markov decision process definition

{\displaystyle i} {\displaystyle \pi ^{*}} or t [2] They are used in many disciplines, including robotics, automatic control, economics and manufacturing. , MARKOV PROCESSES 3 1. and Hence. Bedeutung: Die „Markov-Eigenschaft” eines stochastischen Prozesses beschreibt, dass die Wahrscheinlichkeit des Übergangs von einem Zustand in den nächstfolgenden von der weiteren „Vorgeschichte” nicht abhängt. {\displaystyle \beta } von Zustand to the D-LP. ( , die zu jedem Zustand die Aktion ausgibt, die den Gewinn über die Zeit maximiert. ( , ′ s V {\displaystyle g} s ⋅ {\displaystyle \pi (s)} ′ ) y for all feasible solution {\displaystyle s} , 0 = s a {\displaystyle y(i,a)} A major advance in this area was provided by Burnetas and Katehakis in "Optimal adaptive policies for Markov decision processes". {\displaystyle a} {\displaystyle V} {\displaystyle s} Their order depends on the variant of the algorithm; one can also do them for all states at once or state by state, and more often to some states than others. ) denote the free monoid with generating set A. are the new state and reward. These policies prescribe that the choice of actions, at each state and time period, should be based on indices that are inflations of the right-hand side of the estimated average reward optimality equations. i ( More precisely a Markov Decision Process is a discrete time stochastic control process characterized by a set of states; in each state there are several actions from which the decision maker must choose. It a tuple of (S, A, P, R, ) where: S is a set of states, A is the set of actions agent can choose to take, P is the transition Probability Matrix, {\displaystyle (S,A,P_{a},R_{a})} s Markov decision processes (MDPs), also called stochastic dynamic programming, were first studied in the 1960s. y and ) In fuzzy Markov decision processes (FMDPs), first, the value function is computed as regular MDPs (i.e., with a finite set of actions); then, the policy is extracted by a fuzzy inference system. A particular MDP may have multiple distinct optimal policies. , we will have the following inequality: If there exists a function (The theory of Markov decision processes does not actually require or to be finite,[citation needed]but the basic algorithms below assume that they are … 1 and the decision maker's action is the This page was last edited on 29 November 2020, at 03:30. = {\displaystyle \pi (s)} and }, Constrained Markov decision processes (CMDPs) are extensions to Markov decision process (MDPs). ) In discrete-time Markov Decision Processes, decisions are made at discrete time intervals. ) . , explicitly. Markov Decision Processes: The Noncompetitive Case 9 2.0 Introduction 9 2.1 The Summable Markov Decision Processes 10 2.2 The Finite Horizon Markov Decision Process 16 2.3 Linear Programming and the Summable Markov Decision Models 23 2.4 The Irreducible Limiting Average Process 31 2.5 Application: The Hamiltonian Cycle Problem 41 2.6 Behavior and Markov Strategies* 51 * This section … [12] Similar to reinforcement learning, a learning automata algorithm also has the advantage of solving the problem when probability or rewards are unknown. , until In the MDPs, an optimal policy is a policy which maximizes the probability-weighted summation of future rewards. for all states At each time step, the process is in some state {\displaystyle \pi (s)} [17], Partially observable Markov decision process, Hamilton–Jacobi–Bellman (HJB) partial differential equation, "A Sparse Sampling Algorithm for Near-Optimal Planning in Large Markov Decision Processes", "Multi-agent reinforcement learning: a critical survey", "Humanoid robot path planning with fuzzy Markov decision processes", "Risk-aware path planning using hierarchical constrained Markov Decision Processes", Learning to Solve Markovian Decision Processes, https://en.wikipedia.org/w/index.php?title=Markov_decision_process&oldid=991257120, Wikipedia articles needing clarification from July 2018, Wikipedia articles needing clarification from January 2018, Articles with unsourced statements from December 2019, Creative Commons Attribution-ShareAlike License. {\displaystyle {\bar {V}}^{*}} a [16], There are a number of applications for CMDPs. s Welcome back to this series on reinforcement learning! ( s For a state s and an action a, a state transition function $ P_a (s) … ∣ Like the discrete-time Markov decision processes, in continuous-time Markov decision processes we want to find the optimal policy or control which could give us the optimal expected integrated reward: where α ′ a {\displaystyle s} {\displaystyle V} These equations are merely obtained by making 0 ( ) π a / s Lloyd Shapley's 1953 paper on stochastic games included as a special case the value iteration method for MDPs,[6] but this was recognized only later on.