Local equational logic
We generalize the Horn clause logic by using arbitrary algebras instead of the free algebras of terms. Local means that deductions are performed in a fixed algebra.
The hypotheses of local Horn clause logic with equality (denoted by HL) are Horn clauses in projective algebras. The deductions are done using atomic formulas in a fixed algebra. Our completeness theorem includes the classical one. For equational logic, we introduce rewriting in an algebra and we prove the equivalence between the unique normal form semantics and the initial algebra semantics. This gives mathematical foundations for rewriting modulo a theory where the rewriting steps are performed in a free algebra of the theory.
Unable to display preview. Download preview PDF.
- 2.R. Diaconescu. The logic of Horn clauses is equational. Technical Report PRG-TR-3-93, PRG Computing Laboratory, Oxford 1993. Written in 1990Google Scholar
- 3.R. Diaconescu. The formal completeness of Equational Logics. Technical Report PRG-TR-12-92, PRG Computing Laboratory, Oxford 1991Google Scholar
- 4.J. Goguen and J. Meseguer. Models and equality for logical programming. Proceedings, TATSOFT 87, Springer-Verlag LNCS 250:1–22, 1987Google Scholar
- 5.J. Goguen and J. Meseguer. Order-sorted algebra solves the constructorselector, multiple representation and coercion problems. In Proceedings, Second Symposium on Logic in Computer Science, 18–29, IEEE Computer Society Press, 1987Google Scholar
- 6.J. Goguen and T. Winkler. Introducing OBJ. Tech. Rept. SRI-CSL-88-9, SRI International, Computer Science Lab., 1988Google Scholar
- 7.J. Goguen and J. Meseguer. Order-sorted algebras I: Equational deduction for multiple inheritance, polymorphism, ovreloading and partial operations. Tech. Rept. SRI International, Computer Science Lab., 1989Google Scholar
- 9.S. MacLane, Categories for the working mathematician, Springer, Berlin 1971Google Scholar