glossary

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A

Abstraction

Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, and generalizing it so that it has wider applications. All engineering designs are abstractions; they tend to include both ontological or structural descriptions of things, followed by appropriately mathematical models conforming to those structures.

Algorithm

In mathematics and computer science, an algorithm is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always unambiguous and are used as specifications for performing calculations, data processing, automated reasoning, and other tasks. A crypto-economic system's minting algorithm, for instance, comprises the sequence of steps required to implement the system's minting policy.

Action

An action ua ∈ U(Xa; x) for an agent with address a is any activity under the control of the agent that may influence the state of the system.

Action space / Action set

The set U(Xa; x) represents the set of feasible actions, given agent a’s local state space Xa, and the global state x.

Address

An address a is an index of a shared state that facilitates actions taken by an agent. The set of addresses for a given realization of agents is denoted by A.

Agent (economics)

In economics, an agent is an actor and unique identity, and more specifically a decision maker in a model of some aspect of the economy. Typically, every agent makes decisions by solving a well- or ill-defined optimization or choice problem. In blockchain-based networks, agents are typically represented by addresses, which perform identity transactions on behalf of their owners.

Agent-based model (ABM)

Class of computational models for simulating the actions and interactions of autonomous agents (both individual or collective entities such as organizations or groups) with a view to assessing their effects on the system as a whole.

Automated system

Closed feedback loop between a computational system and another system being controlled that dynamically enforces a set of rules with the intent to drive a particular system behavior.

Autonomous system

Closed feedback loop at a sufficiently high level of abstraction as to require no additional inputs in order to act. The goals and behaviors of an autonomous system are intrinsic.

B

Block diagram

A block diagram is a diagram of a system in which the principal parts or functions are represented by blocks connected by lines that show the relationships of the blocks. They are heavily used across engineering disciplines. Block diagrams are typically used for higher level, less detailed descriptions that are intended to clarify overall concepts without concern for the details of implementation.

Blockchain

Open, distributed ledger that can record transactions between two parties efficiently and in a verifiable and permanent way. A growing list of records, called blocks, are linked using cryptography and each block contains a cryptographic hash of the previous block, a timestamp, and transaction data (generally represented as a Merkle tree). For use as a distributed ledger, a blockchain is typically managed by a peer-to-peer network collectively adhering to a protocol for inter-node communication and validating new blocks.

Bonding curve

Bonding Curves are continuous liquidity mechanisms which are used in market design for cryptographically-supported token economies. Academic literature increasingly refers to bonding curves as "configuration spaces" as Bonding Curves are part of a larger theory of scalar functions that remain invariant under legal changes in state.

Boundary conditions

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Byzantine fault tolerance (BFT)

The property of a system that is able to resist the class of failures derived from the Byzantine Generals’ Problem. This means that a BFT system is able to continue operating even if some of the nodes fail or act maliciously.

There is more than one possible solution to the Byzantine Generals’ Problem and, therefore, multiple ways of building a BFT system. Likewise, there are different approaches for a blockchain to achieve Byzantine fault tolerance and this leads us to the so-called consensus algorithms.

C

Chaos theory

Is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. “Chaos” is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions.

Complex system

System in which behavior of the whole within a given environment cannot be understood directly by understanding the behavior of the parts and the environment as separate additive entities. Rather, the system’s behavior is deeply dependent on the relations among the parts. Systems that are “complex” have distinct properties that arise from these relationships, including non-linearity (the output of a complex system is greater than the sum of its parts), emergence (the behavior of a complex system cannot be understood simply by looking at its components), spontaneous order (the apparent chaos of a complex system produces order or stability without outside imposition), adaptation (systemic behavior alters when it alteration finds, responding to change to aid the success of the system), and feedback loops (some portion of the system’s output becomes fresh input for the system).

In practice, complex systems exhibit unintuitive behaviors when they exist. Because complex systems appear in a wide variety of fields, the commonalities among them have become the subject of their own independent area of research. Cryptoeconomic systems can typically be categorized as complex systems.

Configuration space

Reachable state space. Subset of a system's (global) state space, representing all achievable states under the designed mechanisms. Any global properties true for all points in the configuration space are true for all possible sequences of actions on the part of agents. A Manifold characterized by the enforced conservation of one or more desired global properties. Resulting induced state space after introducing internally consistent restrictions to the global state space. Serves the role of enforcing desirable macro-economic properties, while retaining sufficient degrees of freedom for the agents at the micro level to act according to their own private preferences.

Consensus algorithm

Approach for solving the consensus problem in the field of Computer Science which is used to achieve agreement on a single data value among distributed processes or systems or the current state of a distributed system. Consensus algorithms are primarily used to achieve reliability in a network involving multiple distributed nodes that contain the same information. The most common implementations in Decentralized Ledger Technology networks are called Proof of Work (PoW) and Proof of Stake (PoS).

