# On the existence of initial models for partial (higher-order) conditional specifications

## Abstract

Partial higher-order conditional specifications may not admit initial models, because of the requirement of extensionality, even when the axioms are positive conditional. The main aim of the paper is to investigate in full this phenomenon.

If we are interested in term-generated initial models, then partial higher-order specifications can be seen as special cases of partial conditional specifications, i.e. specifications with axioms of the form ∧Δ ⊃ ɛ, where Δ is a denumerable set of equalities, ɛ is an equality and equalities can be either strong or existential. Thus we first study the existence of initial models for partial conditional specifications.

The first result establishes that a necessary and sufficient condition for the existence of an initial model is the emptiness of a certain set of closed conditional formulae, which we call “naughty”. These naughty formulae can be characterized w.r.t. a generic inference system complete w.r.t. closed existential equalities and the above condition amounts to the impossibility of deducing those formulae within such a system. Then we exhibit an inference system which we show to be complete w.r.t closed equalities; the initial model exists if and only if no naughty formula is derivable within this system and, when it exists, can be characterized, as usual, by the congruence associated with the system.

Finally, applying our general results to the case of higher-order specifications with positive conditional axioms, we obtain necessary and sufficient conditions for the existence of term-generated initial models in that case.

## References

- [AC1]Astesiano, E.; Cerioli, M. “Free objects and equational deduction for partial (higher-order) conditional specifications”, (Technical report, February 1989).Google Scholar
- [AR1]Astesiano, E.; Reggio, G. “SMoLCS-Driven Concurrent Calculi”, (invited paper)
*Proc. TAPSOFT 87*, vol. 1, Berlin, Springer Verlag, 1987 (Lecture Notes in Computer Science n. 249), pp. 169–201.Google Scholar - [AR2]Astesiano, E.; Reggio, G. “An Outline of the SMoLCS Methodology”, (invited paper)
*Mathematical Models for the Semantics of Parallelism, Proc. Advanced School on Mathematical Models of Parallelism*(Venturini Zilli, M. ed.), Berlin, Springer Verlag, 1987 (Lecture Notes in Computer Science n. 280), pp. 81–113.Google Scholar - [B]Burmeister, P.
*A Model Theoretic Oriented Approach to Partial Algebras*, Berlin, Akademie-Verlag, 1986, pp. 1–319.Google Scholar - [BW1]Broy, M.; Wirsing, M. “Partial abstract types”,
*Acta Informatica*18 (1982), 47–64.Google Scholar - [BW2]Broy, M.; Wirsing, M. “On the algebraic specification of finitary infinite communicating sequential processes”,
*Proc. IFIP TC2 Working Conference on "Formal Description of Programming Concepts II"*, Garmisch 1982.Google Scholar - [K]Keisler, H.J.
*Model Theory for Infinitary Logic*, Amsterdam — London, North-Holland Publishing Company, 1971, pp. 1–208.Google Scholar - [M]Möller, B. “Algebraic Specification with Higher-Order Operations”,
*Proc. IFIP TC 2 Working Conference on Program Specification and Transformation, Bad Tolz F.R.G. 1986*(Meertens, L.G.L.T. ed.), Amsterdam-New York-Oxford-Tokyo, North-Holland Publ. Company, 1987.Google Scholar - [MG]Meseguer, J.; Goguen, J.A. “Initiality, Induction and Computability”,
*Algebraic Methods in Semantics*, Cambridge, edited by M.Nivat and J.Reynolds, Cambridge University Press, 1985, pp.459–540.Google Scholar - [MTW]Möller B., Tarlecki A., Wirsing M. “Algebraic Specification with Built-in Domain Constructions”,
*Proceeding of CAAP '88 (Nancy France, March 1988)*, edited by Dauchet M. and Nivat M., Berlin, Springer-Verlag, 1988, pp. 132–148.Google Scholar - [R]Reichel H.
*Initial Computability, Algebraic Specifications, and Partial Algebras*, Berlin (D.D.R.), Akademie-Verlag, 1986.Google Scholar - [T]Tarlecki A. “Quasi-varieties in Abstract Algebraic Institutions”, Journal of Computer and System Science, n. 33 (1986), pp. 333–360.Google Scholar
- [WB]Wirsing, M.; Broy, M.
*An analysis of semantic models for algebraic specifications*, International Summer School Theoretical Foundation of Programming Methodology, Munich. Germany 28/7–9/8, 1981.Google Scholar