Specified complexity is a creationist argument introduced by William Dembski, used by advocates to promote the pseudoscience of intelligent design.^{[citation needed]} According to Dembski, the concept can formalize a property that singles out patterns that are both specified and complex, where in Dembski's terminology, a specified pattern is one that admits short descriptions, whereas a complex pattern is one that is unlikely to occur by chance. An example cited by Dembski is a poker hand, where for example the repeated appearance of a royal flush will raise suspicion of cheating.^[1] Proponents of intelligent design use specified complexity as one of their two main arguments, along with irreducible complexity.

Dembski argues that it is impossible for specified complexity to exist in patterns displayed by configurations formed by unguided processes. Therefore, Dembski argues, the fact that specified complex patterns can be found in living things indicates some kind of guidance in their formation, which is indicative of intelligence. Dembski further argues that one can show by applying no-free-lunch theorems the inability of evolutionary algorithms to select or generate configurations of high specified complexity. Dembski states that specified complexity is a reliable marker of design by an intelligent agent—a central tenet to intelligent design, which Dembski argues for in opposition to modern evolutionary theory. Specified complexity is what Dembski terms an "explanatory filter": one can recognize design by detecting complex specified information (CSI). Dembski argues that the unguided emergence of CSI solely according to known physical laws and chance is highly improbable.^[2]

The concept of specified complexity is widely regarded as mathematically unsound and has not been the basis for further independent work in information theory, in the theory of complex systems, or in biology.^[3][4][5] A study by Wesley Elsberry and Jeffrey Shallit states: "Dembski's work is riddled with inconsistencies, equivocation, flawed use of mathematics, poor scholarship, and misrepresentation of others' results."^[6] Another objection concerns Dembski's calculation of probabilities. According to Martin Nowak, a Harvard professor of mathematics and evolutionary biology, "We cannot calculate the probability that an eye came about. We don't have the information to make the calculation."^[7]

Dembski formulates and proposes a law of conservation of information as follows:

This strong proscriptive claim, that natural causes can only transmit CSI but never originate it, I call the Law of Conservation of Information.

Immediate corollaries of the proposed law are the following:

1. The specified complexity in a closed system of natural causes remains constant or decreases.
2. The specified complexity cannot be generated spontaneously, originate endogenously or organize itself (as these terms are used in origins-of-life research).
3. The specified complexity in a closed system of natural causes either has been in the system eternally or was at some point added exogenously (implying that the system, though now closed, was not always closed).
4. In particular any closed system of natural causes that is also of finite duration received whatever specified complexity it contains before it became a closed system.^[15]

Dembski notes that the term "Law of Conservation of Information" was previously used by Peter Medawar in his book The Limits of Science (1984) "to describe the weaker claim that deterministic laws cannot produce novel information."^[16] The actual validity and utility of Dembski's proposed law are uncertain; it is neither widely used by the scientific community nor cited in mainstream scientific literature. A 2002 essay by Erik Tellgren provided a mathematical rebuttal of Dembski's law and concludes that it is "mathematically unsubstantiated."^[17]

In a more recent paper,^[18] Dembski provides an account which he claims is simpler and adheres more closely to the theory of statistical hypothesis testing as formulated by Ronald Fisher. In general terms, Dembski proposes to view design inference as a statistical test to reject a chance hypothesis P on a space of outcomes Ω.

Dembski's proposed test is based on the Kolmogorov complexity of a pattern T that is exhibited by an event E that has occurred. Mathematically, E is a subset of Ω, the pattern T specifies a set of outcomes in Ω and E is a subset of T. Quoting Dembski^[19]

Thus, the event E might be a die toss that lands six and T might be the composite event consisting of all die tosses that land on an even face.

Kolmogorov complexity provides a measure of the computational resources needed to specify a pattern (such as a DNA sequence or a sequence of alphabetic characters).^[20] Given a pattern T, the number of other patterns may have Kolmogorov complexity no larger than that of T is denoted by φ(T). The number φ(T) thus provides a ranking of patterns from the simplest to the most complex. For example, for a pattern T which describes the bacterial flagellum, Dembski claims to obtain the upper bound φ(T) ≤ 10²⁰.

Dembski defines specified complexity of the pattern T under the chance hypothesis P as

${\displaystyle \sigma =-\log _{2}[R\times \varphi (T)\times \operatorname {P} (T)],}$

where P(T) is the probability of observing the pattern T, R is the number of "replicational resources" available "to witnessing agents". R corresponds roughly to repeated attempts to create and discern a pattern. Dembski then asserts that R can be bounded by 10¹²⁰. This number is supposedly justified by a result of Seth Lloyd^[21] in which he determines that the number of elementary logic operations that can have been performed in the universe over its entire history cannot exceed 10¹²⁰ operations on 10⁹⁰ bits.

Dembski's main claim is that the following test can be used to infer design for a configuration: There is a target pattern T that applies to the configuration and whose specified complexity exceeds 1. This condition can be restated as the inequality

${\displaystyle 10^{120}\times \varphi (T)\times \operatorname {P} (T)<{\frac {1}{2}}.}$

Dembski's explanation of specified complexity

Dembski's expression σ is unrelated to any known concept in information theory, though he claims he can justify its relevance as follows: An intelligent agent S witnesses an event E and assigns it to some reference class of events Ω and within this reference class considers it as satisfying a specification T. Now consider the quantity φ(T) × P(T) (where P is the "chance" hypothesis):

Think of S as trying to determine whether an archer, who has just shot an arrow at a large wall, happened to hit a tiny target on that wall by chance. The arrow, let us say, is indeed sticking squarely in this tiny target. The problem, however, is that there are lots of other tiny targets on the wall. Once all those other targets are factored in, is it still unlikely that the archer could have hit any of them by chance?

