B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical report no. 94-21, FM Informatik, Technische Universitat Berlin, May 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....over the resulting domain of the accumulating parameter c. There are several ways to identify these associative operators in our programs: limiting application scope by requiring all associative operators to be made explicit, e.g. in [FG94] or adopting AI techniques likeanti unification [Hei94] to synthesize them, or more interestingly, deriving them from the resulting domain types. For the last, it is known [SF93] that every linear type R that has a zero constructor CZ (a constructor with only only a don t care value like[ for lists) has a function 8, which is associative and has ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical report no. 94-21, FM Informatik, Technische Universitat Berlin, May 1994.

....these claims as auxiliary lemmas during the proof of 8x; y 9z val(x) val(y) val(z) they are implicitly generated by the sort algo rithms. The sort calculus allows the recognition of new concepts so to speak, although only within the rather limited framework given by the sort language. In [8], an approach to the automatic generation of more complex auxiliary lemmas is presented based on E generalization using regular sorts, too. A prototype support system written in Quintus Prolog takes a total of 41 seconds user time on a Sparc 1 to automatically conduct the 9 induction proofs, with ....

Heinz, B., Lemma discovery by anti-unification of regular sorts, TU Berlin, Technical Report 94-21, 1994

....of Omega . As the first step, we must be able to recognize them in a program. There are several ways. We may restrict our application scope so that all associative and distributive operators can be made explicit, e.g. in [FG94, CTT97] Or, wemay adopt some artificial methods likeanti unification [Hei94] to synthesize them. However, these approaches are not so satisfactory to be used practically in a parallelization system. In this paper, rather than recognizing all associative and distributive operators, we are interested in the associative and distributive operators that are derivable from the ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical report no. 94-21, FM Informatik, Technische Universitat Berlin, May 1994.

....t G :oe 1 E t 1 and t G :oe 2 E t 2 . Here, E denotes the semantic equality (conversion relation) in the equational theory E , and t :oe denotes the result of applying substitution oe to term t . Generalization is the dual of unification and is sometimes also called anti unification [35]. Let us assume that function h is a homomorphism, i.e. 4) holds, and t H denotes a term over u and v that defines fi : u fi v t H (8) The following two terms, constructed from t H by substitutions: t L = t H :fu 7 h( a] v 7 h(y)g and t R = t H :fu 7 h(x ) v 7 h( b] g, are ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical Report 94-21, TU Berlin, May 1994.

....8 as well as distributivity of Omega . So we must be able to recognize them in a program. There are several ways; limiting application scope so that all associative and distributive operators can be made explicit, e.g. in [FG94, CTT97] or adopting some artificial methods like anti unification [Hei94] to synthesize them. To be constructive, we shall focus on the associative and distributive operators that are derivable from the target type of a sequential program to be parallelized. In fact every type R which has a zero constructor CZ (a constructor with no arguments like [ for lists) has ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical report no. 94-21, FM Informatik, Technische Universitat Berlin, May 1994.

....oe Generalization R fy 7 [b]g 6 rw TR : h (x ) fi h [b] E oe TS Our ultimate goal is to find term TH from given TC and TS , e.g. for scan function, TC and TS are the right hand sides of (3) and (4) correspondingly. We use so called generalization or anti unification of terms [16]. Definition 7. Generalization in equational theory E of terms T 1 and T 2 w.r.t. substitutions oe 1 and oe 2 is term TG = GenE fT 1 j oe 1 ; T 2 j oe 2 g, which satisfies TG :foe 1 g E Omega T 1 and TG :foe 2 g E Omega T 2 . Theorem 8. If an lw term TC and an rw term TS , both for ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical Report 94-21, TU Berlin, May 1994.

....since x 0 :oe 1 = t 1 and x 0 :oe 2 = t 2 where oe 1 = def fx 0 7 t 1 g and oe 2 = def fx 0 7 t 2 g. So it is obvious that people prefer the most special generalizer, provided there is one. In this respect, generalization is the dual to unification and is sometimes also called anti unification [14]. Plotkin has closely studied the special case relation between terms [18] In the case where E is empty, there are most general syntactic unifiers and most special syntactic generalizers. In contrast to unification, properties and methods for generalization for nonempty E are hardly known. A few ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical Report 94-21, TU Berlin, May 1994.

....So it is obvious that people prefer the most special generalizer, provided there is one. In this respect, generalization is the dual to unification where, modulo conversion, a most general common special case is wanted. For this reason, generalization is sometimes also called anti unification [13]. Plotkin has closely studied the special case relation between terms [14] In the case where E is empty, the special case relation forms a lattice, i.e. there are most general syntactic unifiers and most special syntactic generalizers. In contrast to unification, an intensively studied subject ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical Report 94-21, TU Berlin, May 1994.

....6, we allow calls to f to be different but logically treat them simply as f calls. program. There are several ways: limiting application scope by requiring all associative and distributive operators to be made explicit, e.g. in [FG94, CTT97] or adopting AI techniques like anti unification [Hei94] to synthesize them. All of them need human insights. Fortunately, we are able to constructively derive such associative operators from the resulting type of the function to be parallelized It is known [SF93] that every type R that has a zero constructor CZ (a constructor with no arguments like ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical report no. 94-21, FM Informatik, Technische Universitat Berlin, May 1994.

....two terms, a variable; it is obvious that people prefer the most special generalizer, provided there is one. In this respect, generalization is the dual to unification where a most general common special case is wanted. For this reason, generalization is sometimes also called anti unification [16]. Let us first study the equality and substitution relations between the terms we deal with. Assume that function h is a fi homomorphism, and t H denotes the goal of extraction a homomorphic term over u and v that defines fi . This term is shown underlined in the left part of the diagram ....

B. Heinz. Lemma discovery by anti-unification of regular sorts. Technical Report 94-21, TU Berlin, May 1994.

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