Notes on Monoids and Automata

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1 Notes on Monoids and Automata Uday S. Reddy November 9, 1994 In this article, I define a semantics for Algol programs with Reynolds s syntactic control of interference?;? in terms of comonoids in coherent spaces also called correletation spaces. 1 Background: Monoids 1.1 Definition A monoid in SET is a triple M = M, 1, where M is a set, 1 M is an element called the unit, and : M M M is a binary operation called multiplication such that the following identities hold: x 1 = x = 1 x, for all x M, and x y z = x y z, for all x, y, z M. Note that the unit is necessarily unique. We often omit the operator and write x y as simply xy. The structure is called a semigroup if we don t insist on the unit being present. It is called a commutative monoid if, in addition, xy = yx, for all x, y M. Suppose only some instances of the commutativity equation hold so that the structure is a partially commutative monoid. Then, we can define a relation M M such that x y xy = yx. Evidently, is reflexive and symmetric. A zero in a monoid is an element 0 M such that 0x = 0 = x0 for all x M. Note that 0 is necessarily unique. A submonoid of M is a subset M M containing 1 and closed under multiplication. If x M, we use the notation x n for the n-fold product x x with x 0 = 1. x denotes the set { x n : n 0 }. Similarly, if S M, S denotes the set {x 1 x n : x 1,...,x n S }. Note that the S is the least submonoid of M including S. If S = M, we say that S is a set of generators for M. 1.2 Examples The following examples will be of much interest to our discussion: i Let Q be a set. The set of partial functions from Q to Q called transformations of Q forms a monoid. The identity transformation 1 Q is the unit and the composition f; g of transformations is the multiplication. We find it convenient to use postfix notation for transformations. Then, the above operations are defined by q1 Q = { q qfg, if qf is defined qf; g = undefined, otherwise 1

2 We denote this monoid by [Q Q]. It has a zero element, viz., the undefined transformation. Any submonoid of [Q Q] is called a transformation monoid. ii More generally, in any category C, the endomorphisms on an object A form a monoid, denoted End C A. So, the set of binary relations over Q and the set of total functions from Q to Q form monoids too. iii Let Σ be a set of symbols. The set of strings over Σ, denoted Σ is a monoid with concatenation as the multiplication and the empty string ǫ as the unit. It is called the free monoid generated by Σ. Similarly, the set of nonempty strings over Σ Σ + is the free semigroup generated by Σ, and the set of finite multisets over Σ is the free commutative monoid generated by Σ. Some more examples of passing interest are as follows: iv The set of natural numbers ω forms a monoid with multiplication as the binary operation and the integer 1 as the unit. This is, in fact, a commutative monoid. v The set of natural numbers, again, forms a commutative monoid with addition as the binary operation and 0 as the unit. This is isomorphic, in the sense to be made precise below, to the monoid {1} the free monoid generated by {1}. vi The set of n n matrices forms a monoid under matrix multiplication as the multiplication and the identity matrix as the unit. This is, of course, a special case of the endomorphisms in a category example mentioned above. The following examples give important constructions on monoids: vii If M = M, 1, is a monoid, its dual monoid is M, 1, where is defined by x y = y x. Thus, the notion of monoid is a symmetric concept. viii Let M be a monoid. For each x M, there corresponds a right multiplication operator R x : M M defined by zr x = zx. The set of right multiplication operators forms a monoid, in fact, a transformation monoid over M. The unit is R 1 and the multiplication is R x R y = R xy. In fact, the right multiplication monoid is isomorphic to the monoid itself. Similarly, the left multiplication operators L x : M M are defined by L x z = xz. They have a unit L 1 and multiplication L x L y = L yx. Thus, the left multiplication operators form a monoid that is isomorphic to the dual of M. ix Let M be a monoid. An equivalence relation M M is called a monoid congruence relation if x x y y = xy x y. A congruence class is a maximal set X M such that all the elements of X are equivalent to each other. The congruence class containing an element x is [x] = { y : y x }. The set of such congruence classes forms a monoid. The unit element is [1] and the multiplication of equivalence classes is defined by: [x] [y] = [xy]. The fact that is a congruence relation ensures that multiplication is well-defined as may be verified. This monoid is called the quotient monoid of M under, denoted M/. x A pair Σ,, where Σ is a set and Σ Σ a symmetric relation, is called an independence alphabet and its independence relation. Consider strings over Σ and the least congruence such that a b = ab ba. A congruence class [a 1 a n ] consists of all strings which only differ in the relative order of consecutive independent symbols. Such congruence classes are called traces and the quotient monoid Σ / is called the free partially commutative monoid generated by Σ. xi Let Σ be an alphabet and S Σ be a set of strings. S induces an equivalence on strings by x y z, z. zxz S zyz S. This is, in fact, a congruence relation, called the syntactic congruence or the Myhill congruence of S. Suppose x x and y y. Then, zxx z S zyx z S from the first assumption and zyx z S zyy z S from the second assumption. Hence, xx yy. Intuitively, x and y are equivalent if they behave 2

