Algebraic specifications : Specification and SPECalgebra

1
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.

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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)

3
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 .

6
Algebraic specifications Specification and
SPEC-algebra

• A rewriting relation gtR is like a congruence
  relation without the reflexivity property.
• Top(push(pop(push(empty,0)), succ(m)))
• top(push(empty, succ(m)))
• succ(m)

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Algebraic specifications Specification

• Definition (Algebra of Terms)
• Given a signature SIG (S, F). We define the
  algebra of terms T (ST, FT) w.r.t. SIG and a
  set of variables X or simply termalgebra as
• ST (TF,s(X))s?S as the family of base sets
• fT f as the constant for f ? s
• fT TF,s1(X) x . . . x TF,sn (X) ? TF,s(X)
  defined by
• fT(t1,..,tn) f(t1,..,tn) for fs1 ...sn ? s
  and
• ti ?TF,si(X)

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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)

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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.

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Algebraic specifications Specification

• The quotient term algebra TSPEC of a
  specification SPEC (S, F, E) has the following
  properties
• The evaluation eval TF ? TSPEC is equal to
• nat TF ? TSPEC, defined by nat(t) t for
  all
• t ?TF, and hence surjective.
• Each equation e (t1, t2) of ground term
• t1, t2 ? TF is valid in TSPEC if and only if it
  is valid in each SPEC-algebra A.
• TSPEC is a SPEC-algebra.

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Algebraic specifications Specification

• Defintions (Equational Rules and Proofs)
• An equational rule (over SIG) is given by a pair
• (E, e)
• Where E is a set of equations and e is a single
  equation w.r.t. SIG. We also write E -- e
• 2. Given a set R of equational rules and a set of
  E of equations w.r.t. SIG. Then an (equational)
  proof
• With rules R and premisses E is a sequence E is a
  sequence e1, ..,er.

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Algebraic specifications Specification

• Definition ( Equational calculus) The
  equational calculus, is defined to contain
  exactly the following equational rules
• for t1, t2, t3 ?TF(X) and t TF(Y).
• R1 -- t1 t1 (identity)
• R2 t1 t2 -- t2 t1 (symmetry)
• R3 t1 t2 and t2 t3 -- t1 t3
  (transitivity)
• R4 (X, t1 t2) -- (X ? Y, h(t1) h(t2) )
• for ass X ?TF(Y) (substitivity).

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Algebraic specifications Specification

• When the equations are used as rewrite rules,
  the symmetric rule is to be droped.
• In rewriting techniques, the process of orienting
  equations is based on the so-called simplication
  orderings a partial order between operations
  extended to terms.
• example add gt succ gt 0 gt add(..,..) gt
  succ(..)
• In order to ensure the termination of rewriting
  of a term, such ordering is required to be
  well-founded (any ordering should have has a
  small element).
• The small element a any term is called the normal
  form of the term.

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Algebraic specifications Specification

• In order to ensure the uniqueness of computation,
  the so-called confluence property is required
• t1 gt t2 gtNf(t1) and
• t1 gt t3
• then t3 should be rewritten to Nf(t1) i.e.
• t3 gtNf(t1)
• This property is ensured by the so-called
  Knuth-Bendix completion procedure. It takes a set
  of equations and an ordering, and it generate a
  set of rewrite rules which terminate and are
  confluent.

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Algebraic specifications Specification

• The confluence property is verified by
  eliminating all ambiguities that may be hidden
  between different rules of the system. These
  ambiguities are called critical-pairs.
• Fot their definition, we need the notion of
  unification
• Two terms t1 and t2 are said to be unfiable is
  there is a substitution ? such ? (t1) ? (t2)
  .
• Example let t1 f(a,g(y)) and t2 f(x,g(h(b))),
  then is it easy to proof that ? x --gt a, y --gt
  h(b) is a unfier of t1 and t2. That is, ? (t1)
  f(a,g(h(b))) ? (t2)

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Algebraic specifications Specification

• Definition (critical pair)
• If l ? r and s ? t are two rewrite rules with
  distinct variables, p is the position of a
  nonvariable subterm of s, and ? is the unifier of
  sp and l, then the equation ?(t) ?(s?(r)p )
  is a critical pair formed from those rules.
• Example suppose we want to add the alternation
  in the stack specification using the following
  two rewrite rules
• alternate(push(x,y),z) ? push(x, alternate(z,y))
• Alternate(y1,?) ? y1
• Then, by applying the above definition, we can
  notice that alternate in the second rule occurs
  at position ? in the first rule. That is, s
  alternate(push(x,y),z) and l aternate(y1, ?).
  So, we have to check for a unification of
  alternate(z,y)) and Alternate(y,?). The unfier
  here is ? y1 ? push(x,y) , z ? ?

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Algebraic specifications Specification

• So, the resulting members of the critical pair
  are
• ?(t) ?(push(x, alternate(z,y)))
  push(x,alternate(?,y)))
• ?(t) ?(s?(r)p ) ?(s?(r)?) ?(r)
• ?(y1) push(x,y)
• And the critical pair is therefore the resulting
  equation
• push(x, alternate(?, y)) push(x,y)

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Advanced Algebraic specifications

• To deal a maximal of cases and errors, subsorts
  may be defined. (S lt S)
• Parametrized specifications are specifications
  based on others stack(string) list(nat) ....
• Parametrized specifications are interpreted using
  category on algebras .
• To go beyong the non-changing or fixed notions of
  algebras, and thereby interpreting state-based
  reactive (information) systems, several
  extensions have been proposed to the algebraic
  semantics.
• Rewriting logic a computation is a functor from
  an algebra to an another.
• Hidden sorted algebra some sorts modelling
  states are hidden.