Algebraic specifications : formal definitions

1
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
2
Algebraic specifications formal definitions

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

3
Algebraic specifications Equations

• By choosing an algebraic semantics to abstract
  data types, algebraic structures (e.g. monoid,
  group, rings, ..) are also endowed with equations
• They express the relationship between different
  operators.
• They allow to describe complex (defined) function
  symbols using elementary (constructor) operation
  symbols.
• They allow us to have a simplified form of any
  term.

4
Algebraic specifications Equations

• Examples
• (a) bool
• Sorts bool
• Ops true, false ? bool
• ? bool ? bool
• ?, ? bool x bool ? bool
• if_then_else_fi bool x bool x bool ? bool
• Vars p, q bool
• Eqs if true then p else q fi p
• if false then p else q fi q
• ? p if p then false else true fi
• p ? q if p then q else false fi
• p ? q if p then true else q fi

5
Algebraic specifications Equations

• (b) nat
• Sorts nat
• Ops 0 ? bool
• succ nat ? nat
• add nat x nat ? nat
• Vars n, m nat
• Eqs add(n, 0) 0
• add(succ(n), m)) succ(add(n,m))

6
Algebraic specifications Equations

• (b) nat1 natbool
• Ops ? nat x nat ? bool
• Vars n, m nat
• Eqs (0 ? n) true
• (succ(n) ? 0) false
• (succ(n) ? succ(m)) (n ? m)

7
Algebraic specifications Equations

• (b) natstack nat1
• Sorts S
• Ops new ? S
• push S x nat ? S
• pop S ? S
• top S ? nat
• Vars s S n nat
• Eqs pop(push(s,n) s
• top(push(s,n)) n
• pop(new) new
• top(new) 0

8
Algebraic specifications Equations

• Definition Given a signature SIG ?S, F? and
  variables X w.r.t. SIG
• A triple e (X , L, R) with L, R ? TF,s (X) for
  some s?S, is called an equation of sort s w.r.t.
  SIG.
• The equation e (X , L, R) is called valid in a
  SIG-algebra A if for all assignments ass X ? A
  we have
• ass(L) ass(R)
• Where ass is the extended assignment of ass. If
  e is valid in A we also say that A satisfies e.
• 3. Ground equations are equations e (X , L, R)
  with X ? (that is when L and R are ground
  terms).

9
Algebraic specifications Equations

• An equation is a universal first order formula
• ? x1 ? s1, ? x2 ? s2 . . . . ? xn ? sn (L R)
  written usally as x1 ? s1, x2 ? s2 . . . . xn ?
  sn (L R)
• X must contains all variables occuring in L and
  R.
• In general, for sake of simplicity the variables
  set is omitted.

10
Algebraic specifications Specification and
SPEC-algebra

• Definition (specification and SPEC-algebra)
• A specification SPEC ?S, F, E? consists of a
  signature SIG ?S, F ? nand a set E of equations
  e w.r.t. SIG variables X w.r.t. SIG
• An algebra A of the specification SPEC, short
  SPEC-algebra , is an algebra A of the signature
  SIG which satisfies all equations in E.

11
Algebraic specifications Specification and
SPEC-algebra

• A specification is also called algebraic
  specification or equational specification.
• If a specification SPEC1 consists of a given
  specification SPEC and additional sorts S1,
  operations F1, and equations E1, we write this in
  the form
• SPEC1 SPEC (S1, F1, E1)
• Which means
• SPEC1 (S, S1, F F1, E, E1)

12
Algebraic specifications Specification and
SPEC-algebra

• Show that NAT (N, , 1, ) is a nat-algebra,
  that is for each assignment ass X ? N with
  Xn,m the extended assignments applied to both
  equations deliver the same values.

13
Algebraic specifications Specification and
SPEC-algebra

• Definition (derivation of rewriting of terms)
• Given a set E of equations for a signature with
  a fixed set of variables X Xe for each
  equation e. (L,R) ? E defines two substitution
  rules
• (1) L gt R (L-R-rule)
• (2) R gt L (R-L-rule)
• A rule t1 gt t2 is applicable to a term t ?
  TF(X) if there is an assignment assX? TF(X) with
  extension ass TF(X) ? TF(X) such that we have
  for t1 ass(t1) and t2 ass(t2)
• (3) t1 is a subterm of t.

