arithmetic until the numeral representations of numbers are learned. A person cannot multiply numbers until addition is learned). Furthermore, previous theories of stage have confounded the stimulus and response in assessing stage by simply scoring responses and ignoring the task or stimulus. The Model of Hierarchical Complexity separates the task or stimulus from the performance. The participants performance on a task of a given complexity represents the stage of developmental complexity.

# Development as vertical complexity of tasks performed
One major basis for this developmental theory is task analysis. The study of ideal tasks, including their instantiation in the real world, has been the basis of the branch of stimulus control called psychophysics. Tasks are defined as sequences of contingencies, each presenting stimuli and each requiring a behavior or a sequence of behaviors that must occur in some non-arbitrary fashion. The complexity of behaviors necessary to complete a task can be specified using the horizontal complexity and vertical complexity definitions described below. Behavior is examined with respect to the analytically-known complexity of the task.
Tasks are quantal in nature. They are either completed correctly or not completed at all. There is no intermediate state. For this reason, the Model characterizes all stages as hard and distinct.  The orders of hierarchical complexity are quantized like the electron atomic orbitals around the nucleus. Each task difficulty has an order of hierarchical complexity required to complete it correctly. Since tasks of a given order of hierarchical complexity require actions of a given order of hierarchical complexity to perform them, the stage of the participants performance is equivalent to the order of complexity of the successfully completed task. The quantal feature of tasks is thus particularly instrumental in stage assessment because the scores obtained for stages are likewise discrete.
Every task contains a multitude of subtasks (Overton, 1990). When the subtasks are carried out by the participant in a required order, the task in question is successfully completed. Therefore, the model asserts that all tasks fit in some sequence of tasks, making it possible to precisely determine the hierarchical order of task complexity. Tasks vary in complexity in two ways: either as horizontal (involving classical information); or as vertical (involving hierarchical information).

### Horizontal (classical information) complexity
Classical information describes the number of yes-no questions it takes to do a task. For example, if one asked a person across the room whether a penny came up heads when they flipped it, their saying heads would transmit 1 bit of horizontal information. If there were 2 pennies, one would have to ask at least two questions, one about each penny. Hence, each additional 1-bit question would add another bit. Let us say they had a four-faced top with the faces numbered 1, 2, 3, and 4. Instead of spinning it, they tossed it against a backboard as one does with dice in a game of craps.  Again, there would be 2 bits. One could ask them whether the face had an even number. If it did, one would then ask if it were a 2. Horizontal complexity, then, is the sum of bits required by just such tasks as these.

### Vertical (hierarchical) complexity
Hierarchical complexity refers to the number of recursions that the coordinating actions must perform on a set of primary elements. Actions at a higher order of hierarchical complexity: (a) are defined in terms of actions at the next lower order of hierarchical complexity; (b) organize and transform the lower-order actions (see Figure 2); (c) produce organizations of lower-order actions that are new and not arbitrary, and cannot be accomplished by those lower-order actions alone. Once these conditions have been met, we say the higher-order action coordinates the actions of the next lower order.
To illustrate how lower actions get organized into more hierarchically complex actions, let us turn to a simple example.  Completing the entire operation 3 x (4 + 1) constitutes a task requiring the distributive act. That act non-arbitrarily orders adding and multiplying to coordinate them. The distributive act is therefore one order more hierarchically complex than the acts of adding and multiplying alone; it indicates the singular proper sequence of the simpler actions. Although simply adding results in the same answer, people who can do both display a greater freedom of mental functioning. Thus, the order of complexity of the task is determined through analyzing the demands of each task by breaking it down into its constituent parts.
The hierarchical complexity of a task refers to the number of concatenation operations it contains, that is, the number of recursions that the coordinating actions must perform. An order-three task has three concatenation operations. A task of order three operates on a task of order two and a task of order two operates on a task of order one (a simple task).

# Stages of development
The notion of stages is fundamental in the description of human, organismic, and  machine evolution.  Previously it has been defined in some ad hoc ways.  Here, it is described formally in terms of the Model of Hierarchical Complexity (MHC).