3 the same way as segments of strings in S. For example, let Σ = {a, b} and S be the regular set ab. Under the Myhill congruence of S, b b n+1 and a 2 a n+2 ba n+1. Thus, the congruence classes are [ǫ], [a], [b] and [a 2 ]. [ǫ] is the unit. [a 2 ] is a zero. For the others, multiplication is defined by [a] [b] = [a] and [b] [a] = [a 2 ]. The monoid Σ / is called the syntactic monoid of S. More generally, a subset of any monoid S M induces a syntactic congruence and a syntactic monoid. xii If S Σ is a set of strings, one can also define a syntactic right congruence on Σ by x r y iff z. xz S yz S. This is a right congruence relation in the sense that x r y = xv yv for all v Σ. For example, for S = ab, the right congruence classes are [ǫ], [a] containing all ab n and [a 2 ] containing all b n. Let the set of right congruence classes be Q. Then, each string x Σ has a corresponding right multiplication operator on Q, defined by [z]r x = [zx]. The set of right multiplication operators R x forms a monoid with unit R 1 and multiplication R x R y = R xy. This is in fact a transformation monoid over Q. As will be seen below, such a transformation monoid is nothing but an automaton. 1.3 Definition A monoid homomorphism h : M, 1, M, 1, is a function h : M M such that i h1 = 1, and ii hx y = hx hy for all x, y M. A one-one homomorphism is called a monomorphism and an onto homomorphism is called an epimorphism. 1 If there is an inverse homomorphism h 1 such that h h 1 = h 1 h = id then h is called an isomorphism, and M and M are said to be isomorphic. Whenever h : M M is a homomorphism, the image of h, defined by Imh = { y : x M. hx = u }, is a submonoid of M. Dually, the equivalence relation on M defined by x y hx = hy is a congruence relation called the kernel of h. The homomorphism h : M/ M given by h [x] = hx, is then a mono. Therefore, M/ and Imh are isomorphic. If f : Σ M is a function, it extends to a unique homomorphism h : Σ M such that h restricted to Σ is f. We can define it explicitly by ha 1 a n = fa 1 fa n. If h is a surjection, M is isomorphic to the quotient monoid Σ / where is the kernel of h. More generally, if M has a set of generators S, to specify a homomorphism h : M M, we only need to specify it on S. 2 Background: Automata 2.4 Definition Let Σ be a set the alphabet or the instruction set of the automaton. A Σ- automaton is a pair A = Q, F where Q is a set the state set and F : Σ [Q Q] is a function called the interpretation mapping instructions to transformations of Q. We also call A an automaton class of type Σ, and denote this fact by writing A = Q, F Σ. This kind of a machine is what is traditionally called a semiautomaton because we have no distinguished start and final states, except that the function F is often treated as a partial function of type F : Q Σ Q and called a transition function. This is just an uncurried version of our interpretation function. More significantly, note that we have no finiteness restrictions on Q or Σ. So, our automata are not necessarily finite state. 1 The latter is not an epi in the categorical sense. 3