Note that the PoW algorithm is not 100% tolerant to the Byzantine faults, but due to the cost-intensive mining process and the underlying cryptographic techniques, PoW has proven to be one of the most secure and reliable implementations for blockchain networks. In that sense, the Proof of Work consensus algorithm, designed by Satoshi Nakamoto, is considered by many as one of the most genius solutions to the Byzantine faults.

Conservation function

Conservation functions are output functions engineered to encode desired global properties. A conservation function is a representation or measure of a property, and must be constructed specifically for each particular property under scrutiny.

Control engineering / Control systems engineering

Control Engineering or Control Systems Engineering is an engineering discipline that applies automatic control theory to design systems with desired behaviors in control environments. It overlaps and is usually taught along with electrical engineering at many institutions around the world. Systems designed to perform without requiring human input are called automatic control systems (such as cruise control for regulating the speed of a car). Professional crypto-economic systems engineering employs a range of well-established tools and methods from Control Engineering.

Crypto-economic system

Crypto-economic systems are data-driven, multiscale, adaptive and dynamic networks with a system-level state available to all agents. These systems use cryptographic tokens as information carriers, allowing for economic activities to emerge on top of a shared distributed ledger technology (DLT) enabled infrastructure such as blockchain. DLT creates the conditions for a digital economic game with enforced state-space restrictions, by providing a tamper-proof universal state layer.

Crypto-economics

Crypto-economics is an emerging research field concerned with economic coordination games in cryptographically secured peer-to-peer networks. It is a multidisciplinary field aimed at creating robust, decentralized P2P networks.

D

Differential equation

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Differential equations are an important tool in mathematical modeling. One of the most basic examples of differential equations is the Malthusian Law of population growth dp/dt = rp shows how the population (p) changes with respect to time. More complicated differential equations can be used to model the relationship between predators and prey. The interactions between the two populations are connected by differential equations.

Differential equation (event-driven)

A set of Differential equations with actions at discrete events are used to model piecewise differential equations with jump discontinuities, or impacts and collisions such as a bouncing ball. They can also model hybrid systems with both continuous and discrete dynamics. The discrete dynamics can come from sampled or digital processes, such as a digital controller controlling a continuous process, or the discrete dynamics can represent modes such as a chemical reactor following a recipe.

Differential game

In mathematical game theory differential games are a group of problems related to the modeling and analysis of conflict in the context of a dynamical system. More specifically, a state variable or variables evolve over time according to a differential equation. Early analyses reflected military interests, considering two actors — the pursuer and the evader — with diametrically opposed goals. More recent analyses have reflected engineering or economic considerations.

Differential games are related closely with optimal control problems. In an optimal control problem there is single control u(t) and a single criterion to be optimized; differential game theory generalizes this to two controls u(t),v(t) and two criteria, one for each player. Each player attempts to control the state of the system so as to achieve its goal; the system responds to the inputs of all players.

Differential game (multi-mechanism)

A differential game in which there is more than one mechanism through which the players may affect the state of the game. Multi-mechanism differential games are generally Hybrid Differential games because strategies must include decision over which set of mechanism two employ which are almost require discrete switching policies.

When considering equilibria multi-mechanism games, it does not suffice to consider the mechanisms stability independently as there are well known results in hybrid systems that indicate that two stable systems composed under a switching policy can be driven unstable.

Differential game (multi-player)

Game theory is the study of decision problems in which there are multiple decision makers, and the quality of a decision maker’s choice depends on that choice as well as the choices of others. While game theory has been studied predominantly as modeling paradigm in the mathematical social sciences, there is a strong connection to control systems in that a controller can be viewed as a decision-making entity. Accordingly, game theory is relevant in settings in which there are multiple interacting controllers

Differential specification

A differential specification is a formal characterization of how a dynamical system evolves over time. It contains the logic required to characterize the system dynamics even when those dynamics are not under the control of the system designer. It is described through formal system modeling syntax (e.g., policies, state update logic, state variables in BlockSciences's "swim-lane chart"). In computational models, data may be used to estimate the current state of the system using a state-space model, but a differential specification is required to make projections about future states or to characterize trends.

Digital twin

A digital representation of a real-world, physical or digital system, reflecting both that system's structure - i.e. its elements and interconnections - and its dynamics - i.e. how the system behaves, performs, and evolves over time. Typically describes a model which comprehensively represents a real-world complex system while maintaining a data link to, and therefore evolving with that complex system over time as a "living model". BlockScience creates Digital Twins of crypto-economic systems using the cadCAD modeling framework.

Discrete event system

In control engineering, a discrete event dynamic system (DEDS) is a discrete-state, event-driven system of which the state evolution depends entirely on the occurrence of asynchronous discrete events over time. Although similar to continuous-variable dynamic systems (CVDS), DEDS consists solely of discrete state spaces and event-driven state transition mechanisms.

