Condensed detachment
Condensed detachment (Rule D) is a method of finding the most general possible conclusion given two formal logical statements. It was developed by the Irish
Informal description
A rule of detachment (often referred to as modus ponens) says:
"Given that implies , and given , infer ."
The condensed detachment goes a step further and says:
"Given that implies , and given an , use a unifier of and to make and the same, then use a standard rule of detachment."
A substitution A that when applied to produces , and substitution B that when applied to produces , are called unifiers of and .
Various unifiers may produce expressions with varying numbers of
If the most general unifier is used in condensed detachment, then the logical result is the most general conclusion that can be made in the given inference with the given second expression. Since any weaker inference you can get is a substitution instance of the most general one, nothing less than the most general unifier is ever used in practice.
Some logics, such as classical propositional calculus, have a set of defining axioms with the "D-completeness" property. If a set of axioms is D-Complete, then any valid theorem of the system, including all of its substitution instances (up to variable renaming), can be generated by condensed detachment alone. For example, if is a theorem of a D-complete system, condensed detachment can prove not only that theorem but also its substitution instance by using a longer proof. Note that "D-completeness" is a property of an axiomatic basis for a system, not an intrinsic property of a logic system itself.[2]
J. A. Kalman proved that any conclusion that can be generated by a sequence of uniform substitution (all instances of a variable are replaced with the same content) and modus ponens steps can either be generated by condensed detachment alone, or is a substitution instance of something that can be generated by condensed detachment alone.[1] This makes condensed detachment useful for any logic system that has modus ponens and substitution, regardless of whether or not it is D-complete.
D-notation
Since a given major premise and a given minor premise uniquely determine the conclusion (up to variable renaming),
This notation, besides being used in some automated theorem provers, sometimes appears in catalogs of proofs. For example, the "shortest known proofs" database of Metamath's mmsolitaire project features 196 theorems with such proofs.[3]
Condensed detachment's use of unification predates the resolution technique of automated theorem proving which was introduced in 1965.[4][5]
Advantages
For automated theorem proving condensed detachment has a number of advantages over raw modus ponens and uniform substitution.
At a Modus Ponens and substitution proof you have an infinite number of choices for what you can substitute for variables. This means that you have an infinite number of possible next steps. With condensed detachment there are only a finite number of possible next steps in a proof.[clarification needed]
The D-notation for complete condensed detachment proofs allows easy description of proofs for cataloging and search. A typical complete 30 step proof is less than 60 characters long in D-notation (excluding the statement of the axioms.)
References
- ^ S2CID 121221548.
- .
- ^ "Shortest known proofs of the propositional calculus theorems from Principia Mathematica". Metamath. Retrieved 9 September 2023.
- .
- S2CID 14389185.
- MR 1231287
- William McCune and Larry Wos (1992). "Experiments in Automated Deduction with Condensed Detachment" (PDF). In D. Kapur (ed.). Proc. 11th International Conference on Automated Deduction (CADE). LNCS. Vol. 607. Springer. pp. 209–223.