Finite-Delay Strategies In Infinite Games

Transcription

1 Finite-Delay Strategies In Infinite Games von Wenyun Quan Matrikelnummer: Diplomarbeit im Studiengang Informatik Betreuer: Prof. Dr. Dr.h.c. Wolfgang Thomas Lehrstuhl für Informatik 7 Logik und Theorie diskreter Systeme Professor Dr. Dr.h.c. Wolfgang Thomas Fakultät für Mathematik, Informatik und Naturwissenschaften Rheinisch-Westfälische Technische Hochschule Aachen

2

3 Erklärung Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe. Aachen, den 26. Juni 29 Wenyun Quan c 29 Wenyun Quan Alle Rechte vorbehalten.

4

5 Abstract In this thesis we study Church's Problem in the framework of innite games between player I input and player O output, and present algorithms for solving the problem of whether nite-delay solutions for player O in innite games exist. Even and Meyer provided an algorithm to determine solutions with nite delay for sequential Boolean equations with safety winning condition. In our work we introduce special game graphs, in which winning strategies with a xed delay and corresponding winning regions for both players can be constructed. Using this approach, solutions with a xed delay for player O in innite games are built up. Furthermore we present algorithms for solving the problem of whether player O has winning strategies with bounded delay in arbitrary regular innite games. As a consequence, we obtain a new proof for the result of Hosch and Landweber that the existence of the solution of Church's Problem by a strategy with a bounded delay is decidable.

6

7 Contents Introduction 4. History Background Motivation Outline Preliminaries 9 2. Languages Innite games Operators Winning strategies with delay Finite-Delay Solutions for Sequential Boolean Equations 2 3. Sequential Boolean equations Representing sequential Boolean equations by graphs Solvable graphs of sequential Boolean equations Solvable graphs of sequential Boolean equations with delay d Solvable graphs of sequential Boolean equations with nite delay 3 4 Solutions with a Fixed Delay in Innite Games Game graphs with a xed delay d Strategies with a xed delay in reachability games Strategies with a xed delay in weak parity games Strategies with a xed delay in Büchi games Strategies with a xed delay in parity games Solutions with Bounded Delay in Innite Games Absorbed game graphs Winning strategies with bounded delay in reachability games. 53 2

8 Contents Winning strategies with bounded delay in weak parity games Winning strategies with bounded delay in parity games Alternative approaches for nding solutions with bounded delay 67 6 Conclusion 7

9 Chapter Introduction. History A central task in computer science is synthesis, which builds from a specication a program which realizes it. An instance of the synthesis problem is the description of a desired condition for an input sequence from the environment and an output sequence from the computer. Given such a condition we construct a program, which generates an output sequence to guarantee that the condition is fullled. Church's Problem. In 957 a master problem was raised by Church regarding the innite behavior of automata and the use of automata as transducers. Church's Problem concerns the synthesis of automata that realizes functions over innite words [5] : Given a requirement which a circuit is to satisfy, we may suppose the requirement expressed in some suitable logistic system which is an extension of restricted recursive arithmetic. The synthesis problem is to nd recursion equivalences representing a circuit that satises the given requirement, or to determine that there is no such circuit alternatively. Church's Problem asks for a nite-state automaton which realizes the transformation of an innite sequence α into another innite sequence β, such that a requirement on α, β, which is expressed in certain logic formula, is satised. Therefore to solve Church's Problem, we determine whether the 4

10 Chapter. Introduction 5 transformations of innite words specied in a system of arithmetic are computable by nite automata. There are classes of specications for which algorithmic synthesis is possible. The proof of the expressive equivalence between nite automata and weak monadic second-order arithmetic over the natural number ordering, was established by Büchi and Elgot [], and also by Trakhtenbrot [9] in 958. This result is considered as a breakthrough on the relation between automata and logic. A few years later Büchi [4] extended the Büchi-Elgot-Trakhtenbrot Theorem to the expressive equivalence between the full monadic second-order theory of the natural number ordering and nite automata over innite sequences. Based on a result on the determinization of Büchi automata from McNaughton [], Church's problem was solved in 967 by Büchi and Landweber [3] for specications in monadic second-order logic MSO logic over N, <..2 Background To study Church's Problem we work with transducers in the format of deterministic Mealy automata, which is introduced in [2]. In the theory of computation, a Mealy machine is a nite state transducer that generates an output bit based on its current state and input bit. Mealy automata. A Mealy automaton has the format A M = S, Σ, Γ, s, δ, τ where S is the nite set of states, Σ and Γ are the input alphabet and output alphabet, respectively, s is the initial state, δ : S Σ S is the transition function and τ : S Σ Γ is the output function. A transition from p S to δ p, a S where a Σ is labeled by a/τ p, a where τ p, a Γ. We present the way to specify languages and the task of synthesis in the system of monadic second-order logic MSO over the successor structure N, +, <. It is also called sequential calculus or SS second-order theory of one successor. Church's Problem can be expressed as follows [7]: Given a specication ϕ stated in sequential calculus, decide whether a Mealy automaton exists that transforms each input sequence α {, } ω into an output sequence β {, } ω such that N, +, < = ϕ α, β, and if this is the case, construct such an automaton. Example.2.. Let α t be the t + st bit of an innite input sequence α {, } ω, and let β t be the t + st bit of an innite output sequence