Dynamical system theory

Dynamical systems theory is an area of mathematics used to describe the behavior of the complex dynamical systems, usually by employing differential equations or difference equations. When difference equations are employed, the theory is called discrete dynamical systems. BlockScience's approach and methods are rooted in Dynamical Systems Theory.

E

Enabled economy

BlockScience terminology for any "real-world" economy (i.e. economic agents exchanging goods or services) enabled by a technological substrate such as Decentralized Ledger Technology. From a network perspective a network of valid transactions whose edges represent transactions and whose nodes are unique addresses used to encode the source and destination of a particular transaction.

Enabling economy

BlockScience terminology for any economy (i.e. economic agents exchanging goods or services) which has the primary purpose of enabling a technological substrate for another "real-world" economy to function. In the blockchain context typically associated with a network's consensus. From a network perspective a protocol as a peer-to-peer computer network whose objective is to enforce and broadcast a valid state of the ledger.

Environmental process driver

Data feed that informs an exogenous process. This may be real or assumed data to provide updated input for a model representing a change of external conditions.

Environmental process model

A modelled generator providing an update to state variables using external input signals from an environmental process driver. User action behavior and control functions should not affect this model. The affected state variables should be purely exogenously based, but may be blended from the internal state if necessary to model the system. The exogenous state variables are the link between the changes from the outside world germane to the modeled system, and provides the information necessary for users and controllers to make decisions.

F

Framework

A framework is, or contains, a (not completely detailed) structure or system for the realization of a defined result/goal. Many frameworks comprise one or more "models". Compared with methods, frameworks give the users much more freedom regarding the (partial or entire) use of the framework and the use of the models or techniques therein. An example of a framework employed by BlockScience is the state-space (time-domain) modeling framework.

G

Game theory

Game theory is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

Generalized energy function

H

Hybrid differential game

Combines differential games with hybrid games. In both kinds of games, two players interact with continuous dynamics. The difference is that hybrid games also provide all the features of hybrid systems and discrete games, but only deterministic differential equations. Differential games, instead, provide differential equations with input by both players, but not the luxury of hybrid games, such as mode switches and discrete or alternating interaction.

Hybrid system

A dynamical system that exhibits both continuous and discrete dynamic behavior — a system that can both flow (described by a differential equation) and jump (described by a state machine or automaton). Often, the term “hybrid dynamical system” is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena.

I

Initial conditions

In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Complex systems often have a very sensitive dependence on initial conditions ("butterfly effect")

Invariant

In Computer Science, an expression whose value doesn't change during program execution. In Mathematics, something unaltered by a transformation, e.g. a conservation function. BlockScience commonly defines configuration spaces within crypto-economic systems, which remain invariant under legal changes in state.

J

K

L

Local state

Local state xa ∈ Xa is a projection of the global state x ∈ X onto Xa.

Local state space

Shared or local state space is a subset Xa ⊆ X, a ∈ A, indicating that part of the global state X that the agent(s) with access to address a can directly influence.

Lyapunov function (generalized energy function)

Lyapunov functions may be interpreted as generalized energy functions for arbitrary state-space dynamical systems. When doing mechanism design where state transition functions are declared rather than observed, one can start with a Lyapunov function, which the designer considers to be a measure of energy within the system, then constrain the mechanism design to include only state transitions which satisfy Lyapunov’s second method, thus ensuring analytical stability of the resulting system. This method does not prevent external actors from pumping energy into the system, but it does allow energy conservation to be asserted, preventing a class of attacks whereby an agent perturbs the state of the system, in order to extract profits from the control action.

M

Manifold

Topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.

Markov chain

A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. In continuous-time, it is known as a Markov process. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo

Mechanism

Natural or established process by which something takes place or is brought about. In engineering, a mechanism transforms input forces and movement into a desired set of output forces and movement. A mechanism is usually a piece of a larger process or system. An example of a voting mechanism is conviction voting.

Mechanism (acad)

Natural or established process by which something takes place or is brought about. In engineering, a mechanism is a device that transforms input forces and movement into a desired set of output forces and movement. A mechanism is usually a piece of a larger process or system. Formally, a mechanism is a mapping f : X × U → X taking the current state x ∈ X and an action u ∈ U and returning a future state x. Denote the set of all mechanisms by F. An agent will thus select from a subset F(a; x) ⊆ F.

Method

A method is a systematic approach to achieve a specific result or goal, and offers a description in a cohesive and (scientific) consistent way of the approach that leads to the desired result/ goal. Minimally a method consists of a way of thinking and a way of working. An example of a method to coordinate human activity is voting.

