Algebraic specifications : formal definitions

1
Algebraic specifications formal definitions

• The signature of an algebraic specification
  consists of sorts of some data and some
  operation symbols on them
• Definition 1 (Signature) A signature is a pair
  ? ?S, F?. With S a set of sorts, and F
  fs1,...,sn,s, with si ? F is a set of
  SxS-indexed operation symbols. Each operation
  is usually represented as f s1 x ...x sn
  s

2
Algebraic specifications formal definitions

• Signature may be graphically represented as
  graphs with
• There are two kinds of nodes each sort in an
  operation is node. Each operation is also
  represented by a node.
• There are two kinds of arcs. Indirected arcs from
  argument sorts nodes to an operation node, and a
  directed node from an opration to the resulting
  sort.

3
Algebraic specifications formal definitions
s1
.
s
.
f s1 ... sn,s
sn
4
Algebraic specifications formal definitions

• Examples
• (a) nat
• Sorts nat
• Ops 0 ? nat
• succ nat ? nat

succ
nat
0
5
Algebraic specifications formal definitions

• Examples
• (b) bool
• Sorts bool
• Ops true, false ? bool
• ? bool ? bool
• ?, ?, ... bool x bool ? bool

?
true
bool
false
?, ?, ...
6
Algebraic specifications formal definitions

• Examples
• (c) nat1 bool nat
• Ops nat x nat ? nat
• ? nat x nat ? bool

succ
true
0
nat
bool
?
false
?, ?, ...
7
Algebraic specifications formal definitions

• Examples
• (c) natstack nat1
• Sorts Stack
• Ops new ? stack
• push stack x nat ? stack
• pop stack ? stack
• top stack ? nat

push
succ
new
8
Algebraic specifications formal definitions

• Examples
• (c) natqueue nat1
• Sorts queue
• Ops empty ? queue
• in queue x nat ? queue
• out queue ? queue
• front queue ? nat

succ
in
empty
9
Algebraic specifications formal definitions

• There is a close relationship between signatures
  ? ?S, F? and context-free grammars.
• G(?)G(?)s0 (N,T,s0 ,P) s0?S
• Non terminal symbols are N S
• The teminal symbols are T F ? (,) ? ,
  where F is the disjunctive union of sets fs,s
• The start symbol for G(?)s0 is s0 .
• The production rules are given by
• P s f(s1 ,..., sn) / f?F
• G(bool) bool true false ?bool
  bool ? bool ...

10
Algebraic specifications formal definitions

• Definition 2 (Terms) given a signature ?
  ?S, F?, With S a set of sorts, and F
  fs1,...,sn,s.
• A set of (closed) terms of a sort s over F,
  denoted TF,s are constructed as follows
• Each constant of sort s is a term in TF,s.
• If t1, .., tn are terms of sorts s1, ..., sn
  respectively and fs1,...,sn,s is in F then
  f(t1,...,tn) is a term in TF,s
• There no terms of sort s except those from 1. and
  2.

11
Algebraic specifications formal definitions

• Examples
• Nat spec. 0 succ(0) succ(succ(0)) ....
  (succ(succ(0)),0) succ((succ(0),succ(succ(succ
  (0))))).
• bool spec. true ?true ?(true, ?true)
  ....
• Natstack spec. new push(push(push(new,0),0),s
  ucc(0)) push(pop(new),top(pop(new))) ....
• In addition of this prefix notations of terms, it
  is also possible to use the usual infix , postfix
  or mixfix notations succ(0) succ(succ(0))
  succ(succ(0))!, if true then succ(0) else
  succ(succ(succ(0)))

12
Algebraic specifications formal definitions

• Terms with variables usually to express and use
  terms we include in terms also variables.
• Given a signature SIG ?S, F?, we generally
  introduce a set Xs called a set of variables of
  sort s. Such variables are assumed to be pairwise
  disjoint and also disjoint with F. The union X
  ?s? S Xs is called set of variables w.r.t. SIG.
• Terms (with variables) of sort s are constructed
  as in definitions 2, with additionally in 1.
  variables of sort s are also terms.