11 Chapter. Introduction 6 β {, } ω for some t. We show a nite automaton from [7] which transforms α into β, such that the following conditions are satised:. t α t = β t = ; 2. t β t = β t+ = ; 3. ω t α t = ω t β t =. To fulll these three conditions we require the following procedure:. For input bit produce output bit ; 2. For input produce: output bit if last output bit was, output bit if last output bit was. This procedure can be executed by a Mealy automaton A M as follows: /, / / s s / Figure.: A Mealy automaton A M If input bit is, the Mealy automaton A M produces output bit at each moment t. If occurs in input sequence, output bit is chosen between and alternately. Remark.2.2. Trakhtenbrot claried in [8] two relevant aspects of Church's Problem. The rst aspect is the computability of the transformation by a nite-state automaton, such as the Mealy automaton mentioned above. The other aspect concerns whether the transformation is nonanticipatory i.e., causal; in other words, whether an output β t is determined by the prex α α... α t of an input α. In this thesis we are interested in anticipatory transformations from input sequence α into output sequence β. Therefore we give some examples of anticipatory transformations to understand this aspect of Church's Problem.

12 Chapter. Introduction 7 Example.2.3. For α, β {, } ω, an anticipatory transformation can be β t = α 3t, i.e., the output sequence β consists of every third letter of the input sequence α. This transformation requires response with an unbounded delay to fulll the condition of the input sequence α and the output sequence β. An instance of anticipatory transformations with a bounded delay can be β t = α t+2, whose transformation requires a bounded delay 3 to generate the output sequence to fulll the certain condition. In this thesis we do not concern anticipatory transformations with unbounded delay, but anticipatory transformations with bounded delay..3 Motivation In the context of reactive systems, we view nite automata with output as nite-state controllers. They are supposed to behave correctly for any behavior of the environment represented by input sequences. This amounts to solutions of innite two-player games with winning conditions given by ω- regular languages. Based on the work of Gale and Stewart [7], McNaughton introduced the treatment of Church's Problem in the framework of innite games in []. Innite games of descriptive set theory are put therefore into an algorithmic framework, and the algorithmic theory of innite games is in a very active research area. In our work we deal with Church's Problem in the framework of innite games. An innite game describes the interaction between a controller and the environment, together with a specication as a winning condition. It is more convenient to understand, therefore we consider it advantageous to transform automaton graphs into game graphs when modeling reactive systems. A specication ϕ as a winning condition for player O denes an innite two-player game between player I "I" denotes "Input" and player O "O" denotes "Output". The two players contribute respectively the input- and output-bits in turn. In applications there are situations in which a perfect synchronized response by a controller is not possible. Hence, it is necessary to consider

13 Chapter. Introduction 8 possibilities with some tolerance bound, i.e., bounded delay. We search for solutions with bounded delay which fullls conditions between two innite sequences i.e., binary relation. In the framework of innite games, we ask whether or not player O wins an innite game over the corresponding nite game graph with a bounded delay, i.e., whether player O wins an innite game with delay d for some < d < +. The result of the work from Even and Meyer [6] provided an algorithm of solving sequential Boolean equations with a nite delay over Even-Meyer automaton graphs. In other words they solve the problem of determining nite-delay solutions in safety games. Based on this result, Hosch and Landweber [8] proved that the existence of the solution of Church's Problem with delay is decidable. We deal with Church's Problem in framework of innite games with nite delay and provide algorithms over game graphs to nd solutions with xed delay in corresponding innite games. The essential problem is how to determine the existence of nite-delay solutions for player O in arbitrary innite games. Our goal is to nd the break condition for searching a nite-delay solution for player O in regular innite games, therefore to prove that that the existence of the solution of Church's Problem by a strategy with nite delay is decidable..4 Outline We start with a preliminary chapter in which we recall basic denitions relevant to our work. The idea of winning strategies with delay in innite games will be presented as a bridge to the following chapters. The algorithm of Even and Meyer [6] is presented in Chapter 3. This algorithm provide a possibility to determine nite-delay solutions for sequential Boolean equations with safety winning condition. Furthermore we study how to construct winning strategies with a xed delay in arbitrary innite games over to corresponding d-delay game graphs in Chapter 4. The major question is how to determine the existence of solutions with nite delay in innite game, i.e., it asks whether one of the players has a winning strategy with nite delay in an innite game. The algorithms for solving this problem are presented in Chapter 5. Hosch and Landweber [8] proved that the problem whether Church's Problem with conditions stated in sequential calculus are solvable with delay is decidable. The conclusion of our work is that the existence of the solution of Church's Problem by a strategy with a bounded delay is decidable.

14 Chapter 2 Preliminaries In this chapter we introduce basic denitions which are referred to in the following chapters. The terms of regular languages, innite games and operators will be recalled, which are followed by the denition of solutions with nite delay. 2. Languages Computers operate on texts consisting of sequences of symbols over a given alphabet. Every program transforms input texts into output texts. Therefore the basic terms are alphabets, words and languages which specify the fundamental notions of computer science. Hence, we recall in this section the denitions of regular language and ω-regular language. Denition 2... An alphabet is a nite set consisting of elements called symbols, it is denoted by Σ. Let Σ be the set of all nite words over Σ and Σ ω be the set of innite words over Σ. A word with length n over an alphabet Σ is a nite sequence of letters a a a 2... a n Σ with a i Σ for i n ; an ω-word is an innite sequence of letters α α α 2... Σ ω with a i Σ for i o. Length of a nite word w is denoted by w where w Σ. In this thesis we refer to regular languages dened by standard regular expression [9] with the operators +,,. Denition An ω-regular language is a nite union of ω-languages U V ω with regular languages U, V Σ. 9