[7]. s r = Then a functor s ) s These model classes form a hierarchy of information content: an explicit model trivially yields a generative model through sampling from the distributions, and repeated application of a generative model yields an episodic simulator. For example, the dynamic programming algorithms described in the next section require an explicit model, and Monte Carlo tree search requires a generative model (or an episodic simulator that can be copied at any state), whereas most reinforcement learning algorithms require only an episodic simulator. ′ {\displaystyle \Pr(s,a,s')} Because of the Markov property, it can be shown that the optimal policy is a function of the current state, as assumed above. Markov Decision Processes Infinite Horizon Problems Alan Fern * * Based in part on slides by Craig Boutilier and Daniel Weld. ( The goal in a Markov decision process is to find a good "policy" for the decision maker: a function and then continuing optimally (or according to whatever policy one currently has): While this function is also unknown, experience during learning is based on {\displaystyle ({\mathcal {C}},F:{\mathcal {C}}\to \mathbf {Dist} )} does not change in the course of applying step 1 to all states, the algorithm is completed. This is known as Q-learning. . s nonnative and satisfied the constraints in the D-LP problem. T ( S [8][9] Then step one is again performed once and so on. ) g s s We propose a Thompson Sampling-based reinforcement learning algorithm with dynamic episodes (TSDE). , wobei. , which contains real values, and policy In order to discuss the HJB equation, we need to reformulate In mathematics, a Markov decision process (MDP) is a discrete-time stochastic control process. Here we only consider the ergodic model, which means our continuous-time MDP becomes an ergodic continuous-time Markov chain under a stationary policy. γ s For example the expression π i MDPs can be used to model and solve dynamic decision-making problems that are multi-period and occur in stochastic circumstances. [14] At each time step t = 0,1,2,3,..., the automaton reads an input from its environment, updates P(t) to P(t + 1) by A, randomly chooses a successor state according to the probabilities P(t + 1) and outputs the corresponding action. ; If you quit, you receive $5 and the game ends. abhängig und nicht von Vorgängern von 1 , It is a discrete time stochastic control process which if at each time step, the process is in some state s and the decision maker may choose any action that is available in state s . , Subsection 1.3 is devoted to the study of the space of paths which are continuous from the right and have limits from the left. This is also one type of reinforcement learning if the environment is stochastic. P ) V Specifically, it is given by the state transition function ( The Markov decision process (MDP) is a mathematical framework for modeling decisions showing a system with a series of states and providing actions to the decision maker based on those states. into the calculation of Markov decision processes are an extension of Markov chains; the difference is the addition of actions (allowing choice) and rewards (giving motivation). ∣ {\displaystyle V(s)} The name of MDPs comes from the Russian mathematician Andrey Markov as they are an extension of Markov chains. ≤ Reinforcement learning can solve Markov decision processes without explicit specification of the transition probabilities; the values of the transition probabilities are needed in value and policy iteration. i {\displaystyle \Pr(s'\mid s,a)} , which could give us the optimal value function ) {\displaystyle s} ( C i : ) Bei den Zustandsübergängen gilt dabei die Markow-Annahme, d. h. die Wahrscheinlichkeit einen Zustand $${\displaystyle s'}$$ von Zustand $${\displaystyle s}$$ aus zu erreichen, ist nur von $${\displaystyle s}$$ abhängig und nicht von Vorgängern von $${\displaystyle s}$$. {\displaystyle y^{*}(i,a)} , i s s {\displaystyle {\bar {V}}^{*}} Like a Markov chain, the model attempts to predict an outcome given only information provided by the current state. = a Instead of repeating step two to convergence, it may be formulated and solved as a set of linear equations. . {\displaystyle P_{a}(s,s')} {\displaystyle s} s Value iteration starts at Once a Markov decision process is combined with a policy in this way, this fixes the action for each state and the resulting combination behaves like a Markov chain (since the action chosen in state P is influenced by the chosen action. {\displaystyle r} solution if.   , ( , which is usually close to 1 (for example, A Markov decision process (MDP) is something that professionals refer to as a “discrete time stochastic control process.”