Abstract
This chapter describes in detail the set-based proof-verification system Ref, developed by the authors, and its underlying design. The chapter falls into two parts: (i) An account of the general syntax and overall structure of proofs acceptable to the verifier. (ii) A listing of the mechanisms actually chosen from the list of candidate inference mechanisms surveyed earlier in the book, for inclusion in the verifier’s initial endowment. The syntax used to invoke each of the verifier’s built-in inference mechanisms is explained.
The most central among the chosen inference primitives is named ELEM. Its use is, often, tacitly combined with other forms of inference. ELEM implements multi-level syllogistic, a decision algorithm which determines whether a given unquantified set-theoretic formula involving individual variables (which designate sets) and a restricted collection of set operators is satisfiable.
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Notes
- 1.
For additional information and more detailed information, cf. http://setl.dyndns.org/EtnaNova/login/Ref_user_manual.html.
- 2.
- 3.
As seen here, we often enclose quantified formulae within ‘〈’ and ‘〉’—instead of within ‘(’ and ‘)’—to make matching parentheses more visible.
- 4.
We keep the sign ↔Def distinct from =Def for clarity, but this distinction is not very important.
- 5.
Case (i) in Sect. 3.7 referred to the operator arb, which, as remarked above, is built into elem. Treatment of the other cases is unimplemented as yet in the Ref verifier.
- 6.
The mechanisms described in the ongoing of this section are unimplemented as yet in Ref.
- 7.
Primitives Set_monot and Pred_monot supporting proof by monotonicity are available in the Ref system as implemented, but the enable_elem directive is not implemented yet.
- 8.
A primitive, ALGEBRA, supporting algebraic deduction is available in the Ref system as implemented, but the enable_algebra directive is not implemented yet.
- 9.
As of today, the directives \(\mbox {$\textsc {watch}$}\) and \(\mbox {$\textsc {dont\_watch}$}\) have not been implemented in Ref.
- 10.
While the accelerated instantiation mechanism is implemented in the current Ref system, the instance facility is not available yet: the instantiating substitution must be indicated explicitly.
- 11.
The feature described in this section is as yet unimplemented in the Ref system.
References
Formisano, A., Omodeo, E.: Theory-specific automated reasoning. In: Dovier, A., Pontelli, E. (eds.) A 25-Year Perspective on Logic Programming—Achievements of the Italian Association for Logic Programming, GULP. LNCS, vol. 6125, pp. 37–63. Springer, Berlin (2010). Chap. 3
Omodeo, E.G., Schwartz, J.T.: A ‘Theory’ mechanism for a proof-verifier based on first-order set theory. In: Kakas, A., Sadri, F. (eds.) Computational Logic: Logic Programming and beyond—Essays in honour of Bob Kowalski, Part II, vol. 2408, pp. 214–230. Springer, Berlin (2002)
Wolfram, S.: The Mathematica Book, 5th edn., p. 1464. Wolfram Media, Champaign (2003)
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© 2011 Springer-Verlag London Limited
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Schwartz, J.T., Cantone, D., Omodeo, E.G. (2011). More on the Structure of the Verifier System. In: Computational Logic and Set Theory. Springer, London. https://doi.org/10.1007/978-0-85729-808-9_4
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