In addition, we need to factor in what I call the replicational resources associated with T, that is, all the opportunities to bring about an event of T's descriptive complexity and improbability by multiple agents witnessing multiple events.

According to Dembski, the number of such "replicational resources" can be bounded by "the maximal number of bit operations that the known, observable universe could have performed throughout its entire multi-billion year history", which according to Lloyd is 10¹²⁰.

However, according to Elsberry and Shallit, "[specified complexity] has not been defined formally in any reputable peer-reviewed mathematical journal, nor (to the best of our knowledge) adopted by any researcher in information theory."^[22]

This section possibly contains synthesis of material that does not verifiably mention or relate to the main topic. (May 2012)

The soundness of Dembski's concept of specified complexity and the validity of arguments based on this concept are widely disputed. A frequent criticism (see Elsberry and Shallit) is that Dembski has used the terms "complexity", "information" and "improbability" interchangeably. These numbers measure properties of things of different types: Complexity measures how hard it is to describe an object (such as a bitstring), information is how much the uncertainty about the state of an object is reduced by knowing the state of another object or system,^[24] and improbability measures how unlikely an event is given a probability distribution.

On page 150 of No Free Lunch Dembski claims he can demonstrate his thesis mathematically: "In this section I will present an in-principle mathematical argument for why natural causes are incapable of generating complex specified information." When Tellgren investigated Dembski's "Law of Conservation of Information” using a more formal approach, he concluded it is mathematically unsubstantiated.^[25] Dembski responded in part that he is not "in the business of offering a strict mathematical proof for the inability of material mechanisms to generate specified complexity".^[26] Jeffrey Shallit states that Demski's mathematical argument has multiple problems, for example; a crucial calculation on page 297 of No Free Lunch is off by a factor of approximately 10⁶⁵.^[27]

Dembski's calculations show how a simple smooth function cannot gain information. He therefore concludes that there must be a designer to obtain CSI. However, natural selection has a branching mapping from one to many (replication) followed by pruning mapping of the many back down to a few (selection). When information is replicated, some copies can be differently modified while others remain the same, allowing information to increase. These increasing and reductional mappings were not modeled by Dembski. In other words, Dembski's calculations do not model birth and death. This basic flaw in his modeling renders all of Dembski's subsequent calculations and reasoning in No Free Lunch irrelevant because his basic model does not reflect reality. Since the basis of No Free Lunch relies on this flawed argument, the entire thesis of the book collapses.^[28]

According to Martin Nowak, a Harvard professor of mathematics and evolutionary biology "We cannot calculate the probability that an eye came about. We don't have the information to make the calculation".^[7]

Dembski's critics note that specified complexity, as originally defined by Leslie Orgel, is precisely what Darwinian evolution is supposed to create. Critics maintain that Dembski uses "complex" as most people would use "absurdly improbable". They also claim that his argument is circular: CSI cannot occur naturally because Dembski has defined it thus. They argue that to successfully demonstrate the existence of CSI, it would be necessary to show that some biological feature undoubtedly has an extremely low probability of occurring by any natural means whatsoever, something which Dembski and others have almost never attempted to do. Such calculations depend on the accurate assessment of numerous contributing probabilities, the determination of which is often necessarily subjective. Hence, CSI can at most provide a "very high probability", but not absolute certainty.

Another criticism refers to the problem of "arbitrary but specific outcomes". For example, if a coin is tossed randomly 1000 times, the probability of any particular outcome occurring is roughly one in 10³⁰⁰. For any particular specific outcome of the coin-tossing process, the a priori probability (probability measured before event happens) that this pattern occurred is thus one in 10³⁰⁰, which is astronomically smaller than Dembski's universal probability bound of one in 10¹⁵⁰. Yet we know that the post hoc probability (probabilitly as observed after event occurs) of its happening is exactly one, since we observed it happening. This is similar to the observation that it is unlikely that any given person will win a lottery, but, eventually, a lottery will have a winner; to argue that it is very unlikely that any one player would win is not the same as proving that there is the same chance that no one will win. Similarly, it has been argued that "a space of possibilities is merely being explored, and we, as pattern-seeking animals, are merely imposing patterns, and therefore targets, after the fact."^[15]

Apart from such theoretical considerations, critics cite reports of evidence of the kind of evolutionary "spontanteous generation" that Dembski claims is too improbable to occur naturally. For example, in 1982, B.G. Hall published research demonstrating that after removing a gene that allows sugar digestion in certain bacteria, those bacteria, when grown in media rich in sugar, rapidly evolve new sugar-digesting enzymes to replace those removed.^[29] Another widely cited example is the discovery of nylon eating bacteria that produce enzymes only useful for digesting synthetic materials that did not exist prior to the invention of nylon in 1935.

Other commentators have noted that evolution through selection is frequently used to design certain electronic, aeronautic and automotive systems which are considered problems too complex for human "intelligent designers".^[30] This contradicts the argument that an intelligent designer is required for the most complex systems. Such evolutionary techniques can lead to designs that are difficult to understand or evaluate since no human understands which trade-offs were made in the evolutionary process, something which mimics our poor understanding of biological systems.

Dembski's book No Free Lunch was criticised for not addressing the work of researchers who use computer simulations to investigate artificial life. According to Shallit:

The field of artificial life evidently poses a significant challenge to Dembski's claims about the failure of evolutionary algorithms to generate complexity. Indeed, artificial life researchers regularly find their simulations of evolution producing the sorts of novelties and increased complexity that Dembski claims are impossible.^[27]

• List of topics characterized as pseudoscience
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• Texas sharpshooter fallacy