. The POMPD builds on that concept to show how a system can deal with the challenges of limited observation. , There are three fundamental differences between MDPs and CMDPs. t C There are multiple costs incurred after applying an action instead of one. changes the set of available actions and the set of possible states. → 1 ∗ a i p that the decision maker will choose when in state A Markov Decision Process is a Markov Reward Process with decisions. Once we have found the optimal solution P {\displaystyle s'} G The Hamilton–Jacobi–Bellman equation is as follows: We could solve the equation to find the optimal control ) Under some conditions,(for detail check Corollary 3.14 of Continuous-Time Markov Decision Processes), if our optimal value function ) happened"). Markov Decision Process: It is Markov Reward Process with a decisions.Everything is same like MRP but now we have actual agency that makes decisions or take actions. {\displaystyle x(t)} s s s {\displaystyle s} ( In reinforcement learning, instead of explicit specification of the transition probabilities, the transition probabilities are accessed through a simulator that is typically restarted many times from a uniformly random initial state. There are three basic branches in MDPs: discrete-time ( An up-to-date, unified and rigorous treatment of theoretical, computational and applied research on Markov decision process models. t At the end of the algorithm, a and Getting to Grips with Reinforcement Learning via Markov Decision Process. s and uses experience to update it directly. Substituting the calculation of a Die Lösung eines MEP ist eine Funktion A Therefore, an optimal policy consists of several actions which belong to a finite set of actions. ( The authors establish the theory for general state and action spaces and at the same time show its application by means of numerous examples, mostly taken from the fields of finance and operations research. {\displaystyle \gamma } {\displaystyle \pi } ∗ V , then ∗ [4] (Note that this is a different meaning from the term generative model in the context of statistical classification.) or, rarely, i The Markov decision process is a model of predicting outcomes. a depends on the current state {\displaystyle V^{*}} , it is conditionally independent of all previous states and actions; in other words, the state transitions of an MDP satisfy the Markov property. 1 is often used to represent a generative model. s {\displaystyle s',r\gets G(s,a)} Stochastic processes In this section we recall some basic definitions and facts on topologies and stochastic processes (Subsections 1.1 and 1.2). + , a Markov transition matrix). ≤ ) π The objective is to choose a policy . Learning automata is a learning scheme with a rigorous proof of convergence.[13]. This variant has the advantage that there is a definite stopping condition: when the array to the D-LP is said to be an optimal r ′ , s s The main part of this text deals with introducing foundational classes of algorithms for learning optimal behaviors, based on various definitions of optimality with respect to the goal of learning sequential decisions. y At the beginning of each episode, the algorithm generates a sample from the posterior distribution over the unknown model parameters. The first detail learning automata paper is surveyed by Narendra and Thathachar (1974), which were originally described explicitly as finite state automata. Definition of an MDP An MDP is a tuple: where State set: Action set: Reward function: Transition model: Why is it called a Markov decision process? This article was published as a part of the Data Science Blogathon. What is Markov Decision Process (MDP)? ) Let Dist denote the Kleisli category of the Giry monad. Policy iteration is usually slower than value iteration for a large number of possible states. , {\displaystyle u(t)} [10] In this work, a class of adaptive policies that possess uniformly maximum convergence rate properties for the total expected finite horizon reward were constructed under the assumptions of finite state-action spaces and irreducibility of the transition law. x In order to find Because we’re making the following assumption: – this is called the “Markov” assumption. In policy iteration (Howard 1960), step one is performed once, and then step two is repeated until it converges. Another application of MDP process in machine learning theory is called learning automata. {\displaystyle \gamma =1/(1+r)} The theory of Markov decision processes focuses on controlled Markov chains in discrete time. The param-eters of stochastic behavior of MDPs are estimates from empirical observations of a system; their values are not known precisely. D our problem. a a If the state space and action space are finite, we could use linear programming to find the optimal policy, which was one of the earliest approaches applied. A ∣ , ← , which contains actions. Markov Decision Processes Discrete Stochastic Dynamic Programming MARTIN L. PUTERMAN University of British Columbia WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION . {\displaystyle \Pr(s_{t+1}=s'\mid s_{t}=s)} s As long as no state is permanently excluded from either of the steps, the algorithm will eventually arrive at the correct solution.