15 Chapter 2. Preliminaries Let Σ I = {, } be an input alphabet and Σ O = {, } be an output alphabet. Church's Problem concerns transformations of an innite word α = α α... {, } ω into an innite word β = β β... {, } ω, fullling a certain condition C X, Y where X and Y are variables for innite sequences, i.e., C α, β holds. Since a condition C X, Y can be described by an ω- language L, Church's Problem asks whether α β := α β α β... L where L {, } ω. In our work we dene conditions C X, Y in the framework of sequential calculus. Denition Sequential calculus is the class of conditions which can be expressed in a formalism which consists of: rst order variables x, y, z,... interpreted as elements of the set of natural numbers N; monadic second order predicate variables X, Y, Z,... interpreted as subsets of natural numbers N; the usual logical connectives,,,...; equality =; quantiers over both rst and second order variables, ; the function symbol for the successor; the constant symbol for the natural number zero. Sequential calculus is the monadic second order theory of the natural numbers with the function successor i.e., SS. Church's Problem is solved by Büchi and Landweber [3] with an algorithm, which decides whether conditions stated in the framework of sequential calculus admit nite automata solutions, and produces such nite automata, if they exist. 2.2 Innite games Compared with automata graphs, game graphs oer a more intuitive view. Therefore it is useful to study Church's Problem in the framework of innite games. Gale and Stewart [7] introduce a form of innite games in 953, which represents Church's Problem for any ω-language L {, } ω.

16 Chapter 2. Preliminaries Denition In a Gale-Stewart game, player I and player O choose symbols from an alphabet Σ = {, } alternately and create an innite sequence α β α β... {, } ω, where α i denotes an input bit from player I and β i denotes an output bit from player O for i. Player O wins the Gale-Stewart game if α β = α β α β... L {, } ω, the innite language L describes the winning condition C X, Y in the innite game where X and Y are variables for innite sequences. Büchi and Landweber show in [3] that a condition C X, Y stated in sequential calculus is computable by a nite automaton. An automaton can be transformed into a game graph by distinguishing the contribution of bits, which are chosen by player I and player O. As shown in Figure 2. we introduce intermediate states between two steps from one step in automaton graphs, processing a single bit. Hence Gale-Stewart games with winning conditions dened in sequential calculus can be considered as two-player innite games with nite-state game graphs. Figure 2.: Transformation from an automaton to a game graph We dene now general game graphs with nite states for solving Church's Problem with conditions stated in sequential calculus. Denition A general game graph is a tuple G = Q, µ with a vertex set Q = Q I Q O where Q I denotes the set of vertices for player I, Q O denotes the set of vertices for player O, and µ := {Q I Σ I Q O } {Q O Σ O Q I } is the function determining moves. The corresponding edge relation E {Q I Q O } {Q O Q I } is dened as q, p E if and only if µ q, u = p for some u Σ I Σ O.

17 Chapter 2. Preliminaries 2 Note that all outgoing edges are connected only with vertices of the opponent. The vertices in the set Q I are depicted by boxes and the vertices in the set Q O by circles in game graphs. The set Q will always be nite, therefore we also speak of nite-state game graphs. Denition A play is an innite sequence ρ = ρ ρ ρ 2... with ρ i, ρ i + E for i. The set of vertices which occur in the play ρ is denoted by Occ ρ: Occ ρ := {q Q i ρ i = q}. Inf ρ denotes the set of vertices which occur in the play ρ innitely often: Inf ρ := {q Q ω i ρ i = q}. If a vertex p Q is reached by an outgoing edge from a vertex q Q in a game graph G, we say that q is a predecessor of p, and p is a successor of q. Denition An innite game for solving Church's Problem is a tuple G, q, ϕ consisting of a game graph G = Q, µ, an initial vertex q Q and a winning condition ϕ for player O. The size of an innite game is the number of vertices in the corresponding game graph G, denoted by Q. For solving Church's Problem with conditions in sequential calculus an appropriate model of automaton is introduced by Muller [3]. The Büchi- Elgot-Trakhtenbrot Theorem shows that any MSO formula ϕ can be transformed into an equivalent Muller automaton. The results in [4] and [] allow to convert an MSO formula into an equivalent non-deterministic Büchi automaton, and hence into an equivalent deterministic Muller automaton. Thus, we recall the denition of a Muller game which is transformed from Muller automaton. Denition A Muller game is an innite game G, q, F with a game graph G = Q, µ, an initial vertex q Q and a winning condition for player O given by a collection F 2 Q. Player O wins a play ρ if and only if Inf ρ F. A weak Muller game is an innite game G, q, F with a game graph G = Q, µ, an initial vertex q Q and a winning condition for player O given by a collection F 2 Q. Player O wins a play ρ if Occ ρ F.

18 Chapter 2. Preliminaries 3 Denition A strategy for player O in an innite game is a function f : Q I Q O + Σ O, i.e., it determines output bits on an innite word α β = α β α β... {, } ω of even length; analogously a function g : Q I Q O Q I Σ I is a strategy for player I which determines input bits on an innite word α β = α β α β... {, } ω of odd length. A positional strategy f : Q O {, } for player O can be described by a subset of the set of edges E: {q, p E q Q O, f q = p Q I } E; analogously for a positional strategy g : Q I {, } for player I. A play ρ = ρ ρ ρ 2... started in ρ is consistent with f if for every ρ i Q O, ρ i + = µ ρ i, f ρ... ρ i holds. A strategy f is a winning strategy from q Q for player O in a game graph G = Q, µ if each play from q consistent with f fullls the winning condition ϕ, analogously for a winning strategy g for player I. Let Win Q ω be the set of the plays won by player O, in a game graph G a winning region of player O is dened as follows: W O := {q Q player O has a winning strategy from q}. A winning region of player I denoted by W I is dened analogously. In our work we concern Church's Problem, thus we are interested in the problem of deciding whether player O has a winning strategy from the initial vertex q in a game graph G = Q, µ, i.e., whether q W O. Denition A game is determined if each vertex of the game graph is either in W I or in W O, i.e., W I W O = Q holds. Büchi and Landweber show in [3] that every innite game with winning condition denable in sequential calculus, is determined. Hereby we recall denitions of determined innite games with dierent winning conditions for player O. A reachability game G, q, ϕ is an innite game together with a game graph G = Q, µ, an initial vertex q Q and a set F Q dening a reachability winning condition as follows: ρ Win : i : ρ i F. Reachability games are a special case of weak Muller games.