## Formal definition of stage
Since actions are defined inductively, so is the function h, known as the order of the hierarchical complexity. To each action A, we wish to associate a notion of that action's hierarchical complexity, h(A). Given a collection of actions A and a participant S performing A, the stage of performance of S on A is the highest order of the actions in A completed successfully at least once, i.e., it is
stage(S, A) = max{h(A) | A (epsilon) A and A completed successfully by S}. Thus, the notion of stage is discontinuous, having the same gaps as the orders of hierarchical complexity. This is in agreement with previous definitions (Commons et al., 1998; Commons & Miller, 2001; Commons & Pekker, 2007).
Because MHC stages are conceptualized in terms of the hierarchical complexity of tasks rather than in terms of mental representations (as in Piagets stages), the highest stage represents successful performances on the most hierarchically complex tasks rather than intellectual maturity. Table 1 gives descriptions of each stage.

## Stages of hierarchical complexity
Table 1. Stages described in the Model of Hierarchical Complexity

# Relationship of MHC with Piagets theory
There are some commonalities between the Piagetian and Commons notions of stage and many more things that are different.  In both, one finds:
1. Higher order actions defined in terms of lower order actions. This forces the hierarchical nature of the relations and makes the higher order tasks include the lower ones and requires that lower order actions are hierarchically contained within the relative definitions of the higher order tasks.
2. Higher order of complexity actions organize those lower order actions. This makes them more powerful. Lower order actions are organized by the actions with a higher order of complexity, i.e., the more complex tasks.
What Commons et al. (1998) have added includes:
3. Higher order of complexity actions organize those lower order actions in a non-arbitrary way.
This makes it possible for the Models application to meet real world requirements, including the empirical and analytic. Arbitrary organization of lower order of complexity actions, possible in the Piagetian theory, despite the hierarchical definition structure, leaves the functional correlates of the interrelationships of tasks of differential complexity formulations ill-defined. The following dimensions are inherent in the application:
- Task and performance are separated
- All tasks have an order of hierarchical complexity
- There is only one sequence of orders of hierarchical complexity.
- Hence, there is structure of the whole for ideal tasks and actions
- There are gaps between the orders of hierarchical complexity
- Stage is defined as the most hierarchically complex task solved.
- There are gaps in Rasch Scaled Stage of Performance.
- Performance stage is different task area to task area.
- There is no structure of the wholehorizontal decalagefor performance. It is not inconsistency in thinking within a developmental stage. Decalage is the normal modal state of affairs.

# Orders and Their Corresponding Stages
The MHC specifies 15 orders of hierarchical complexity and their corresponding stages, showing that each of Piagets substages, in fact, are hard stages. Commons also adds four postformal stages: Systematic stage 11, Metasystematic stage 12, Paradigmatic stage 13, and Crossparadigmatic stage 14.  It may be the Piaget's consolidate formal stage is the same as the systematic stage.  There is one other difference in the orders and stages.  At the suggestion of Biggs and Biggs, the sentential stage 5 was added.  The sequence is as follows: (0) computory, (1) sensory & motor, (2) circular sensory-motor, (3) sensory-motor, (4) nominal, the new (5) sentential, (6) preoperational, (7) primary, (8) concrete, (9) abstract, (10) formal, and the four postformal: (11) systematic, (12) metasystematic, (13) paradigmatic, and (14) cross-paradigmatic.  The first four stages (0-3) correspond to Piagets sensorimotor stage at which infants and very young children perform.  The sentential stage was added at Fischers suggestion (1981, personal communication) citing Biggs & Collis (1982).  Adolescents and adults can perform at any of the subsequent stages.  MHC stages 4 through 5 correspond to Piagets pre-operational stage; 6 through 8 correspond to his concrete operational stage; and 9 through 11 correspond to his formal operational stage.
The three highest stages in the MHC are not represented in Piagets model.  These stages from the Model of Hierarchical Complexity have extensively influenced the field of Positive Adult Development.  Few individuals perform at stages above formal operations. More complex behaviors characterize multiple system models (Kallio, 1995; Kallio & Helkama, 1991). Some adults are said to develop alternatives to, and perspectives on, formal operations. They use formal operations within a higher system of operations and transcend the limitations of formal operations. In any case, these are all ways in which these theories argue for and present converging evidence that adults are using forms of reasoning that are more complex than formal operations with which Piagets model ended.