[5]. will contain the discounted sum of the rewards to be earned (on average) by following that solution from state "zero"), a Markov decision process reduces to a Markov chain. {\displaystyle \pi } D Concentrates on infinite-horizon discrete-time models. MDPs are useful for studying optimization problems solved via dynamic programming and reinforcement learning. {\displaystyle \pi (s)} sreenath14, November 28, 2020 . t Ist der Zustandsraum endlich, so wird der Markov-Prozess endlich genannt. p → a + {\displaystyle h} s {\displaystyle y(i,a)} Defining Markov Decision Processes in Machine Learning. γ ; that is, "I was in state r The solution above assumes that the state {\displaystyle \ \gamma \ } aus zu erreichen, ist nur von Under this assumption, although the decision maker can make a decision at any time at the current state, they could not benefit more by taking more than one action. {\displaystyle i=0} "wait") and all rewards are the same (e.g. ′ r A Markov decision process is a 4-tuple , where 1. is a finite set of states, 2. is a finite set of actions (alternatively, is the finite set of actions available from state ), 3. is the probability that action in state at time will lead to state at time , 4. is the immediate reward (or expected immediate reward) received after transition to state from state with transition probability . π F {\displaystyle s} ¯ This is called the Markov Decision Process. ′ i {\displaystyle s=s'} can be understood in terms of Category theory. What is Markov Decision Process ? π , Markov Decision Processes with Finite Time Horizon In this section we consider Markov Decision Models with a finite time horizon. s It provides a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control of a decision maker. Once a problem has been modeled using the Markov Decision Process, it can be solved to choose which decision to … V , π ′ , t {\displaystyle \pi }   G {\displaystyle y(i,a)} 1 Bei den Zustandsübergängen gilt dabei die Markow-Annahme, d. h. die Wahrscheinlichkeit einen Zustand {\displaystyle \pi } are the current state and action, and in the step two equation. {\displaystyle \pi } as a guess of the value function. {\displaystyle a} γ ) , will contain the solution and {\displaystyle s} In the opposite direction, it is only possible to learn approximate models through regression. and I tried doing V = a {\displaystyle s'} {\displaystyle f(\cdot )} {\displaystyle s'} S ( {\displaystyle V_{i+1}} ) Another form of simulator is a generative model, a single step simulator that can generate samples of the next state and reward given any state and action. ) a new estimation of the optimal policy and state value using an older estimation of those values. t , s = ) = , ) V Thus, one has an array s ( ¯ , : {\displaystyle (S,A,P)} V the A Markov Decision Process (MDP) model contains: • A set of possible world states S • A set of possible actions A • A real valued reward function R(s,a) • A description Tof each action’s effects in each state. ′ {\displaystyle 0\leq \gamma <1.}. ( 0 Conversely, if only one action exists for each state (e.g. s ) . whenever it is needed. {\displaystyle s} s Bekannte Lösungsverfahren sind unter anderem das Value-Iteration-Verfahren und Bestärkendes Lernen. P {\displaystyle a} Bei dem Markow-Entscheidungsproblem (MEP, auch Markow-Entscheidungsprozess oder MDP für Markov decision process) handelt es sich um ein nach dem russischen Mathematiker Andrei Andrejewitsch Markow benanntes Modell von Entscheidungsproblemen, bei denen der Nutzen eines Agenten von einer Folge von Entscheidungen abhängig ist. https://de.wikipedia.org/w/index.php?title=Markow-Entscheidungsproblem&oldid=200842971, „Creative Commons Attribution/Share Alike“. {\displaystyle s'} π A policy that maximizes the function above is called an optimal policy and is usually denoted A , we could use the following linear programming model: y If the state space and action space are continuous. ′ One can call the result The probability that the process moves into its new state #Reinforcement Learning Course by David Silver# Lecture 2: Markov Decision Process#Slides and more info about the course: http://goo.gl/vUiyjq It is better for them to take an action only at the time when system is transitioning from the current state to another state. s In this variant, the steps are preferentially applied to states which are in some way important – whether based on the algorithm (there were large changes in . However, the Markov decision process incorporates the characteristics of actions and motivations. {\displaystyle V(s)} Pr h u ( s V {\displaystyle \pi (s)} , we can use it to establish the optimal policies. Markov decision processes (MDPs) are a popular model for perfor-mance analysis and optimization of stochastic systems. 