19 Chapter 2. Preliminaries 4 A safety game G, q, ϕ is an innite game together with a game graph G = Q, µ, an initial vertex q Q and a set F Q dening a safety winning condition as follows: ρ Win : i : ρ i F. An innite game G, q, ϕ given by a game graph G = Q, µ, an initial vertex q Q and F Q is called a Büchi game if the winning condition of player O is given by: ρ Win : Inf ρ F. The dual co-büchi winning condition of player O is: ρ Win : Inf ρ F, an innite game with such a winning condition is called a co-büchi game. Let G = Q, µ be a game graph and c be a coloring function c : Q {,,..., k} for some k N. A play ρ = ρ ρ ρ 2... delivers a sequence of colors c ρ = c ρ c ρ c ρ 2... A weak parity game is an innite game G, q, ϕ, a game graph G = Q, µ with a coloring function c : Q {,,..., k} for some k N, an initial vertex q Q and a weak parity condition dening the set of plays ρ won by player O: ρ Win : max Occ c ρ is even. Given a game graph G = Q, µ with a coloring function c : Q {,,..., k} for some k N, and an initial vertex q Q. A parity game is an innite game G, q, ϕ with a parity winning condition dening the set of plays ρ won by player O: ρ Win : max Inf c ρ is even. The parity winning condition is the most fundamental one of all winning conditions for innite two-player games. Every other winning condition can be reduced to the parity condition.

20 Chapter 2. Preliminaries Operators Denition Given two nite sets, Σ I as an input alphabet and Σ O as an output alphabet, Σ ω I and Σω O denote the sets of innite words over Σ I and Σ O respectively. Let X be a variable for innite sequences in Σ ω I and Y be a variable for innite sequences in Σ ω O. Let C X, Y be a binary relation between X and Y, and op be an operator mapping Σ ω I into Σω O. We say that an operator op : Σ ω I Σω O is a solution for a condition C X, Y for Y if {α, op α α Σ ω I } C X, Y. if An operator op : Σ ω O Σω I is a solution for a condition C X, Y for X {op β, β β Σ ω O} C X, Y. Denition An operator op for innite words is called nonanticipatory or causal, if for any innite words α = α α... α t... and α = α α... α t... for t, the corresponding outputs op α = β = β β... β t... and op α = β = β β... β t... fulll the following condition: α = α, α = α,..., α t = α t,... β = β, β = β,..., β t = β t,... ; i.e., the output β t at time t is uniquely determined by the input sequence α, α,..., α t, and is independent of the input sequence α t+, α t+2,... in the future. Otherwise an operator op for innite words is an anticipatory operator.

21 Chapter 2. Preliminaries 6 Trakhtenbrot and Barzdin introduce in [5] a type of operators with delay as solutions to fulll conditions C X, Y between innite words. Denition An operator op : Σ ω I Σω O is continuous if for each α Σ ω I, each bit β t of op α depends only on a nite prex of α for t < +. Hence, every nonanticipatory operator is continuous. Example The following operator op is causal: {, if α α op α t =... α t contains more ones than zeros;, otherwise. A continuous operator op 2 for some k with < k < + is shown as follows: {, if α α op 2 α t =... α t... α t+k contains more ones than zeros;, otherwise. And the following operator op 3 is a typical non-continuous one: {, if α α op 3 α t =... α t... contains innitely many ones;, otherwise. We are interested in the class of continuous operators op mapping Σ ω I into Σ ω O for which β t = δ α α... α ft holds, where β t is the t + st element of the sequence β, f is a function f : N N, and δ maps Σ I into Σ O where Σ I denotes a set of all nite sequences over Σ I. Denition A continuous operator op is called deterministic if f t t, i.e., if for d < + it can be given in the form β t = δ α α... α t+d. Denition A deterministic operator is a d-shift operator for some d if f t = t d for t d, otherwise f t =, i.e., a d-shift operator produces β t according to the sequence α... α t d. An operator is a d-delay operator for some d > if f t = t + d, which means a d-delay operator will not generate β t until a nite sequence α... α t+d is seen.

22 Chapter 2. Preliminaries 7 Example We have following examples for d-shift operator and d-delay operator:. 3-shift operator generates {, if α α β t =... α t 3 contains even many ones;, otherwise. In other words player O has to produce the output bit β t depending only on the input sequence up to time t 3. t Player I Player O delay operator generates {, if α α β t =... α t+2 contains even many ones;, otherwise. In other words player O can wait until the sequence of input bits from player I till time t + 2 is seen. t Player I Player O Denition A condition C X, Y is determined if and only if there exists either a -delay solution β = op α of the condition C X, Y for Y or a -shift solution op of the condition C X, Y for X. Büchi and Landweber [3] give a synthesis algorithm for sequential calculus with respect to deterministic operators by showing that every condition C X, Y of sequential calculus is determined. Let X be a variable for innite sequences, we write X t for the t + st position in the corresponding sequence, analogously for innite words Y and Z.

