# Representing Reversible Cellular Automata with Reversible Block Cellular Automata

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, Representing Reversible Cellular Automata with Reversible Block Cellular Automata Jérôme Durand-Lose Laboratoire ISSS, Bât. ESSI, BP 145, Sophia Antipolis Cedex, France received February 4, 2001, revised April 20, 2001, accepted May 4, Cellular automata are mappings over infinite lattices such that each cell is updated according to the states around it and a unique local function. Block permutations are mappings that generalize a given permutation of blocks (finite arrays of fixed size) to a given partition of the lattice in blocks. We prove that any d-dimensional reversible cellular automaton can be expressed as the composition of d 1 block permutations. We built a simulation in linear time of reversible cellular automata by reversible block cellular automata (also known as partitioning CA and CA with the Margolus neighborhood) which is valid for both finite and infinite configurations. This proves a 1990 conjecture by Toffoli and Margolus (Physica D 45) improved by Kari in 1996 (Mathematical System Theory 29). Keywords: Cellular automata, reversibility, block cellular automata, partitioning cellular automata. 1 Introduction Cellular automata (CA) provide the most common model for parallel phenomena, computations and architectures. They operate as iterative systems on d-dimensional infinite arrays of cells (the underlying space is d ). Each cell takes a value from a finite set of states S. An iteration of a CA is the synchronous replacement of the state of each cell by the image of the states of the cells around it according to a unique local function. The same local function is used for every cell. A block is a (finite) d-dimensional array of states. A block permutation (BP) is a generalization of a permutation of blocks, over a regular partition of the lattice d into blocks. A reversible block cellular automaton is the composition of various BP which use the same permutation of blocks e. If e is just a mapping not necessary a permutation we speak of block cellular automaton. Block cellular automata (BCA) are also known as CA with the Margolus neighborhood or partitioning cellular automata. Let us notice that BCA are not partitioned CA as introduced by Morita [Mor95]. Reversible cellular automata (R-CA) are famous for modeling non-dissipative systems as well as for being able to backtrack a phenomenon to its source. Reversibility is also conceived as a way to reduce heating and save energy. We refer the reader to the 1990 article by Toffoli and Margolus [TM90] for a wide survey of the R-CA field (history, aims, uses, decidability... ) and a large bibliography (even though it is quite old). In this paper, the authors made the following conjecture about R-CA: c 2001 Maison de l Informatique et des Mathématiques Discrètes (MIMD), Paris, France

2 146 Jérôme Durand-Lose Conjecture 1 [TM90, Conjecture 8 1] All invertible cellular automata are structurally invertible, i.e., can be (isomorphically) expressed in spacetime as a uniform composition of finite logic primitives. A finite logic primitives is a representation of a permutation of blocks e. Kari [Kar96] proved Conj. 1 for dimensions 1 and 2. At the end, he conjectures that: Conjecture 2 [Kar96, Conjecture 5 3] For every d 1, all reversible d-dimensional cellular automata are compositions of block permutations and partial shifts. In our definition, the shift is included in the definition of the block permutation. It should be noted that a reversible CA are quite tricky to design while reversibility is very simple to achieve with BCA. Moreover, reversibility of CA is undecidable in dimension greater than 1 [Kar94] whereas it can be checked easily for BCA. This does not contradict the conjecture since only invertible / reversibility CA are concerned. In [DL95], Conj. 1 is proved for any dimension d with 2 d 1 1 BP of width 4r, where r is the greater of the radii of the neighborhood of the CA and of its inverse. Here, Conj. 1 is proved to be true for any dimension d with a lower number of permutations, d 1, than proposed by Kari (exponential in d). The construction presented here is done by progressively erasing previous states and adding next states. The d 1 partitions are not regularly displayed, their origins are aligned; one goes from a partition to the next one by a constant translation of (3r, 3r,... 3r). All definitions and proofs in this paper can be read without any previous knowledge of the subject. The paper is structured as follows. The definitions of cellular automata, block partitions, reversibility and simulation are given in Section 2. In Section 3, for any reversible CA A, we exhibit d 1 BP such that the global function of A corresponds to their composition. During the simulation, some cells have to store their previous and next states. This is achieved by embedding the states in S 2 (S is the set of states of the simulated R-CA). All permutations of blocks are compatible, i.e., the useful parts of their domains and ranges do not overlap. In Sect. 4, we collapse the set of states of the BP from S 2 to S S 2 and prove that all the local functions of the BP are compatible. This allows to iterate the composition of the BP and to built the corresponding BCA. This yields a simulation where encoding and decoding functions are identities, what we call a representation. It should be noted that Kari also used S S 2 as the set of states for intermediate configurations. 2 Definitions Let d be a natural number. Cellular automata and block permutations define mappings over d-dimensional d infinite arrays over a finite set of states S. Let C S be the set of all configurations. For any configuration c and any subset E of d, c E is the restriction of c to E. The set of integers i i 1 i 2 j is denoted i j. For any x d, σ x is the shift by x (i.e. c C i d σx c i c i x). 2.1 Cellular automaton A cellular automaton (CA) A is defined by d S r f, where the radius r is a natural number and the local function f maps S 2r 1 d into S. The global function of A G A maps configurations into configurations as