13
Algebraic specifications formal definitions

• Examples
• Nat spec. Xnat x, y Tnat(X) 0 succ(0)
  succ(succ(x)) .... (succ(succ(0)),y)
  succ((succ(x),succ(succ(succ(x))))) ... .
• bool spec. Xbool z z, true ?true ?(z,
  ?true) ....
• Natstack spec. Xstackst new
  push(push(push(st,0),0),succ(0))
  push(pop(new),top(pop(st))) ....
• Terms without variables are called ground or
  closed terms. 0 succ(0) ...

14
Algebraic specifications formal definitions

• Definition 3 An algebra A (SA, FA) of a
  signature SIG ?S, F? , also called SIG-algebra,
  is given by two families SA (As)s?S and FA
  (fA) f? F , where
• As are sets for all s?S, called base sets or
  domains of A.
• fA are elements fA ? As for all constant symbols
  in F i.e.
• f ? s and s?S , called constant.
• fA As1 x As2 x ... x Asn ? As are functions
  for all operation symbols f s1...sn,s from F,
  called operations of A, where x denotes the
  cartesian product of sets.
• If a signature SIG ?S, F? is given as a list
  s1,..,sn of sorts and a list f1,..,fn of
  constants and operation symbols a SIG-algebra is
  represented as a list
• A (As1,....,Asn, f1A,..., fnA)

15
Algebraic specifications formal definitions

• Example
• Nat-algebra NAT (Snat, Fnat) (natnat, 0nat,
  succnat)
• natnat 0, 1 , 2 , 3 , ..... , n,...
  naturals
• 0nat 0 (of the natural)
• succnat successor (of the naturals) naturals
  ? naturals .

16
Algebraic specifications formal definitions

• Binary trees
• Binarytree-base
• sorts alphabet
• bintree
• opns K1,...,Kn alphabet
• LEAF alphabet ? bintree
• RIGHT bintree alphabet ? bintree
• BOTH bintree alphabet bintree ? bintree
• BINTREE-BASE (A , B , a1 ,.., an ,leaf,left,
  right, both) where
• A a1,...,an B the set of binary trees
• leaf A ? B is defined by leaf(ai) tree(ai)
  for i 1,...,n
• left B x A ? B adds a left subtree
• right A x B ? B adds a right subtree, and
• both B x A x B ? B adds a left and a right
  subtree .

17
Algebraic specifications formal definitions

• Define the terms of bintree, and give the term
  corresponding to

18
Algebraic specifications formal definitions

• Now we are going to define the evaluation of
  terms with and without variables in a given
  algebra A. For terms with variables we have to
  start an assignment for the variables. These
• Definition 4 (evaluation of terms)
• Let TF be the set of terms of a signature SIG
  ?S, F? and A a SIG-algebra. The evaluation eval
  TF ? A is recursively defined by
• (i) Eval (f) fA for all constant symbols f ?
  F
• (ii) Eval(f(t1,...,tn)) fA(eval(t1),...,eval(tn
  )) for all terms f(t1,...,tn) ? TF

19
Algebraic specifications formal definitions

• Given a set of variables X and SIG (S,F) and an
  assignment ass X ? A with ass(x) ?As for x ?
  Xs and s ? S. The extended assignment, or simply
  extension ass TF(X) ? A of the assignment ass
  X ? A is recursively defined by
• ass(x) ass(x) for all variables x
• ass(f) fA for all constant
  symbols f ? F
• (ii) ass(f(t1,...,tn)) fA(ass(t1),...,ass(tn))
  for all f(t1,...,tn) ? TF(X).

20
Algebraic specifications formal definitions

• For X ? there is exactly one assignment ass
  which the empty assignment, and we have ass
  eval .
• We have then families of functions
• eval (evals TF,s ? As)s?S
• ass (asss Xs ? As)s?S and
• ass (asss TF,s (X)? As)s?S
• The diagrams (1) and (2) commute
• That is TF,s ? TF (X) A TF A
• and X ? TF(X) A X A

TF(X)
TF(X)
X
(1)
(2)
ass
eval
A
ass
ass
ass
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Algebraic specifications formal definitions

• Evaluate add(succ(n),m)), with Xnat n,m and
  Assx(n)5 and assx(m) 3