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By Suel Memon

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2. t. e. A 3. ~ ADT(SPEC). The classical semantics, also called loose semantics, of SPEC is the class Alg(SPEC) = {A/A SPEC-algebra} of all SPEC-algebras. 4. t. K if K is equal to Alg(SPEC). 2. t. t. 8. For the specification have constructed already the quotient term algebra T . 3. t. 2 such that both are defining the same abstract data type ADT(STRING). 4 and 5: =MAKE(ca) =LADD(ca,EMPTY) b CONCAT(ts1,ts2) =CONCAT(cs1,cs2) =cs3 MAKE (ta) Cstring (eqns (4) & (2) of b Cstring ~~~~~~) (by (8) above) LADD(ta,ts1) _ LADD(ca,cs1) b Cstring RADD(ts1,ta) _ RADD(cs1 ,cal _ CONCAT(cs1 ,MAKE(ca» _ CONCAT(cs1,LADD(ca,EMPTY» (eqn (5) of (eqns (4) & (2) of This completes the proof of (6) by structural induction.

These are O,SUCC for K1, ••• ,Kn, EMPTY, LADD for ~~~~~~ ~~~, O,SUCC,PRED for ~~~ and (we could also use RADD instead of LADD). Next we construct all terms generated by these constant and operation symbols. If there are no equations between these constant for for and ~~~ SPEC. ~~~~~~ we can consider all these terms as canonical terms COP for If there are equations, like SUCC(PRED(n» ~~~, and operation symbols, like = nand PRED(SUCC(n» =n we have to designate a suitable subset of all these terms as canonical terms.

DEFINITION (Semantics, Abstract Data Type, Correctness) Let SPEC be a specification with signature SIG: 1. The initial semantics, or short semantics, of SPEC is the class ADT(SPEC) = {A/A ~ TSPEC } of all algebras isomorphic to the quotient term algebra TSPEC • ADT(SPEC) is also called the (initial) abstract data type defined by SPEC. 2. t. e. A 3. ~ ADT(SPEC). The classical semantics, also called loose semantics, of SPEC is the class Alg(SPEC) = {A/A SPEC-algebra} of all SPEC-algebras. 4. t. K if K is equal to Alg(SPEC).

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Algorithms for Delta Compression and Remote File Synchronization by Suel Memon


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