23 Chapter 2. Preliminaries 8 Remark A condition C X, Y stated in sequential calculus has a d-delay solution for Y if and only if the condition dened by the formula C d X, Y := Z C Z, Y t Z t Y t + d has a -delay solution for Y. It is equivalent to C X, Y having a d-delay solution for Y. A condition C X, Y in sequential calculus has a d-shift solution for X if and only if the condition dened by the formula C d X, Y := Z t Z t = Y t + d C Z, Y has a -shift solution for X. It is equivalent to C X, Y having a d-shift solution for X. For any xed d the existence of a d-delay solution is determined according to the Büchi-Landweber algorithm [3]. Because conditions in sequential calculus are determined, C d X, Y does not have a -delay solution for β if and only if C d X, Y has a -shift solution for X. It also means that for every condition C X, Y which is dened in sequential calculus and for every xed d, either C X, Y has a d-delay solution for Y or C X, Y has a d-shift solution for X [8]. Remark If C X, Y has a d-delay solution for Y for some d >, it also has d -delay solutions for Y for all d > d; and if C X, Y has a d-shift solution for X, it also has d -shift solutions for all d < d. 2.4 Winning strategies with delay An operator op for solving a condition C X, Y can be considered as a winning strategy for player O in an innite game, which is specied by the ω-language described by C X, Y. We consider a certain innite game specied by the condition C X, Y. Player I selects an element α t of an input alphabet Σ I at time t, and then player O makes a move by selecting an element β t of an output alphabet Σ O. Both players have complete information about all previous moves of their opponent, i.e., player I knows the output sequence β β... β t when making a move at time t, and player O replies with the input sequence α α... α t. A play α, β of the game creates an in- nite sequence α β α β... consisting of all moves of each player. Player O wins the game if C X, Y is satised by a play α, β, otherwise player I

24 Chapter 2. Preliminaries 9 wins. Hence in the framework of innite games, a strategy for player I is a deterministic -shift operator η : Σ ω O Σω I and a strategy for player O is a deterministic -delay operator δ : Σ ω I Σω O. Remark A condition C X, Y is determined if and only if one of the players has a winning strategy in the innite game with the winning condition C X, Y, i.e., player I has a d-shift operator or player O has a d-delay operator, with which they can beat all strategies of their opponent. An operator δ : Σ ω I Σ ω O is a d-delay winning strategy for player O in an innite game dened by a condition C X, Y, if and only if it is a deterministic d-delay solution of C X, Y for Y. As mentioned in the previous section, player O therefore has winning strategies with all delay d for d > d. Analogously for player I, an operator η is a d-shift winning strategy for player I in an innite game described by a condition C X, Y, if and only if it is a deterministic d-shift solution of C X, Y for X. Player O has no d-delay winning strategy in this innite game and player I has winning strategies with all shift d for d < d. Example We consider the operator op 3 in Example It shows us that player O wins the game only with an innite delay, i.e., player O produces a sequence β Σ ω O according to an entire input sequence α Σω I of player I. It is clear that no nite delay suces for player O to win. Because if player O chooses output bits with a delay d, which means a bit β t is determined after the choice of α t α t+... α t+d by player I, player I may decide the continuing sequence α t+d α t+d+... ω for β t+d =, or α t+d α t+d+... ω for β t+d = respectively. In another aspect, player I has a winning strategy with an arbitrary nite shift d, or player I loses the game with an innite shift. In fact there are innite games in which player I wins with an innite shift, i.e., player O does not have a winning strategy even with an innite delay in these innite games, see the operator op 3 in Example Our main purpose in this thesis is to determine whether a condition C X, Y stated in sequential calculus has a solution with a nite delay for Y or an innite delay, or C X, Y has a solution with an innite shift for X. In other words we look for winning strategies with delay for player O in an innite game specied by a winning condition C X, Y for player O. There are three situations:

25 Chapter 2. Preliminaries 2. Player O wins an innite game with a nite delay d i.e., a bounded delay, therefore player O wins also with delay d for all d > d; i.e., player I has winning strategies with shift d for all d < d in the game; 2. Player O has no winning strategy with delay in an innite game, i.e., player I wins an innite game with an arbitrary shift, and also with an innite shift; 3. Player O wins an innite game, but only with an innite delay, i.e., player I has an arbitrary nite shift winning strategy in the game. Since we are interested only in continuous operators, we do not consider the third case. We study in the following chapters the algorithms to distinguish the rst two situations in dierent innite games, and focus on the problem of deciding whether a nite delay solution for player O exists in an innite game.

26 Chapter 3 Finite-Delay Solutions for Sequential Boolean Equations In this chapter we present an algorithm of Even and Meyer which determines nite delay solutions for sequential Boolean equations. 3. Sequential Boolean equations Even and Meyer [6] solve the nite delay problem for a fragment of sequential calculus called sequential Boolean equations. They show that the problem to decide whether such an equation has a solution with a nite delay can be reduced to the problem of deciding whether a given nite automaton graph is solvable with a nite delay. For a nite Even-Meyer automaton graph, they provide an algorithm determining whether or not a sequential Boolean equation has a nite-delay solution. Because of the equivalence between solving sequential Boolean equations and solving problem for nite automaton graphs, the problem whether a corresponding sequential Boolean equation has a solution with a nite delay is solvable. Denition 3... Let F X, Y, dy be a quantier free formula of sequential calculus without constants, where X is a vector x, x,..., x m consisting of independent Boolean variables, Y is a vector y, y,..., y n consisting of dependent binary variables and dy = dy, dy,..., dy n. F is a Boolean expression with: Boolean addition '+'; 2

