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Experiments in Automated Deduction with Condensed Detachment 1 Exp eriments in Automated Deduction with Condensed Detachment William McCune and Larry Wos Mathematics and Computer Science Division Argonne National Lab oratory Argonne Illinois USA Abstract This pap er contains the results of exp eriments with several search strategies on problems involving condensed detachment The problems are taken from nine dierent logic calculi three versions of the twovalued sentential calculus the manyvalued sentential calculus the implicatio nal calculus the equivalential calculus the R calculus the left group calculus and the right group calculus Each problem was given to the theorem prover Otter and was run with at least three strategies a basic strategy a strategy with a more rened metho d for selecting clauses on which to fo cus and a strategy that uses the rened selection mechanism and deletes deduced formulas according to some simple rules Two new features of Otter are also presented the rened metho d for selecting the next formula on which to fo cus and a metho d for controlling memory usage Intro duction The aim of this pap er is to examine the role of strategy in the study of logic calculi with condensed detachment We present results of exp eriments with the theorem proving program Otter on problems all of whichcontain the axiom or from another p oint of view inference rule condensed detachment All of the problems concern axiomatizations of various logic calculi including the twovalued sentential calculus and two of its variations the manyvalued senten tial calculus the implicational fragment of sentential calculus equivalential calculus and three subsystems of the equivalential calculus the R calculus the left group calculus and the right group calculus The problems should also servewell as test problems for evaluating other search strategies and other theoremproving programs Wehave exp erimented extensively with most of the problems and wehavedevel op ed sp ecialized strategies for particular logic calculi For the exp eriments presented in this pap er however we sought strategies that p erform well on all of the problems This work was supp orted by the Applied Mathematical Sciences subprogram of the Oce of Energy Research US DepartmentofEnergy under Contract WEng Aside from a default basic strategywe exp erimented with a guidance strategy and a deletion strategy The guidance strategywhichwe call the ratio strategy Section combines b estrst search with breadthrst search when selecting the next for mula on which to fo cus The deletion strategy Section causes derived formulas that are instances of simple patterns to b e deleted Condensed Detachment All of the problems use C A Merediths condensed detachment a rule of inference that combines detachment mo dus p onens and instantiation Let C b e the binary op eration of concern If C and are b oth theorems renamed so that they havenovariables in common and if and unify with most general unier then is deduced by condensed detachment The formula C isthemajor premiseand is the minor premise The binary op eration is usually interpreted as implies equivalent or some variation of the group op eration dep ending on the calculus The logic calculi can b e studied as rstorder theories by a trivial transformation First a unary predicate P interpreted as is a theorem or is the group iden tity is intro duced Then each axiom of the calculus is preceded by P with its variables universally quantied Finally condensed detachment b ecomes an axiom of the theory xy P C x y P x P y An application of hyp erresolution with the axiom condensed detachment corresp onds directly to an application of the inference rule condensed detachment Although we used hyp erresolution exclusively for the exp eriments presented in this pap er any inference rule for rstorder logic is applicable The AN calculus whichisavariation of the twovalued sentential calculus has a binary op eration o whichcanbeinterpreted as disjunction and a unary op eration n which can b e interpreted as negation For the AN calculus the following variation of condensed detachment is used xy P onxy P x P y The study of logic calculi with condensed detachment has b een one of the rst and most successful applications of automated theorem proving Original research has b een conducted and op en questions have b een answered by relying heavily on automated theorem proving programs Otter and Simple Strategies Otter is a resolutionparamo dulation theoremproving program for rstorder logic with equality Its basic algorithm restricted to hyp erresolution with condensed detachment is shown in Figure Selecting the Given Clause Our default strategy for selecting the given clause in Step of the basic algorithm has traditionally b een to select a clause with the fewest symb ols if there is more Start with sos list containing all axioms and with usabl e list containing the axiom for condensed detachment Lo op G selectgivenclausesos move G from sos to usabl e apply condensed detachmentasmuch as p ossible with G as one premise taking the other premise from usabl e app end to sos the results that are not subsumed byanything in sos or usabl e end lo op Figure Otters Basic Algorithm with Condensed Detachment than one clause of minimum length the rst of those is selected We call the default strategy selecting the smallest clause as given However some problems in the logic calculi yield quickly to a breadthrst search which is accomplished byselectingthe rst clause in the sos list as the given clause The metho d we use for most of the exp eriments presented in this pap er combines those two metho ds In every fourth iteration of the lo op the rst clause is selected and in the remaining iterations the rst clause of minimum length is selected We call this rened metho d selecting the given clause with ratio The renement allows large clauses to enter the search while the fo cus remains mainly on small clauses It is similar to a selection strategy used by J Kalman in one of his early programs Deleting Derived Formulas In the equivalential calculus the R calculus and the left and right group calculi all of whichhave binary op erator e we found that formulas containing subformulas that are instances of ex x are generally not as useful or as p owerful as formulas without such instances Searches in which those formulas are deleted are generally more eective although they can result in longer pro ofs The strategy also applies to the implicational calculi by deleting deduced formulas with instances of ix x although it app ears to b e less eective there The strategy applies to the AN calculus in which the binary op eration is disjunction by deleting formulas with instances of onxx In the calculi with unary op eration n meaning negation we found that deduced nx caused redundancy in the search spaces and formulas containing instances of n that deleting those formulas generally improved the searches We also ran exp eriments deleting formulas with instances of nnnx We used demo dulation of derived formulas to implement the deletion strategy When the strategy was in use demo dulation usually accounted for b etween one third and one half of the CPU time Controlling Memory Usage We limited the Otter jobs to Mbytes of memoryinwhich Otter can store roughly formulas Even with the deletion strategy of the preceding subsection Otter quicklyllsMbytes The list sos typically grows much faster than do es the numb er of given clauses that are removed from it Thus most formulas in sos never enter the search and memory is wasted Our current solution to that problem is the following When one third of available memory has b een lled we imp ose a limit on the number of symb ols in deduced clauses The limit say nissuch that of all formulas in sos have n symbols Every tenth iteration of the main lo op after the initial limit has b een set calculate a prosp ective new limit n in the same wayIfn n then the limit is reset to n We arrived at the values and by trial and error Although this metho d is incomplete its use with condensed detachmentproblemstypically do es not havea great eect on the sequence of given clauses or therefore on the search Wehavenot exp erimented heavily with this metho d on other problems The Exp eriments We ran all of the exp eriments on SPARCstation computers with megabytes of Otter can infer several thousand formulas p er main memory In that environment second most of which are deleted b ecause they are subsumed by existing formulas or by the deletion strategy Back subsumption in which newly kept formulas cause the deletion of weaker existing formulas was not used Here is an example of the wayinwhich problems are presented Given the equiv alential calculus formulas EC eeex y ez xey z EC eex y ey x EC eeex y zex ey z the problem ECEC EC is to nd a refutation of the following set of clauses Symbols x y and z are variables and a bandc are Skolem constants P ex y jP x j P y Condensed Detachment P eex y ey x EC P eeex y zex ey z EC P eeea bec aeb c Denial of EC Each problem was run with several strategies with a time limit of four hours each In the tables that follow Fail indicates that no pro of was found within four hours and indicates that no pro of is p ossible b ecause the goal would b e deleted by the deletion strategy All of the times are given in seconds The strategies are the following Basic The smallest formula is selected as given Ratio Given clauses are selected with ratio Re Given clauses are selected with ratio and deduced clauses containing an in stance of
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