3 Representing Reversible Cellular Automata with Reversible Block Cellular Automata 147 follows: c C i d GA c i f c i r r d The next state of a cell depends only on the states of the cells which are at distance at most r. All cells are updated simultaneously and no global variable is used. 2.2 Block permutation A block permutation (BP) is defined by d S w o e where the width w is a natural number. We define the volume V to be the following subset of d : V 0 w 1 d. The coordinate o belongs to V. We call block a mapping from V to S, or, equivalently, an array of states whose underlying lattice is V. The block function e is a permutation of the blocks, e : S V S V. The block permutation T is the following mapping over C: for any c C, for any i d, let a idivw and b imodw (a d and b 0 w 1 ) so that i a w b, then T c i e c a w V b. In other words, the configuration is partitioned into regularly displayed blocks, then each block is replaced by its image by the permutation of blocks e as in Fig. 1. o 1 o 2 w w w w b 0 1 b 1 1 b 0 0 b 1 0 T o1 o 2 Fig. 1: T o1 o 2, the block permutation of width w and origin o 1 o 2. e b 0 1 e b 1 1 e b 0 0 e b 1 0 The block permutation of origin o, T o is σ o T σ o. It is the same as before but the partition is shifted by o. An example of a two-dimensional block permutation is given in Fig. 1. We call reversible block cellular automaton (R-BCA) the composition of various BP with the same volume w and permutation e. If e is a mapping not necessarily a permutation of blocks, the composition is referred as just a block cellular automaton (BCA). The reversible term is explained in the next subsection. 2.3 Reversibility Both cellular automata, block CA and block permutations define synchronous and massively parallel d mappings G over S. An automaton A is reversible (or invertible) if and only if G A is bijective and there exists another 1 automaton B such that G B G A. The automaton B is called the inverse of A. Reversible CA are denoted R-CA. Amoroso and Patt [AP72] provided an algorithm to check whether a 1-dimensional cellular automaton is reversible or not. Kari [Kar94] proved that the reversibility of CA is undecidable in greater dimensions. By construction, BP are reversible; one simply uses the inverse permutation on the same partition to get the inverse block partition. By composition, it is easy to prove that a block CA is reversible if and only if its block function e is a permutation.