14
Algebraic specifications Specification and
SPEC-algebra

• The replacement of t1 in t by t2 yields a term
  t, the replacement of t1 by t2 in t is denoted
  by
• (4) t t(t1 / t2)
• In this case we write
• (5) t gt t, called direct
  derivation from t to t via E using the rule
  t1gt t2 and assignement ass.
• (6) t gt t represent any sequence
• t0gtt1 gt ....gt tn with t t0 and t
  tn. It is called derivation from t to t via E
  and it is correct w.r.t. SIG-algebra A if for
  each assignment ass X ? A
• (7) ass(t) gt ass(t)

15
Algebraic specifications Specification and
SPEC-algebra

• Definition (occurrence or positions in terms)
• Given a term t, the set of positions in t,
  denoted by Dom(t), is the set of sequences of
  natural numbers defined as
• If t is constant or variable, then Dom(t) ?
• If t is of the form f(t1, ..., tn) then
• Dom(t) ? ? i.p / i ? 1,..,n and p ?
  Dom(ti)
• Definition (subterms)
• Given a term t, and a position p ? Dom(t) we
  define a subterm of t rooted at a position
  denoted tp as
• p ?, then tp t
• If p i.pthen t f(t1, ...,ti,...)i.p
  tip
• A term t is said to be a subterm of a t is there
  is a position p such that t tp

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Algebraic specifications Specification and
SPEC-algebra

• Definition (Term replacement)
• Given a term t, a position p, and a term t, we
  define tp t as
• If p ? then tp t t
• If p i.p then t f(t1, ..., ti-1,ti,
  ti1...)i.p t
• f(t1, ..., ti-1,ti p t,
  ti1...)
• Definition (Rewriting term)
• Given a system of rules (oriented equations), R,
  we define a rewrite relation by gtR , as t gt
  t, if
• There is a rule r l gt r is R there is an
  assignement (substitution) ? X ? TF(X) and a
  position p in t such that tp ? (l) and t
  tp ? (r)

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Algebraic specifications Specification and
SPEC-algebra

• Definition (Congruence on Ground Terms)
• Given a specification SPEC (S, F, E) the
  relation ? on ground terms defined for all t1, t2
  ? TF by
• t1 ? t2 if and only if evalA(t1) evalA(t2)
  for all SPEC-algebra A is called congruence on
  ground terms.
• It satisfies the following conditions for all t1,
  t2, t3 ? TF
• - t1 ? t1 (reflexivity) t1 ? t2 implies t2 ?
  t1 (symmetry)
• t1 ? t2 and t2 ? t3 implies t1 ? t3
  (transitivity)
• - t1 ? t1 ,..., tn ? tn implies f(t1,...tn) ?
  f(t1,....,tn) (congruence)
• - each derivation t1 gt t2 via E between ground
  terms t1, t2 ? TF implies t1 ? t2 .

18
Algebraic specifications Specification and
SPEC-algebra

• A rewriting relation gtR is like a congruence
  relation without the reflexivity property.

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Algebraic specifications Specification and
SPEC-algebra

• Definition (Quotient Term Algebra TSPEC)
• Given a specification SPEC (S, F, E) the
  quotient term algebra
• TSPEC ((Qs) s?S, (fQ) f?F) is defined by
• 1. For each s ? S, we have a base set
• Qs t / t ? TF,s
• where the congruence class t is defined by
• t t / t ? t
• 2. For each constant symbol f ? s in F the
  constant Qs is the congruence class generated
  by f fQ f
• 3. For each operation symbol fs1 ...sn ? s in F
  the operation
• fQ Qs1 x ... x Qsn ? Qs is defined by
• fQ(t1, ...,tn) f(t1,...,tn)

20
Algebraic specifications Specification and
SPEC-algebra

• Example (Quotient Term Algebra Tnat)
• Tnat (Qnat , 0Q, SUCCQ, ADDQ)
• With
• - Qnat SUCCn(0) / n ? 0
• - 0Q 0, and for n, m ? 0
• - SUCCQ(SUCCn(0)) SUCCn1(0)
• - ADDQ(SUCCn(0), SUCCm(0)) SUCCnm(0)
• Fact TSPEC is a SPEC-Algebra and it is called
  the initial semantics with ADT(SPEC) A / A ?
  TSPEC is called the (initial) abstract data
  type defined by SPEC.