Algorithmic Proof with Diminishing Resources, Part 1

This paper examines the following theme. A~ume we are given a consequence relation lof the form A1,. . . , An tB and an algorithmic proof system Stfor deciding (either in a recursive of RE manner) whether for given A1,.. . ,An,B the relation A1,. . . ,A, JB holds. Is it possible to perturb S~ into a proof system LS~ which is still complete for t-, and such that each A~ is used in the proof at most once? The above problem requires clarification. We need to define the notion of "used" in the proof system S~. This will probably not be too difficult to do. We also note that Ican be any logic, and hence the further restriction imposed on S~ may be only sound, but not complete. Further, it is not at all clear whether such a restriction has a corresponding logical meaning. The resulting system S~, which is obtained from S~ by restricting the use of assumptions to at most once, may not be a logic at all. Furthermore, for two proof systems for the same consequence relation, say Sb and T~, the corresponding systems obtained by the above restriction, namely S~ and T~ may not be the same. These matters will become clearer later in the paper. To consider some examples, assume that k is the implicational fragment of Girard's linear logic (see section 5) then any S~will probably do as an LS~. For other logics, for example intuitionistic implication, our theme would seek to compensate for the lack of completeness of S~, by adding new rules to the system S~ to obtain S~* which is complete. We thus want to systematically study the trade off between bounded resource and new clever deduction rules. Of course we must be careful not to replace the resource by a rule which will eventually bring the resource back. So some measure of complexity of resources must be available in the system and during the algorithmic proof execution the total complexity (or weight) of the available resource must be continually reduced. When no more resource is available and the algoirthmic proof execution stops then we decide whether to consider the "run" as success or failure. There are several good reasons for considering our "diminishing resource" theme. The first is that it may provide efficiency and implementational advantage without losing the character of the algorithmic system. Algorithmic systems are proof theoretic and are put forward for conceputal e.s well as computational reasons. Many of them are non-deterministic and allow for looping and non-termination when implemented naively. The natural response for eliminating loops which is conceputally clean is to provide for a historical loop checker. This is nice but computationally expensive. There are probably other alternatives which are likely to be very specific to the particular algoirthmie system at hand but may change the character of the system beyond recognition. The presence of additional side effects having to do with the efficiency considerations may overshadow the original system itself, which will sacrifice clarity and adaptability. The diminishing resource theme allows in principle for efficiency while retaining the character of the algorithmic system. The second advantage is conceputal. Diminishing resources is a natural theme, (as we cannot expect to be using a resource too many times) and because of that can be applied to practically any system. This 'allows us to develop a methodology which can be meaningful and intuitive. We know that a system S~ which is complete, may become only sound (and not necessarily complete) when perturbed into a system S~ of diminishing resource. We also know how to compensate. The difference between Sb and $I~ is that there may be occasions in S~ where a resource is about to be used again and is blocked. By looking historically at what happened in the system when the resource was used in the first time we may be able to find the additional rules which compensate.