27 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 22 Boolean multiplication ' '; Complementation ' '; dz t = z t + is the value of z one time unit later under the condition that time is discrete. A sequential Boolean equation is written in the form F X, Y, dy =. Denition The equation F X, Y, dy = has a solution if and only if for every nite sequence of values for X = X, X,..., X T there exists a sequence of values of Y = Y, Y,..., Y T, Y T + such that the equation will hold for every t =,,..., T. Denition The equation has a solution with a nite delay d if the knowledge of X, X,..., X d is sucient to determine a value for Y ; the knowledge of Y t, X t, X t,..., X t + d for every t > is sucient to determine a value for Y t such that the determined part of the sequence of Y is part of a solution for the given sequence of X. An alternative statement is that there exists operators op and op 2, such that for any T d and any sequence X, X,..., X T the sequence Y, Y,..., Y T d which is given by Y = op X, X,..., X d Y t + = op 2 Y t, X t, X t +,..., X t + d for t =,,..., T d, can be augmented by some values for Y T d +,..., Y T to satisfy the equation F X, Y, dy = for every t =,,..., T. In the following example, we show that equations of the form F X, Y, dy = are enough to represent even more general equation of the form F X, dx, Y, dy =.

28 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 23 Example Given the following equation with an independent variable x and a dependent variable y: xȳdȳ + xydy + xȳdx + xyd x =. We can replace dx with dz and form the simultaneous equations as follows: { xȳdȳ + xydy + xȳdz + xyd z = ; x = z. These two simultaneous equations are equivalent, because x = and y = x + y = ; x = y xȳ + xy =. The resulting equation in the form F x, y, z, dy, dz = is xȳdȳ + xydy + xȳdz + xyd z + x z + xz = where x is an independent variable, y and z are both dependent variables. 3.2 Representing sequential Boolean equations by graphs Denition A directed graph G over a nite alphabet Σ is a system V, V, E where V is the nite set of vertices, V V is the set of initial vertices and E V Σ V is the set of labeled directed edges. We can extend E to edges labeled by nite words from Σ as follows: E v i, ε, v j v i = v j with ε denotes empty word and E v i, ua, v j v k E v i, u, v k E v k, a, v j with u Σ and a Σ for all i, j, k < V. A path in G with labels l l... l n Σ is a sequence of edges of the form v, l, v, v, l, v 2,..., v n, l n, v n.

29 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 24 Let G be a directed graph with n vertices V = {v, v,..., v n } and each edge labeled with l Σ. The only restriction on G is that there exists at most one edge from v i to v j with label l Σ for i, j < n, i.e., on each path there is a unique word consisting of labels which appear on the path. Let U V be any set of vertices and u Σ be a nite word, U, u is the set of all vertices which are reachable from vertices in U through all paths labeled with u U, u = {v V v E w, u, v with w U}. The representation of equations by graphs is best shown by means of an example. Example We consider the equation from Example 3..4 xȳdȳ + xydy + xȳdz + xyd z + x z + xz =. We rst show how to expand the left side on y, z, dy and dz. For each possible value of y, z, dy and dz we build a term, for instance yzdydz when y, z, dy and dz are all. The value of the term yzdydz in brackets is determined by the above equation as follows: + x x =, thus the value in bracket for the term yzdydz is. Furthermore we obtain the result of the expansion on the left side of this equation: yzdydz + yzdyd z + yzdȳdz x + yzdȳd z x + y zdydz x + y zdyd z + y zdȳdz x + y zdȳd z + ȳzdydz x + ȳzdyd z x + ȳzdȳdz + ȳzdȳd z + ȳ zdydz + ȳ zdyd z x + ȳ zdȳdz + ȳ zdȳd z x =. This equation can be represented as a graph. Consider the term yzdȳdz x for instance, it means that yzdȳdz = or in other words that the state yz = may be followed by the state dȳdz = if and only if x = or x =. y, z has 4 possible values,,,,, and,, hence a graph with 4 vertices can be constructed to represent these 4 states. The vertex y, z is connected to the vertex dȳ, dz with an edge labeled by x if and

30 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 25 only if the left side of the equation becomes zero. For the term yzdȳdz x, vertex, is connected to, with an edge labeled by. The resulting graph is shown in Figure 3. and the table representing the graph G is shown in Table 3.. Figure 3.: Graph G,,,, Table 3.: Table of graph G For any graph we can choose codes for vertices of the graph arbitrarily as long as the code of each vertex is unique. We require codes with length k with 2 k n for a graph with n vertices. By 2 k > n, the codes which is not assigned to vertices are considered as assigned to the vertices which are not connected by any edges. As those isolated vertices never appear in a sequence of values for Y, we can just ignore them. When 2 k = n we can assign each code to a vertex. Example Let G 2 in Figure 3.2 be a given graph, the graph is represented in Table 3.2 with the chosen codes for vertices of the graph G. a : b :

31 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 26 a b c d Figure 3.2: Given graph G 2 c : d :,,, Table 3.2: Table of graph G 2 We can construct a corresponding equation from the table of the graph G 2 in Figure 3.2 which is equivalent to the equation in the form F X, Y, y =. yzdydz + yzdyd z x + yzdȳdz x + yzdȳd z + y zdydz + y zdyd z + y zdȳdz x + y zdȳd z x + ȳzdydz x + ȳzdyd z x + ȳzdȳdz x + ȳzdȳd z x + ȳ zdydz + ȳ zdyd z + ȳ zdȳdz x + ȳ zdȳd z =. A directed graph representing an equation in the form F X, Y, dy = has 2 n vertices and 2 m edges, where X = X, X,..., X m and Y =