Model

A graphical, mathematical (symbolic), physical, verbal representation, or otherwise simplified version of a concept, phenomenon, relationship, structure, system, or aspect of the real world. The objectives of a model include: (1) Facilitating understanding by eliminating unnecessary components, (2) Aiding decision making by simulating ‘what if’ scenarios, and (3) Explain, controlling, and predicting events on the basis of past observations. Since most interesting objects and phenomena are both too complicated (they have numerous parts) and too complex (those parts are densely interconnected) to be understood in their entirety, a model contains only those features that are of primary importance to the model maker’s purpose.

Model (formal)

A formal model is a model in which the description is complete and non-ambiguous. It has well-formed syntax and semantics, such that it is amenable to systematic (usually automatable) processing and analysis subject to logical rules. It is based on rigorous methods and formats; it often enables analysis to be performed on the models. A mathematical model is a special type of formal model, and a description of a system using mathematical concepts and language. A computational model is a special type of a mathematical model which is expressed in a programming language. It allows one to run simulations, or backtest against data.

N

Node

The definition of a node may vary significantly according to the context it is used. When it comes to computer or telecommunication networks, nodes may offer distinct purposes, acting either as a redistribution point or as a communication endpoint. Usually, a node consists of a physical network device, but there are some specific cases where virtual nodes are used. Simply put, a network node is a point where a message can be created, received, or transmitted. There are different types of nodes in blockchain: full nodes, supernodes, miner nodes, and SPV clients.

Non-linear system

A system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Crypto-economic systems are typically nonlinear.

O

P

Policy

A policy is a deliberate system of principles to guide decisions and achieve rational outcomes. A policy is a statement of intent, and is implemented as a procedure or protocol. Policies are generally adopted by a governance body within an organization. As part of the BlockScience system modeling syntax, policies determine the inputs to the system dynamics regardless of whether they come from the outside environment, user behavior or algorithmic decision systems.

Potential function

In game theory, a game is said to be a potential game if the incentive of all players to change their strategy can be expressed using a single global function called the potential function.

Potential game

In game theory, a game is said to be a potential game if the incentive of all players to change their strategy can be expressed using a single global function called the potential function. An equilibrium is guaranteed to exist, and there is a wide array of distributed learning algorithms that guarantee convergence.

Primitive

Well established, generic building block. Designed to do one very specific task in a highly reliable fashion. An example of a crypto-economic primitive are automated market makers such as Bonding Curves.

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R

S

Socio-technical system

Sociotechnical systems in organizational development is an approach to complex organizational work design that recognizes the interaction between people and technology in workplaces. The term also refers to the interaction between society's complex infrastructures and human behavior. In this sense, society itself, and most of its substructures, are complex sociotechnical systems.

Stability (analytical)

Analytical stability is defined in contrast to the colloquial use of stability to refer to a set of historical observations about a signal which was over the period of observation remaining within some bounds of an apparent equilibrium. Unlike empirical stability, analytical stability implies future stability but does so by leveraging a formal abstraction for which such stability properties may be proven through real analysis. Strong methods for such proofs often rely on Lyapunov functions.

State space

Set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. May be interpreted as that collection of variables which serve to define the system at any point in time.

State space representation (time-domain approach)

In Control Engineering, a state-space representation is a mathematical model of a system as a set of input, output and state variables related by first-order differential equations or difference equations. The state-space method is characterized by significant algebraization of general system theory, which makes it possible to use advanced mathematical techniques. Also known as "time-domain approach", it provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. The state-space model is used in many different areas. In econometrics, for instance, the state-space model can be used for forecasting stock price.

State transition (acad)

A state transition at time t is the selection of a future state in response to a valid transaction (a, ut, ft). This exposition is sufficient to identify transitions in the state of the system with selections by agents of 1) mechanisms and 2) actions, relying upon local information, but influenced by (and depending upon) the global state. A state transitions describes a dynamical system.

State variable

The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system is usually equal to the order of the system's defining differential equation, but not necessarily.

Stochastic process

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner.

System state (acad)

The state x ∈ X summarizes the system at a given point in time, in the sense that a state-dependent action or outcome mapping to the immediate future state need only condition upon x.

T

Token

Atomic unit of state information which is cryptographically verifiable in peer-to-peer networks.

Token engineering

The emerging professional discipline around the design, modelling, analysis, implementation and steering of crypto-economic systems, rooted in and drawing from the well-established scientific body of knowledge of System Theory, Control Systems Engineering, Operations Research, Mathematics, and several other relevant fields.

Transaction (acad)

A transaction is a tuple (a, u, f) ∈ A ×U×F. A transaction is said to be valid if, given global state x, 1) u ∈ U(Xa; x), 2) f ∈ F(a; x).

Transfer function

In engineering, a transfer function (also known as system function or network function) of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In control engineering and control theory the transfer function is derived using the Laplace transform. The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of multiple-input multiple-output (MIMO) systems, and has been largely supplanted by state space representations for such systems.

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