The model of hierarchical complexity is a framework for scoring how complex a behavior is, such as verbal reasoning or other cognitive tasks.^[1] It quantifies the order of hierarchical complexity of a task based on mathematical principles of how the information is organized, in terms of information science.^[2] This model has been developed by Michael Commons and others since the 1980s.

Overview[edit source]

The model of hierarchical complexity (MHC) is a formal theory and a mathematical psychology framework for scoring how complex a behavior is.^[3]Developed by Michael Lamport Commons and colleagues,^[4] it quantifies the order of hierarchical complexity of a task based on mathematical principles of how the information is organized,^[5] in terms of information science.^[6][7][8] Its forerunner was the general stage model.^[6]

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 hierarchical complexity of task accomplishment in any domain.^[9] It is based on the very simple notions that higher order task actions:^[2]

1. are defined in terms of the next lower ones (creating hierarchy);
2. organize the next lower actions;
3. organize lower actions in a non-arbitrary way (differentiating them from simple chains of behavior).

It is cross-culturally and cross-species 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.^[2] It specifies 16 orders of hierarchical complexity and their corresponding stages. It is different from previous proposals about developmental stage applied to humans;^[10] instead of attributing behavioral changes across a person’s age to the development of mental structures or schema, this model posits that task sequences of task behaviors 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 complex tasks. (For example, a person cannot perform arithmetic until the numeral representations of numbers are learned. A person cannot operationally multiply the sums of numbers until addition is learned).

The creators of the MHC claim that previous theories of stage have confounded the stimulus and response in assessing stage by simply scoring responses and ignoring the task or stimulus.^[2] The MHC separates the task or stimulus from the performance. The participant’s performance on a task of a given complexity represents the stage of developmental complexity.

Vertical complexity of tasks performed[edit source]

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 (tertium non datur). For this reason, the model characterizes all stages as P-hard and functionally distinct. The orders of hierarchical complexity are quantized like the electron atomic orbitalsaround the nucleus: each task difficulty has an order of hierarchical complexity required to complete it correctly, analogous to the atomic Slater determinant. Since tasks of a given quantified order of hierarchical complexity require actions of a given order of hierarchical complexity to perform them, the stage of the participant’s task 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.^[11] 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 configured 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).^[2]

Horizontal complexity[edit source]

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 complexity[edit source]

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 qualitatively 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 × (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. Additional layers of abstraction can be applied. 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 one or more tasks of vertical order two and a task of order two operates on one or more tasks of vertical order one (the simplest tasks).

Stages of development[edit source]

Stage theories describe human organismic and/or technological evolution as systems that move through a pattern of distinct stages over time. Here development is described formally in terms of the model of hierarchical complexity (MHC).

Formal definition of stage[edit source]

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 ∈ A and A completed successfully by S}. Thus, the notion of stage is discontinuous, having the same transitional gaps as the orders of hierarchical complexity. This is in accordance with previous definitions.^[4][12][3]

Because MHC stages are conceptualized in terms of the hierarchical complexity of tasks rather than in terms of mental representations (as in Piaget’s stages), the highest stage represents successful performances on the most hierarchically complex tasks rather than intellectual maturity.

Stages of hierarchical complexity[edit source]

The following table gives descriptions of each stage in the MHC.

Stages described in the model of hierarchical complexity (adapted from Commons, Crone-Todd, & Chen, 2014)

Order or stage	What they do	How they do it	End result
0 – calculatory	Exact computation only, no generalization	Human-made programs manipulate 0, 1, not 2 or 3.	Minimal human result. Literal, unreasoning computer programs (at Turing‘s alpha layer) act in a way analogous to this stage.
1 – automatic	Engage in a single “hard-wired” action at a time, no respondent conditioning	Respond, as a simple mechanism, to a single environmental stimulus	Single celled organisms respond to a single stimulus in a way analogous to this stage
2 – sensory or motor	Discriminate in a rotefashion, stimuligeneralization, move	Move limbs, lips, toes, eyes, elbows, head; view objects or move	Discriminative establishing and conditionedreinforcing stimuli
3 – circular sensory-motor	Form open-ended proper classes	Reach, touch, grab, shake objects, circular babble	Open ended proper classes, phonemes, archiphonemes
4 – sensory-motor	Form concepts	Respond to stimuli in a class successfully and non-stochastically	Morphemes, concepts
5 – nominal	Find relations among concepts	Use names for objects and other utterances as successful commands	Single words: ejaculatives & exclamations, verbs, nouns, number names, letter names
6 – sentential	Imitate and acquire sequences; follow short sequential acts	Generalize match-dependent task actions; chain words	Various forms of pronouns: subject (I), object (me), possessive adjective (my), possessive pronoun (mine), and reflexive (myself) for various persons (I, you, he, she, it, we, y’all, they)
7 – preoperational	Make simple deductions; follow lists of sequential acts; tell stories	Count event roughly events and objects; connect the dots; combine numbers and simple propositions	Connectives: as, when, then, why, before; products of simple operations
8 – primary	Simple logical deduction and empirical rules involving time sequence; simple arithmetic	Adds, subtracts, multiplies, divides, counts, proves, does series of tasks on own	Times, places, counts acts, actors, arithmetic outcome, sequence from calculation
9 – concrete	Carry out full arithmetic, form cliques, plan deals	Does long division, short division, follows complex social rules, ignores simple social rules, takes and coordinates perspective of other and self	Interrelations, social events, what happened among others, reasonable deals, history, geography
10 – abstract	Discriminate variables such as stereotypes; logical quantification; (none, some, all)	Form variables out of finite classes; make and quantify propositions	Variable time, place, act, actor, state, type; quantifiers (all, none, some); categorical assertions (e.g., “We all die”)
11 – formal	Argue using empirical or logical evidence; logic is linear, 1-dimensional	Solve problems with one unknown using algebra, logicand empiricism	Relationships (for example: causality) are formed out of variables; words: linear, logical, one-dimensional, if then, thus, therefore, because; correct scientific solutions
12 – systematic	Construct multivariate systems and matrices	Coordinate more than one variable as input; consider relationships in contexts.	Events and concepts situated in a multivariate context; systems are formed out of relations; systems: legal, societal, corporate, economic, national
13 – metasystematic	Construct multi-systems and metasystems out of disparate systems	Create metasystems out of systems; compare systems and perspectives; name properties of systems: e.g. homomorphic, isomorphic, complete, consistent (such as tested by consistency proofs), commensurable	Metasystems and supersystems are formed out of systems of relationships, e.g. contractsand promises
14 – paradigmatic	Fit metasystems together to form new paradigms; show “incomplete” or “inconsistent” aspects of metasystems	Synthesize metasystems	Paradigms are formed out of multiple metasystems
15 – cross-paradigmatic	Fit paradigms together to form new fields	Form new fields by crossing paradigms, e.g. evolutionary biology + developmental biology = evolutionary developmental biology	New fields are formed out of multiple paradigms
16 – meta-cross-paradigmatic (performative-recursive)	Reflect on various properties of cross-paradigmatic operations	Explicate the dynamics of, and limitations of, cross-paradigmatic thinking	The dynamics and limitations of cross-paradigmatic thinking are explained as they are recursively enacted