32 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 27 Y, Y,..., Y n are second order variables. With a sequence of X, namely X, X,..., X T, a sequence of Y, namely Y, Y,..., Y T is determined such that the equation holds for t =,,..., T. We want to nd a path of length T in the graph with the same labeled word as a given one. In such a path, a sequence of vertices will specify a sequence of values of Y which satises the requirement and vice versa. Thus we say that the problem of deciding whether an equation has a solution is equivalent to the problem of deciding whether the corresponding graph is solvable. With a graph G and label-alphabet Σ we can construct a corresponding equation which has a solution if and only if the graph is solvable. 3.3 Solvable graphs of sequential Boolean equations The problem of deciding whether a given graph is solvable is equivalent to the problem of deciding whether the corresponding nite-state automaton accepts all words. Denition An automaton A corresponding to a graph G with labels from Σ is dened as follows:. The set of states is V ; 2. The input alphabet is Σ; 3. The set of next states from the state v i with an input letter l i is δ {v i }, l i ; 4. The set of initial states is V ; 5. The set of nal states is V. If a word is accepted by an automaton A, then there exists a path labeled with this word in G. We show how to nd a path in G with a given word l l... l T Sigma :. Let V be V the set of all vertices of G; 2. Let V t = V t, l t for every t =, 2,..., T ;

33 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations The result is a sequence V, V,..., V T, V T such that all vertices in V t are reachable by a path labeled with l l... l t ; 4. Choose any v t from V t and nd a v t V t such that there is an edge labeled with l t from v t to v t for every t =, 2,..., T ; 5. The sequence v v... v t v t is one of the solution. The following example illustrates the procedure of the above-mentioned construction. Example Given a directed graph G 2 in Figure 3.2, the transition table of this graph is shown in Table 3.3. a b b d a, b, c c a, b d c b Table 3.3: Table T 2 of G 2 The paths on A labeled with the word are: {a, b, c, d} {a, b, c} {a, b, d} {a, b, c} {a, b, d}. Therefore we nd one of the solutions with the sequence bcabd from the construction. Denition A graph G is solvable with respect to Σ, if and only if for every nite word in Σ there exists a path in G whose sequence of labels is a given word, i.e., G is solvable with respect to Σ V, u for all u Σ. This means a graph is not solvable if and only if an empty set is generated during the construction. Even and Meyer show that the problem of solving such an equation can be reduced to the problem of deciding whether a given graph is solvable with respect to some alphabet Σ. The problem of deciding whether G is solvable is reduced to the problem of deciding whether A accepts all words over Σ. A graph G has a solution without delay if and only if there exists a nonempty set of vertices V V such that {v}, l V for all v V and l Σ.

34 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 29 Example Let's consider a graph given in Figure 3.2, Table 3.3 represents this graph. We eliminate states a and c since a has no transition with label and c has no outgoing edge with label, then d has no transition with label and b is therefore eliminated. Hence G has no solution with delay. b d b d b Table 3.4: Eliminated table T 2 of G 2 b b Table 3.5: Eliminated table T 2 of G Solvable graphs of sequential Boolean equations with delay d Denition A graph G is solvable with a nite delay d with respect to Σ if the knowledge of rst d labels of a word l l... l d is sucient to determine the initial vertex v, and for every t the knowledge of v t, l t l t+... l t+d is sucient to determine a vertex v t+, such that the determined part of the sequence of vertices is a part of the path with the given word of labels. Denition For a solution with a nite delay d in G the set of graphs G d for d is dened as follows:. G = G; 2. The vertices of G d are d + -tuples v, l, l,..., l d where v V and l, l,..., l d Σ; 3. The vertex v, l, l,..., l d is joined to vertex v, l, l 2,..., l d, l with the label l if and only if v δ {v}, l ;

35 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 3 4. There are no other edges in G d. Each vertex in G d contains information of the next d letters. A transition in G d is equal to transitions of d + letters in G. To decide whether G has a solution with delay d is equivalent to decide whether G d has a solution with delay. Example We construct G from Table 3.3 and represent the result in Table 3.6. We start with eliminating states a and c, because they both have no outgoing edge with neither label nor label. Through the same procedure all the states on Table 3.6 are eliminated, which means G does not have a solution with delay either. G 2 is constructed in Table 3.7. All states in G 2 are also eliminated. The same happens to G 3 in Table 3.8. Hence G has a solution neither with delay 2 nor with delay 3. a a b b b d d b a, b, c a, b, c c a, b a, b c d c c d b b Table 3.6: Table of G Hereto an algorithm for deciding whether or not a sequential Boolean equation has solution with delay d is provided. Unfortunately we are not willing to go through the same procedure with any given delay d, but seek an ecient method for nding the upper bound of a nite-delay solution. In the following section, we introduce a method to determine the existence of a nite-delay solution for a sequential Boolean equation.

36 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 3 a a a b b a b b b d d b d d b a, b, c a, b, c b a, b, c a, b, c c a, b a, b c a, b a, b c c d c c d c c d b b d b b Table 3.7: Table of G Solvable graphs of sequential Boolean equations with nite delay In the previous section we introduced a method to decide whether an automaton graph is solvable with a given delay d. Now we raise another problem of deciding whether an automaton graph is solvable with a nite delay. We look for an upper bound of a nite-delay solution in a graph G. In order to solve the problem we dene a d-merge graph G d. Denition Let G = V, V, E be a graph with labels from Σ, a d-merge of G, denoted by G d, is dened as follows:. The vertices of G d are V from the graph G; 2. Σ d denotes the set of all label sequences of Σ with length d; 3. Between v i V and v j V there is an edge in G d with label l l... l d for l, l,..., l d Σ, if and only if a path from v i to v j in G exists with the same label sequence.