4 We often write Fσ as F σ. The application of transformations is written in postfix: F σ q as qf σ. These conventions achieve much economy and will also be seen to be very natural with respect to monoids. F extends to a unique monoid homomorphism Σ [Q Q]. We denote this by F as well: F a1 a n = F a1 F an Note that the image of F is a transformation monoid over Q. We also extend F to sets S Σ. F S is a function F S : Q PQ defined by qf S = { q : x S, qf x = q }. So, qf S is the set of states reachable from q via instruction sequences from S. An automaton is said to be monogenic if there is a state q 0 Q such that all states are reachable from q 0, i.e., q 0 F Σ = Q. 2.5 Definition Let A = Q, F be a Σ-automaton. An A-machine is a pair M = A, q where q Q. A is called the class of the machine and q its current state. The function of a machine is to define a set: LM = qf Σ = { x Σ : qf x defined } LM is traditionally called the language of the machine, but one can also think of it as the set of instruction sequences which the machine successfully executes. Note that LM is prefix-closed. Another kind of machine is a difference machine M = A, i, t where i and t are states, called the initial and terminal states respectively. The language of a difference machine is LM = { x Σ : qf x = t } A recognizer M = A, i, T is like a difference machine except that there is a set of terminal states T Q. The language of a recognizer is defined similarly: LM = { x Σ : qf x T } 2.6 Let L Σ be a set. Consider the syntactic right congruence of L: x r y z, xz L yz L. Let Q be the set of right congruence classes. As noted in 1.2, the right multiplication operators of Σ form a transformation monoid over Q. If R : Σ [Q Q] is defined by [z]r x = [zx], then A = Q, R is a Σ-automaton. It is monogenic with q 0 = [ǫ]. By considering a suitable set of terminal states T = { [z] : z L }, we have a recognizer M = A, q 0, T with LM = L. This is, in fact, the minimal recognizer with language L. Thus, we obtain a recognizer for any language L Σ. This recognizer is, in fact, unique upto renaming of states. 2.7 Next, we show that L uniquely determines the automaton class itself. The effect of an instruction sequence x Σ is to take the automaton from one state to another. So, if two sequences have the same interpretation in the automaton, they are equivalent for the automaton. This equivalence coincides with the syntactic congruence of L. If A = Q, F is an automaton, we have that F x = F y = x LM y for all machines of class A. Suppose F x = F y. To say that zxz LM is to say that there are states q, q Q such that if z = q, qf x = q and q F z T. Since F x = F y, clearly, zxz LM zyz LM. Thus, x LM y. If L Σ, consider the right-congruence automaton A = Q, R constructed in 2.6. Suppose x L y, i.e., zxz L zyz L. Considering a particular z, we have zx r zy, and, so, [zx] = [zy] Q. Next, by varying z, we find that [z]r x = [z]r y for all z, which means R x = R y. 4

5 2.8 Definition The above considerations suggest that, in general, the alphabet of an automaton should be regarded as a monoid. Let X be a monoid. An X-automaton is a pair Q, F where Q is a set and F : X [Q Q] is a monoid homomorphism. Thus, Σ-automata are nothing but Σ -automata with Σ regarded as a monoid. We also call an X-automaton an abstract automaton of type X. Machines for X-automata can be defined similar to the above. If F is mono, i.e., F x = F y = x = y then the image of X under F is a submonoid of [Q Q]. In other words, X is isomorphic to a transformation monoid over Q. By extension, the automaton Q, F is itself called a transformation monoid. We also call it a fully abstract automaton. Our notion of automata allows us to make a smooth transition between syntax and semantics or, as will be seen later, between intensional and extensional semantics of programs. Given a concrete automaton, Q, F Σ, we can make it abstract by using the monoid X = Σ / where is the kernel of the homomorphism F : Σ [Q Q]. The factor F : X [Q Q] gives a fully abstract automaton Q, F X. Conversely, given an abstract automaton Q, F X, we can generate a concrete automaton by taking some set of generators S M to be Σ. 2.9 Examples i Let EL = {up, dn} be an alphabet. Consider the automaton El 3 with states Q = {0, 1, 2} and the interpretation F up = 1 2 F dn = 0 1 El 3 can be thought of as the behavior of primitive elevators running between three floors. up dn is interpreted as instruction to move up down to the next floor. The up instruction at the top floor and the dn instruction at the bottom floor are undefined. The automaton is monogenic with any state as q 0. A particular elevator is then an El 3 -machine. ii A more friendly elevator has the alphabet FEL = {up.clink, up.beep, dn.clink, dn.beep} where clink and beep are understood as the sounds emitted by the elevator in response to an instruction. The friendly version of the above elevator FEl 3 has the same state set but the following interpretation: F up.clink = F dn.clink = F up.beep = F dn.beep = 2 0 Note that up.clink is a single symbol of the alphabet. We discuss below how to decompose such symbols into their input and output parts. iii A counter object has the alphabet CNT = {val.i : i N } {inc}. The automaton Counter has states Q = N and interpretation: val.i : i i inc : i i + 1 for all i N It is monogenic with q 0 = 0. iv A stepper object that successively steps through a sequence of numbers has the alphabet STEP = {next.i : i N }. A stepper Pos that steps through the positive integers has states Q = N and interpretation next.i : i 1 i for all i 0. 5