4 148 Jérôme Durand-Lose 2.4 Simulation Since we are dealing with iterative systems, we use the following definition. For any two functions f : F F and g : G G, we say that g simulates f in linear time τ if there exist two encoding functions α : F G and β : G F, space and time inexpensive compared to f and g, such that: x F n f n x β g τn α x (1) This corresponds to the commuting diagram of Fig. 2. The function g can be used instead of f for iterating. n 0 n α F G f n g τn Fig. 2: g simulates f in linear time τ. F G β For n 0, we get x β α x x. If both f and g are invertible and g simulates f, by the uniqueness of predecessors, (1) still holds for n negative. An automaton simulates another if and only if its global function simulates the global function of the other. We speak of a representation if F is included in G, α is the identity and β is undefined out of F. The domain of g does not have to be included in F, but g τ F must be included in F. If factor τ is 1 then the simulation is in real time. 3 Simulation of a R-CA by a composition of BP Let A S d r f be a reversible cellular automaton. In this section, we prove that it can be simulated by the composition of d 1 block permutations. The inverse CA of A, A 1, can be effectively computed: it is easy to built the composition of two CA and check whether it is the identity. Since there are countably many CA, it is possible to test every CA until the inverse is found. The algorithm stops in finite time; but in any dimension greater than one its complexity cannot be bounded by any computable function because of the undecidability of reversibility. We consider that the radius r of A is large enough for both A and A 1. Let w 3r d 1 be the common width of all the BP. 3.1 Block partitions The following sets are finite sub-arrays of d. They are used to locate previous and next states during the iterations. We denote r the vector r r r of d. For every λ in 0 d 1, let Eλ 3µr r 3 d 1 r λ µ r 1 d and d 1 3µr r 3 d 1 r r 1 d E λ 0 µ λ

5 Representing Reversible Cellular Automata with Reversible Block Cellular Automata 149 These sets are supposed to be completed by and close under all 3 d will never be indicated to ease the presentation. 1 r shifts in every direction. This Lemma 3 These sets verify the symmetry E λ 3rd Ed 1 λ Ed 1 E 0 /0 and E 0 E d 1 d. for 0 λ d 1; and the equalities: Proof The symmetry and the equality with /0 are obvious. Let us prove that E d 1 d, the last equality follows by symmetry. Let x be any element of the underlying lattice d. The d 1 sets (of ) 3λr r r 1 (for λ 0 d ) are non-empty and disjoint. Since x has d coordinates, there exists λ 0 such that none of the coordinates of x belongs to 3λ 0 r r r 1. This means that all the coordinates of x belong to 3λ 0 r r 3 d 1 r r 1 remember that E λ is closed by 3 d 1 r shifts thus x E d 1. The BP use the set of states S 2 to store the previous configuration in the first component and the next one in the second component. The state stand for the missing parts. Let us define the following configurations: c C λ 0 d 1 E λ c c E λ G c Eλ The states are completed by everywhere they are not in of the restrictions; also denotes the configuration where all cells are in state. Lemma 3 implies that: E 0 c c and E d 1 c G c. Let B λ be a BP of width 3 d 1 r and origin 3λr. The width of B λ matches the length of the shift closure of the sets Eλ and E λ. The partition of blocks e λ is defined so that B λ maps reversiblely E λ c into E λ 1 c. The aim is to get the commuting diagram of Fig. 3. c G A G A c B 0 B 1 B d c E 0 c E 1 c E 2 c E d c E d 1 c G A c Fig. 3: Simulation commuting diagram. The BP B λ add next states and erase previous states. Let us prove that there is enough data in E λ c to compute the added next states, and then that the erasing of previous states is done reversiblely. Lemma 4 For any λ in 0 d, there is enough information in E λ c to compute E λ 1 c. Proof The next states added belong to: 3λr r 3 d 1 r r 1 λ E λ 1 E λ d d 3µr r 3 d 1 r 0 µ r 1 λ