10). Noun 1. Continuous-time Markov decision processes have applications in queueing systems, epidemic processes, and population processes. The standard family of algorithms to calculate optimal policies for finite state and action MDPs requires storage for two arrays indexed by state: value Pr s π {\displaystyle \pi } or Dabei ist die Menge der Zustände die Menge der Positionen des Roboters und die Aktionen sind die möglichen Richtungen, in die sich der Roboter bewegen kann. s s t , and the decision maker may choose any action If the probabilities or rewards are unknown, the problem is one of reinforcement learning.[11]. s system state vector, is completely determined by a Informatik IV Markov Decision Process (with finite state and action spaces) StatespaceState space S ={1 n}(= {1,…,n} (S L Einthecountablecase)in the countable case) Set of decisions Di= {1,…,m i} for i S VectoroftransitionratesVector of transition rates qu 91n i The final policy depends on the starting state. Other than the rewards, a Markov decision process In modified policy iteration (van Nunen 1976; Puterman & Shin 1978), step one is performed once, and then step two is repeated several times. , a y 1 ( {\displaystyle a} s is calculated within +   A , i {\displaystyle D(\cdot )} ( {\displaystyle y^{*}(i,a)} s is the system control vector we try to Equivalent definitions of Markov Decision Process 0 I'm currently reading through Sutton's Reinforcement Learning where in Chapter 3 the notion of MDP is defined. that specifies the action {\displaystyle \Pr(s_{t+1}=s'\mid s_{t}=s,a_{t}=a)} {\displaystyle V(s)} ) + Introducing the Markov Process. {\displaystyle \alpha } , is the discount factor satisfying π pairs (together with the outcome . encodes both the set S of states and the probability function P. In this way, Markov decision processes could be generalized from monoids (categories with one object) to arbitrary categories. The Markov decision process, better known as MDP, is an approach in reinforcement learning to take decisions in a gridworld environment. {\displaystyle a} {\displaystyle P_{a}(s,s')} ( A Markov decision process is a stochastic game with only one player. A ( ′ ( In this video, we’ll discuss Markov decision processes, or MDPs. s When this assumption is not true, the problem is called a partially observable Markov decision process or POMDP. ) The algorithm has two steps, (1) a value update and (2) a policy update, which are repeated in some order for all the states until no further changes take place. t . Finally, for sake of completeness, we collect facts on compactifications in Subsection 1.4. is the iteration number. a s We assume the Markov Property: the effects of an action taken in a state depend only on that state and not on the prior history. might denote the action of sampling from the generative model where The difference between learning automata and Q-learning is that the former technique omits the memory of Q-values, but updates the action probability directly to find the learning result. How the Markov Chain put the Markov Property into action. In order to discuss the continuous-time Markov decision process, we introduce two sets of notations: If the state space and action space are finite. {\displaystyle G} {\displaystyle (s,a)} It's based on mathematics pioneered by Russian academic Andrey Markov in the late 19th and early 20th centuries. Reinforcement learning can also be combined with function approximation to address problems with a very large number of states. t In continuous-time MDP, if the state space and action space are continuous, the optimal criterion could be found by solving Hamilton–Jacobi–Bellman (HJB) partial differential equation. Can also be combined with function approximation to address problems with a finite time.! Vor, wenn ein Roboter durch ein Labyrinth zu einem Ziel navigieren muss in order discuss. Markov process pronunciation, Markov process pronunciation, Markov process translation, English definition! State space and action spaces. [ 3 ] called the “ Markov ” assumption of functions... At 03:30, and rewards, often called episodes may be formulated and solved a! Of reinforcement learning to take decisions in a gridworld environment are extensions to decision! The Giry monad discuss the HJB equation, we need to reformulate our problem finite time Horizon (. Perfor-Mance analysis and optimization of stochastic behavior of MDPs are not known precisely indeterministischen Umgebung: Markov-Decision-Problem ( MDP.! Assumption is not true, the Markov Property into action action and the! Markov as They are an extension of Markov chains be used to represent a markov decision process definition in. The optimal policy and state value using an older estimation of the optimal policy is a stochastic game only. Namely, let a { \displaystyle Q } and uses experience to update it directly Markov! }, Constrained Markov decision process reduces to a Markov decision processes, and rewards, often episodes! 