# Empirical research using the model of hierarchical complexity
The MHC has a broad range of applicability.  The mathematical foundation of the model makes it an excellent research tool to be used by anyone examining performance that is organized into stages.  It is designed to assess development based on the order of complexity which the individual utilizes to organize information.  The MHC offers a singular mathematical method of measuring stages in any domain because the tasks presented can contain any kind of information. The model thus allows for a standard quantitative analysis of developmental complexity in any cultural setting.  Other advantages of this model include its avoidance of mentalistic or contextual explanations, as well as its use of purely quantitative principles which are universally applicable in any context.
The following can use the Model of Hierarchical Complexity to quantitatively assess developmental stages:
- Cross-cultural developmentalists;
- Animal developmentalists;
- Evolutionary psychologists;
- Organizational psychologists;
- Developmental political psychologists;
- Learning theorists;
- Perception researchers;
- History of science historians;
- Educators;
- Therapists;
- Anthropologists.
The following list shows the large range of domains to which the Model has been applied.  In one representative study, Commons, Goodheart, and Dawson (1997) found, using Rasch (1980) analysis, that hierarchical complexity of a given task predicts stage of a performance, the correlation being r = .92. Correlations of similar magnitude have been found in a number of the studies.

## List of examples
List of examples of tasks studied using the Model of Hierarchical Complexity or Fischers Skill Theory (1980):
- Algebra (Commons, in preparation)
- Animal stages (Commons & Miller, 2004)
- Atheism (Commons-Miller, 2005)
- Attachment and Loss (Commons, 1991; Miller & Lee, 2000)
- Balance beam and pendulum (Commons, Goodheart, & Bresette, 1995; Commons, Pekker, et al, 2007)
- Contingencies of reinforcement (Commons, in preparation)
- Counselor stages (Lovell, 2004)
- Empathy of Hominids (Commons & Wolfsont, 2002)
- Epistemology (Kitchener & King, 1990; Kitchener & Fischer, 1990)
- Evaluative reasoning (Dawson, 2000)
- Four Story problem (Commons, Richards & Kuhn, 1982; Kallio & Helkama, 1991)
- Good Education (Dawson-Tunik, 2004)
- Good Interpersonal (Armon, 1989)
- Good Work (Armon, 1993)
- Honesty and Kindness (Lamborn, Fischer & Pipp, 1994)
- Informed consent (Commons & Rodriguez, 1990, 1993; Commons, Goodheart, Rodriguez, & Gutheil, 2006; Commons, Rodriguez, Adams, Goodheart, Gutheil, & Cyr, 2007).
- Language stages (Commons, et al., 2007)
- Leadership before and after crises (Oliver, 2004)
- Loevinger=s Sentence Completion task (Cook-Greuter, 1990)
- Moral Judgment, (Armon & Dawson, 1997; Dawson, 2000)
- Music (Beethoven) (Funk, 1989)
- Orienteering (Commons, in preparation)
- Physics tasks (Inhelder & Piaget, 1958)
- Political development (Sonnert & Commons, 1994)
- Relationships (Armon, 1984a, 1984b)
- Report patient=s prior crimes (Commons, Lee, Gutheil, et. al., 1995)
- Social perspective-taking (Commons & Rodriguez, 1990; 1993)
- Spirituality (Miller & Cook-Greuter, 2000)
- Tool Making of Hominids (Commons & Miller 2002)
- Views of the Agood life@ (Armon, 1984c; Danaher, 1993; Dawson, 2000; Lam, 1995)
- Workplace culture (Commons, Krause, Fayer, & Meaney, 1993)
- Workplace organization (Bowman, 1996a, 1996b)
- Writing (Commons & DeVos, 1985)

## Copyright permissions
Portions of this article are from Applying the Model of Hierarchical Complexity by  Commons, M. L., Miller, P. M., Goodheart, E. A., Danaher-Gilpin, D., Locicero, A., Ross, S. N. Unpublished manuscript. Copyright 2007 by Dare Association, Inc. Available from Dare Institute, commons@tiac.net. Reproduced and adapted with permission of the publisher. Portions of this article are also from Introduction to the Model of Hierarchical Complexity by M. L. Commons, in the Behavioral Development Bulletin, 13, 1-6 (/).  Copyright 2007 Martha Pelaez. Reproduced with permission of the publisher.