6 150 Jérôme Durand-Lose For any x λ, x 3λr r 3 d 1 r r 1 d. All the cells of the neighborhood of x should still hold their previous states in order to compute the next state of x. Cell x and its neighbors are all in the block 3λr 0 3 d 1 r 1 d. It corresponds to the block of the partition of B λ since its origin is 3λr. Now, it only remains to verify that previous states needed to compute the next state of x are still present. For any µ in 0 λ 1, since x 3µr r 3 d 1 r r 1 d, there is some index j µ such that x jµ 3µr r 3 d 1 r r 1. So x jµ is in 3µr r r 1 (remember that all is 3 d 1 r periodic). Since the sets 3µr r r 1 are disjoint, all j µ must be different and there are λ of them. Let y be any cell needed to compute x, y belongs to x r r, then, for all µ in 0 d 1, y jµ must be in 3µr 2r 2r 1. By contradiction, let us assume that there exists such a y which does not belong to Eλ then for all ν λ d 1, there exists some k ν such that y kν does not belong to νr r 3 d 1 r r 1, or equivalently, y kν 3νr r r 1. Since the sets 3νr r r 1 are disjoint, all the k ν must be different and there are d 1 λ of them. Altogether, there are d 1 j µ and k ν for d values so there exist µ 0 and ν 0 such that j µ0 k ν0. Then the intersection of 3µ 0 r 2r 2r 1 and 3ν 0 r r r 1 is not empty. This means that µ 0 ν 0, but by construction, µ 0 ν 0. Thus y belongs to Eλ and all the previous states needed to compute the next state of x are still present in the block. The next state of x can be computed with the information held inside the block. Using the symmetry between E and E, the previous states erased can be computed from the next states in E λ 1 c. This means that the corresponding blocks of E λ c and E λ 1 c in the partition of B λ can be uniquely determined one from the other. The partial function e λ can be completed so that e λ is a permutation. The two BP for dimension 1 are given in Fig. 4. This corresponds to the construction of Kari [Kar96]. G A r e 1 e 1 e 1 e 2 e 2 Previous states, Next states. Fig. 4: The two steps in dimension 1. In the 2-dimensional case, the partitions are given in Fig. 5. The three BP are detailed in Fig. 6. It does not correspond to the construction of Kari any more. 4 Collapsing the states and the permutations We collapse the states on S S 2 to make iterations of the simulating R-BCA possible and to get a representation. We also show that the partial definitions of functions e λ are compatible so that they can be merged into a unique e to define a reversible block cellular automaton.

7 Representing Reversible Cellular Automata with Reversible Block Cellular Automata 151 Fig. 5: The three partitions in dimension 2. Previous states, Next states. Fig. 6: The 3 steps in dimension 2. Lemma 5 The current BP B λ can be identified by the position of the double states inside the blocks of partition λ. Proof If all cells are single, then λ 0.

8 152 Jérôme Durand-Lose For i in 0 d, let ε i be the following element of d : ε 0 3λr ε 1 3λr 3r 3r 3r 3r ε 2 3λr 3r 6r 6r 6r 6r ε i 3λr 3r 6r 9r 3 i 1 r 3ir 3ir ε d 3λr 3r 6r 9r 3dr E Let us prove that the cell at ε i holds two states only if it belongs to both E λ and λ. The equation 3λr 3r 6r 9r 3 i 1 r 3ir 3ir 3λr r 3 d 1 r r 1 d E λ λ d 1 implies that ε i belongs to Eλ for λ d which is always the case. The point ε i belongs to E λ ε i 0 µ λ 3µr r 3 d 1 r r 1 3r 6r 3 i 1 r 3ir d that is (by Lemma 3 and a shift): 3ir λ µ 0 3µr r 3 d 1 r r 1 d only if The situation is depicted in Fig. 7. The fact that 3r is a coordinate of ε i implies that the shifted intervals are not to be considered. Since 3r, 6r, 9r,... and 3ir have to be in the interval, the maximum i such that the equality is true is λ 1. Then λ is the maximum i such that ε i holds two states, plus one. If there is no such i then λ 0. µ λ µ 2 µ 1 3 d 1 r 3 λ 1 r 9r 6r 3r 0 3 d 1 r Fig. 7: Graphical representation of the inclusion of ε i following any direction. Inside a block of the λ partition, ε i simplifies to 3r 6r 9r 3 i 1 r 3ir 3ir, which is independent of λ. It is enough to know the position of the double states in a block to know which permutation of blocks e λ to use. Remark 6 Thanks to the symmetry (Lem. 3), it also holds that the position of double states after the BP indicate which e λ was used. Above Lemma and Remark show that all the partial definitions of the permutations of blocks of the BP are compatible for domains and ranges. They can be grouped in a unique bijective local function and states can be collapsed. Altogether:

9 Representing Reversible Cellular Automata with Reversible Block Cellular Automata 153 Theorem 7 In any dimension d, any R-CA of radius r can be expressed as a composition of d width 3 d 1 r and with the same permutation of blocks e, or as a R-BCA with d 1 partitions. 1 BP of The origins of the partitions are: 0, 3r, 6r, 9r... and 3dr. 5 Conclusion Conjectures 1 and 2 are true. It should be noted that even if states in S 2 are used in intermediate configurations during the simulation, it perfectly works since the input and output are restricted to S. Since the BP are compatible and the states are collapsed, the composition can be iterated directly. This defines a reversible block cellular automaton (with d 1 BP) which simulates the R-CA in real time. The fact that the BCA representation can be effectively constructed does not contradict the undecidability of reversibility of CA because the inverse CA is needed for the construction. The proof of Th. 7 is not as explicit and visible as the one in [DL95]. Nevertheless, the number of BP needed is lowered from 2 d 1 1 to d 1. Generation and erasement are done concurrently, not one after the other as in [DL95]. We believe that it is not possible to make a representation with less that d 1 BP. Compare to [DL95], the main drawback is that the volume of the blocks is 3r d 1 d instead of 4r d. The complexity of a BP (or a BCA) is the size of the table of its local function e. It should be noted that if the number of BP is decreasing, the complexity is increasing. The expression with BP allows one to use reversible circuitry in order to build R-CA. This was done in [DL95] to prove that, for 2 d, there exists d-dimensional R-CA (based on the the Billiard ball model) able to simulate any d-dimensional R-CA in linear time on infinite configurations. This result was extended in [DL97] to the first dimension. For more about the relation between R-CA and BCA, we refer to the 1999 article by Kari [Kar99] which refers to an unpublished earlier version of the present article. References [AP72] [DL95] S. Amoroso and Y. Patt. Decision procedure for surjectivity and injectivity of parallel maps for tessellation structure. Journal of Computer and System Sciences, 6: , J. Durand-Lose. Reversible cellular automaton able to simulate any other reversible one using partitioning automata. In LATIN 95, number 911 in LNCS, pages Springer-Verlag, [DL97] J. Durand-Lose. Intrinsic universality of a 1-dimensional reversible cellular automaton. In STACS 97, number 1200 in LNCS, pages Springer-Verlag, [Kar94] J. Kari. Reversibility and surjectivity problems of cellular automata. Journal of Computer and System Sciences, 48(1): , [Kar96] J. Kari. Representation of reversible cellular automata with block permutations. Mathematical System Theory, 29:47 61, [Kar99] J. Kari. On the circuit depth of structurally reversible cellular automata. Fundamenta Informaticae, 38(1 2):93 107, 1999.

10 154 Jérôme Durand-Lose [Mor95] K. Morita. Reversible simulation of one-dimensional irreversible cellular automata. Theoretical Computer Science, 148: , [TM90] T. Toffoli and N. Margolus. Invertible cellular automata: A review. Physica D, 45: , 1990.

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### (Notice also that this set is CLOSED, but does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.)

Definition 3.1 Group Suppose the binary operation p is defined for elements of the set G. Then G is a group with respect to p provided the following four conditions hold: 1. G is closed under p. That is,

### Reliably computing cellular automaton, in 1-sparse noise

Reliably computing cellular automaton, in 1-sparse noise Peter Gács Boston University Peter Gács (Boston University) TRG Spring 2010 1 / 23 Ths is the second one of a series of three lectures on reliable

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

### Properties of Stabilizing Computations

Theory and Applications of Mathematics & Computer Science 5 (1) (2015) 71 93 Properties of Stabilizing Computations Mark Burgin a a University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Sets, Relations and Functions

Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete

### FORMAL LANGUAGES, AUTOMATA AND COMPUTATION

FORMAL LANGUAGES, AUTOMATA AND COMPUTATION REDUCIBILITY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 1 / 20 THE LANDSCAPE OF THE CHOMSKY HIERARCHY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 2 / 20 REDUCIBILITY

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.

Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### Notes on Complexity Theory Last updated: August, 2011. Lecture 1

Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

### Review for Final Exam

Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 6 Permutation Groups Let S be a nonempty set and M(S be the collection of all mappings from S into S. In this section,

quired. The ultimate form of a pattern is the result of an infinite number of steps, corresponding to an infinite computation; unless the evolution of the pattern is computationally reducible, its consequences

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.

MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### Finite dimensional C -algebras

Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Testing LTL Formula Translation into Büchi Automata

Testing LTL Formula Translation into Büchi Automata Heikki Tauriainen and Keijo Heljanko Helsinki University of Technology, Laboratory for Theoretical Computer Science, P. O. Box 5400, FIN-02015 HUT, Finland

### Theory of Computation

Theory of Computation For Computer Science & Information Technology By www.thegateacademy.com Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, Context-Free Grammar s

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Chapter 10. Abstract algebra

Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS

FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure

### 1.1 Logical Form and Logical Equivalence 1

Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic

### Topology and Convergence by: Daniel Glasscock, May 2012

Topology and Convergence by: Daniel Glasscock, May 2012 These notes grew out of a talk I gave at The Ohio State University. The primary reference is [1]. A possible error in the proof of Theorem 1 in [1]

### Automata and Languages

Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful

### The Language of Mathematics

CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,

### Genetic Algorithm Evolution of Cellular Automata Rules for Complex Binary Sequence Prediction

Brill Academic Publishers P.O. Box 9000, 2300 PA Leiden, The Netherlands Lecture Series on Computer and Computational Sciences Volume 1, 2005, pp. 1-6 Genetic Algorithm Evolution of Cellular Automata Rules

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### Classical Information and Bits

p. 1/24 Classical Information and Bits The simplest variable that can carry information is the bit, which can take only two values, 0 or 1. Any ensemble, description, or set of discrete values can be quantified

### The Inverse of a Square Matrix

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### SETS, RELATIONS, AND FUNCTIONS

September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four

### Geometric models of the card game SET

Geometric models of the card game SET Cherith Tucker April 17, 2007 Abstract The card game SET can be modeled by four-dimensional vectors over Z 3. These vectors correspond to points in the affine four-space

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### 3. Prime and maximal ideals. 3.1. Definitions and Examples.

COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### 4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,

### The Halting Problem is Undecidable

185 Corollary G = { M, w w L(M) } is not Turing-recognizable. Proof. = ERR, where ERR is the easy to decide language: ERR = { x { 0, 1 }* x does not have a prefix that is a valid code for a Turing machine

### Omega Automata: Minimization and Learning 1

Omega Automata: Minimization and Learning 1 Oded Maler CNRS - VERIMAG Grenoble, France 2007 1 Joint work with A. Pnueli, late 80s Summary Machine learning in general and of formal languages in particular

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction

italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### CSE 135: Introduction to Theory of Computation Decidability and Recognizability

CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 04-28, 30-2014 High-Level Descriptions of Computation Instead of giving a Turing

### Cellular objects and Shelah s singular compactness theorem

Cellular objects and Shelah s singular compactness theorem Logic Colloquium 2015 Helsinki Tibor Beke 1 Jiří Rosický 2 1 University of Massachusetts tibor beke@uml.edu 2 Masaryk University Brno rosicky@math.muni.cz

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### Lecture 2: Universality

CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

### 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

### Integrating Benders decomposition within Constraint Programming

Integrating Benders decomposition within Constraint Programming Hadrien Cambazard, Narendra Jussien email: {hcambaza,jussien}@emn.fr École des Mines de Nantes, LINA CNRS FRE 2729 4 rue Alfred Kastler BP

### Coefficient of Potential and Capacitance

Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that

### On an algorithm for classification of binary self-dual codes with minimum distance four

Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory June 15-21, 2012, Pomorie, Bulgaria pp. 105 110 On an algorithm for classification of binary self-dual codes with minimum

### ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS

ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract If n is a positive integer, write F n for the nth Fibonacci number, and ω(n) for the number

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

### Fundamentele Informatica II

Fundamentele Informatica II Answer to selected exercises 1 John C Martin: Introduction to Languages and the Theory of Computation M.M. Bonsangue (and J. Kleijn) Fall 2011 Let L be a language. It is clear

### THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER

THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### 1 Limiting distribution for a Markov chain

Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check