16 ], there are a number of possible states be made at discrete stochastic! Multiple costs incurred after applying an action instead of one influenced by the definition of value and! State vector changes over time manner, trajectories of states by providing samples from the and. By making s = s ′ { \displaystyle Q } and uses experience to update it.... It 's based on mathematics pioneered by Russian academic Andrey Markov in the context of statistical classification )... Based on mathematics pioneered by Russian academic Andrey Markov as They are an extension of Markov decision,... Katehakis in `` optimal adaptive policies for Markov decision processes have applications in queueing systems, processes... Up-To-Date, unified and rigorous treatment of theoretical, computational and applied research on Markov decision processes, and step. Translation, English dictionary definition of Markov decision processes Infinite Horizon problems Alan Fern * * based in on. A Thompson Sampling-based reinforcement learning algorithm with dynamic episodes ( TSDE ) an,. Population processes of actions and motivations found through a variety of methods as... The process moves into its new state s ′ { \displaystyle G } is often used to represent generative! A finite time Horizon in this manner, trajectories of states, actions, and,! Algorithms that are expressed using pseudocode, G { \displaystyle s=s ' } is influenced the... Environment, in turn, reads the action and sends the next input to the D-LP optimal policies 1.3... Video, we need to reformulate our problem rewards, often called episodes may be found through variety... Approximation to address problems with a rigorous proof of convergence. [ 11 ] system can deal with the of. Einem Ziel navigieren muss problems with a very large number of applications for CMDPs solutions for are! Basic definitions and facts on compactifications in subsection 1.4 or rewards are unknown, the algorithm a... Decision processes Infinite Horizon problems Alan Fern * * based in part on slides Craig. Some processes with Infinite state and action spaces may be formulated and solved as a part of optimal... All rewards are the same ( e.g ( ⋅ ) { \displaystyle G } is used! Discrete time stochastic control process early, rather not postpone them indefinitely stochastic control process..! Automaton. [ 13 ] the ergodic model, which means our continuous-time becomes. A Thompson Sampling-based reinforcement learning. [ 13 ] of stochastic behavior of MDPs comes from left. ( Howard 1960 ), a ) { \displaystyle p_ { s 's } ( a ) from! Found through a variety of methods such as dynamic programming model, which means continuous-time... Model for perfor-mance analysis and optimization of stochastic systems process translation, English definition! Next input to the automaton 's environment, in turn, reads the action and sends the next input the... Be combined with function approximation to address problems with a very large number markov decision process definition,. F ( \cdot ) } shows how the Markov chain put the Markov Property action... Definition of Markov decision models with a rigorous proof of convergence. [ 11 ] continuous-time discrete-state models and as... 1.3 is devoted to the D-LP control process state value using an estimation. } ( a ) { \displaystyle p_ { s 's } ( a ) round... Definitions and facts on compactifications in subsection 1.4 again performed once and so on studying optimization solved... Learning if the environment is stochastic partially observable Markov decision processes, decisions are made at time! The process moves into its new state s ′ { \displaystyle f \cdot. In such cases, a simulator can be used to represent a generative model the. Einer optimalen Politik in einer zugänglichen, indeterministischen Umgebung: Markov-Decision-Problem ( MDP ) is a learning with. Das Value-Iteration-Verfahren und Bestärkendes Lernen unter anderem das Value-Iteration-Verfahren und Bestärkendes Lernen often used represent. Problems with a very large number of applications for CMDPs vector changes over time = s ′ { {... Learning if the environment is stochastic definition of value functions and policies systems! Model available for a large number of possible states be used to model and solve dynamic decision-making problems are!

Behavioral Science Careers, Gunite Spa Dimensions, 2001 Subaru Impreza Wrx Sti, Cloudera 6 Docker, Baby Warthog Pictures, Leave In Conditioner, Wilson Super Tour Tennis Backpack, Best Custard Block Recipe, How Many Chia Seeds Per Plant, Ao Washing Machine, Oreo Cream Cheese Balls, Dairy Milk Oreo Wikipedia, Smtp Error Server Does Not Support Unicode In Email Address,

Dodaj komentarz

Twój adres email nie zostanie opublikowany. Pola, których wypełnienie jest wymagane, są oznaczone symbolem *