6 3 The structure of automata 3.10 The first question to be asked is what are the morphisms for automata? This is to be considered carefully. If one naively interprets automata as structures, one would be led to morphisms that preserve such structure. On the other hand, automata are not structures; they are cocrete implementations. The state sets of automata are part of theri internal structure. Morphisms can only preserve externally visible structure. Suppose a function from automata of type Σ to automata of type Γ maps an automaton Q, F Σ to an automaton Q, F Γ. It is essentially constructing a new automaton Q, F Γ from an original automaton Q, F Σ. The following considerations arise: i The function must construct the states of the new automaton from the states of the original automaton. Thus, we need a partial function φ : Q Q. ii It should interpret each instruction of the new automaton by issuing instructions to the original automaton. This leads to a function f : Γ Σ. Note that this function runs backwards. iii The behavior of the original automaton under these instructions must be compatible with what is expected for the new automaton, i.e., the following diagram must commute: Q F fb Q φ Q F b φ Q iv Finally, since this must hold for every instruction b of the new automaton, we require that, for every state in Q, there corresponds some state in Q. In other words, φ must be surjective These considerations lead to the following definition definition An implementation of automata is a pair φ f : Q, F Σ Q, F Γ where f : Γ Σ is a function and φ : Q Q is a surjective partial function such that the above diagram commutes. We say that Q, F Σ implements Q, F Γ via φ f. Notice that an implementation uniquely determines the behavior of the new automaton F. To determine q F b, find a q φ 1 q which is nonempty by definition. Then, q F b = qf fb Examples i Consider function f : EL FEL given by up.clink up.beep dn.clink dn.beet f = up ǫ dn ǫ An implementation φ f : El 3 FEl 3 of friendly elevators in terms of ordinary elevators is given by the mapping φq = q. ii There is an implementation of two-storey elevators in terms of three-storey elevators: φ f : El 3 El 2 : f is id EL and φ is the partial assignment 0 0, 1 1. Clearly, there can be no implementation of El 3 using El 2. iii An implementation of the stepper Pos in terms of Counter is φ f where φ is id N and fnext.i = val.i 1 inc. 6

7 3.13 Next, we must add outputs to automata themselves. The traditional mechanism for this is in terms of Mealy machines. These machines have additional components for an output alphabet Θ and an output function G : Q Σ Θ. While this is is essentially what we need, the explicit framework turns out to be a little cumbersome. For instance, to define a Counter, we need val and inc in the input alphabets and N in the output alphabet. But, the machine should not produce an integer in response to inc. To circumvent such issues, we define the following variant of Mealy machines Definition A modular automaton is an automaton Q, F Σ equipped with two projection functions π i : Σ Σ i and π o : Σ Σ o to appropriate alphabets Σ i and Σ o, such that for any q Q, if qf a and qf b are defined π i a = π i b then a = b. An object is a modular automaton such that, for all q Q and a Σ i, there exists a Σ such that π i a = a and qf a is defined. For example, for counters, we can define Σ i = {val, inc} and Σ 0 = N + { }. Counter is then a modular automaton. In fact, it is an object because every state has transitions for val and inc operations. An implementation of modular automata is a triple φ f, g : Q, F Σ Q, F Γ where f : Γ i Σ i and g : Σ Γ are functions 3.15 Examples i An implementation of Counter in terms of storage cells is as follows: φ f : Cell Counter where φ is id N and val.i inc f = get.i get.i put.i + 1 7