37 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 32 a a a a a b b a b b a b b a b b b d d b d d b d d b d d b a, b, c a, b, c b a, b, c a, b, c b a, b, c a, b, c b a, b, c a, b, c c a, b a, b c a, b a, b c a, b a, b c a, b a, b c c c c d c c d c c d c c d c c d b b d b b d b b d b b Table 3.8: Table of G 3

38 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 33 Even and Meyer declare that there exists an algorithm for deciding whether a given graph with n vertices is solvable with a nite delay. It is proved by the following lemmata [6]: Lemma If G is solvable with delay d then G d is solvable with delay. Since a label-alphabet on G k denoted by Σ k is the set of all label sequences of Σ with length k, it is obvious that d transitions in G is equal to a transition in G k. Lemma If G d is solvable with delay then G is solvable with delay 2d. Each d-merge graph G d which is solvable with delay guarantees 2 transitions with each label sequence of length d in G. Therefore the corresponding graph G is solvable with delay 2d. Denition Two graphs G and G are similar if they have the same set of vertices, i.e., V = V ; and for every letter l Σ, with Σ denoting the set of labels of G, there exists a letter l Σ, with Σ denotes the set of labels of G, such that for every v V, {v}, l = {v}, l, and for every l Σ there exists an l Σ for which the same condition holds. Notice that the number of letters in Σ is not necessarily the same as the number of letters in Σ. Lemma If G and G are similar and G is solvable with delay d, then G is solvable with delay d. Proof. We dene a relation between letters of Σ and letters of Σ for l Σ and l Σ as follows: l l v V {v}, l = {v}, l. This means for every l Σ there exists an l Σ, and for every l Σ there exists an l Σ, such that l l. We assume that a word l l... l n is given, G is solvable with delay is demonstrated in the following steps:. Let the initial vertex be one of the initial vertices which is chosen in G, if the rst label on the word is l where l l. Let v be the initial vertex.

39 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations a b, c a, c b, c a, b, c b c a c a a, b b Table 3.9: Table of G 2. Let l i be any letter of Σ which satises l i l i. For all < i < n choose v i as the next vertex in G, when the present vertex is v i and the rst two letters are l i and l i. Because of the similarity, a safe transition for G implies a safe transition for G. This lemma can be generalized to any delay d. Lemma If G i is similar to G j then, for every k >, G i+k is similar to G j+k. Proof. Let {v}, l l... l j+h be the set of vertices connected to v with a path labeled l l... l j+h in G j+h, therefore {v}, l l... l j+h = {v}, l l... l j, l j l j+... l j+h. Since G i is similar to G j there exists a word l l... l i such that for all v V {v}, l l... l j = {v}, l l... l i. Thus, {v}, l l... l j+h = {v}, l l... l i, lj l j+... l j+h = {v}, l l... l i l j l j+... l j+h. The same argument may be repeated with the roles of i and j reversed. Example Let G be a graph given in the Table 3.9. With eliminating of label 3 we obtain a graph G represented in Table 3.. According to the above-mentioned denition, G is not solvable. Table 3. represents a 2-merge graph G 2. G 2 is obtained by elimination of the labels,, 2 and, 2, and represented in Table 3.2.

40 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations 35 2 a a, c a, c b, c b c c a a, b Table 3.: Table of G a a, c a, b a, b, c a, c a, b, c a, c a, b b a a, b c b, c a, c b, c b, c a, c b, c Table 3.: Table of G 2 22 a a, c a, b b c a, c b, c Table 3.2: Table of G a a, b, c a, c a, b, c b, c a, c b, c b c a, b, c a, c a, b, c a, c a, b Table 3.3: Table of G a a, c b, c b c a, c a, b Table 3.4: Table of G 3

41 Chapter 3. Finite-Delay Solutions for Sequential Boolean Equations a a, b, c a, c a, b, c a, c a, b b c a, b, c a, c a, b, c b, c a, c b, c Table 3.5: Table of G a a, c a, b b c a, c b, c Table 3.6: Table of G 4 G 2 has no solution with delay. Therefore we construct G 3 with the words of length 3 consisting a label of G 2 and a label of G as shown in Table 3.3., 2,, 2, 22 and 22 are eliminated, therefore we obtain Table 3.4 representing the 3-merge graph G 3. We continue with the construction of 4-merge graph G 4, Table 3.5 represents G 4 which is constructed from the labels of G 3 and G. The eliminated 4-merge graph G 4 is represented in Table 3.6. Since G 4 is similar to G 2 and G 4 has no solution with delay, k-merge graph has no solution with delay for all k > 4. Hence, G has no solution with nite delay. The algorithm of Even and Meyer provides algorithms solving not only the problem of deciding whether there exist solutions with nite delay for sequential Boolean equations, but also the problem of deciding whether player O has nite-delay winning strategies in safety games. In Even-Meyer automaton graphs, we consider player I choosing labels in a path as an input word, and player O determining the corresponding output word by making moves according to the input bits from player I. A safety condition is fullled if there is always a safe transition in the automaton graph according to a input sequence from player I. Based on the result from Even and Meyer, Hosch and Landweber [8] proved that the problem whether Church's Problem with conditions stated in sequential calculus are solvable with delay is decidable.