Abstract:
A method for simplifying resolution proofs in DAG format where each leaf node represents a clause and each internal node represents a resolution between its children includes representing a SAT proof as a stripped proof, analyzing pivots to identify redundant resolutions, and constructing a simplified proof without the redundant resolutions.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    This invention relates to the simplification of satisfiability problem proofs using double pivot reduction. 
         [0003]    2. Description of Background 
         [0004]    Satisfiability (“SAT”) problems are decision problems, i.e., given an expression, determine if there is some assignment of variables that make the entire expression true. SAT problems have applications in a variety of areas including computer science, automation, and artificial intelligence. 
         [0005]    A well known algorithm for solving SAT problems is the Davis-Putnam-Logemann-Loveland algorithm (“DPLL”). DPLL-based SAT solvers progress by implicitly applying binary resolution. They generate resolution proofs that are useful for a variety of purposes in formal verification and elsewhere including: extracting an unsatisfiable core in case the formula is unsatisfiable, extracting an interpolant, and detecting clauses that can be reused in an incremental satisfiability setting such as the one used in Bounded Model-Checking. 
         [0006]    SAT problem solutions are typically presented as proofs and can be presented in a form called Directed Acyclic Graph (“DAG”). When a prior art SAT solver produces a proof, it can contain redundant resolutions. While there are methods such as Run till fix, Core Trimmer and Proof Tightening that are known in the art as methods of simplifying proofs, the prior art does not disclose a method for simplifying proofs by identifying and removing redundant resolutions. 
       SUMMARY OF THE INVENTION 
       [0007]    The shortcomings of the prior art are overcome and additional advantages are provided through the use of a method for simplifying resolution proofs in DAG format where each leaf node represents a clause and each internal node represents a resolution between its children by representing a SAT proof as a stripped proof, analyzing pivots to identify redundant resolutions, and constructing a simplified proof without the redundant resolutions. 
         [0008]    Additional features and advantages are realized through the techniques of the present invention. Other embodiments and aspects of the invention are described in detail herein and are considered a part of the claimed invention. For a better understanding of the invention with advantages and features, refer to the description and to the drawings. 
       TECHNICAL EFFECTS 
       [0009]    As a result of the summarized invention, SAT proofs are simplified This simplification results in a savings of processing time and cost in use of the simplified proof in place of the original proof. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]    The subject matter which is regarded as the invention is particularly pointed out and distinctly claimed in the claims at the conclusion of the specification. The foregoing and other objects, features, and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings in which: 
           [0011]      FIG. 1  shows an illustrative example in accordance with the invention; 
           [0012]      FIG. 2  shows an illustrative example of a stripped proof; 
           [0013]      FIG. 3  shows an illustrative example of a stripped pivot proof; 
           [0014]      FIG. 4  shows an illustrative example of a reduced proof; 
           [0015]      FIG. 5  shows an illustrative example of a reconstructed proof; 
           [0016]      FIG. 6  shows an illustrative example of a final proof. 
       
    
    
       [0017]    The detailed description explains the preferred embodiments of the invention, together with advantages and features, by way of example with reference to the drawings. 
       DETAILED DESCRIPTION OF THE INVENTION 
       [0018]    The invention herein involves simplification of SAT proofs using a method called double pivot reduction. Double pivot reduction relies on the exploitation of the double pivot phenomena. When two resolutions occur on the same branch of a proof in a tree form, the bottom one, i.e., the one closest to the leafs is redundant. This bottom resolution can be removed from the proof and thus enable its simplification. 
         [0019]    In a DAG presentation, let node n 1  rule node n 2  if every path from root to n 2  passes through n 1 . Given a DAG form of a SAT proof, the preferred embodiment identifies double resolutions on the same pivot where one is ruling the other and removes the bottom resolution. 
         [0020]    It is well known in the art that some reordering of resolutions in a SAT proof is possible. A preferred embodiment involves reordering to increase the double pivot effect and using it for further simplification. 
         [0021]    With reference to the accompanying drawings,  FIG. 1  shows an illustrative environment  30  for managing the processes in accordance with the invention. To this extent, the environment  30  includes a computer infrastructure  32  that can perform the processes described herein. In particular, the computer infrastructure  32  is shown including a computing device  34  operable to perform the processes described herein. 
         [0022]    The computing device  34  is shown including a processor  38 , a memory  40 , an input/output (I/O) interface  42 , and a bus  44 . Further, the computing device  34  is shown in communication with an external I/O device/resource  46  and a storage system  48 . As is known in the art, in general, the processor  38  executes computer program code, which is stored in memory  40  and/or storage system  48 . While executing computer program code, the processor  38  can read and/or write data, such as the range boundary  50 , to/from memory  40 , storage system  48 , and/or I/O interface  42 . The bus  44  provides a communications link between each of the components in the computing device  34 . The I/O device  46  can comprise any device that enables an individual to interact with the computing device  34  or any device that enables the computing device  34  to communicate with one or more other computing devices using any type of communications link. 
         [0023]    The computing device  34  can comprise any general purpose computing article of manufacture capable of executing computer program code installed thereon (e.g., a personal computer, server, handheld device, etc.). However, it is understood that the computing device  34  is only representative of various possible equivalent computing devices that may perform the processes described herein. Similarly, the computer infrastructure  32  is only illustrative of various types of computer infrastructures for implementing the invention. For example, in one embodiment, the computer infrastructure  32  comprises two or more computing devices (e.g., a server cluster) that communicate over any type of communications link, such as a network, a shared memory, or the like, to perform the process described herein. 
         [0024]    A preferred embodiment is disclosed in the example provided in  FIGS. 2-6  and the pseudo code contained herein.  FIG. 2  is an original “stripped proof.” A stripped proof in this embodiment is a DAG where each node leaf  111 ,  121 ,  131 ,  141  represents a clause and every non-leaf node  110 ,  120 ,  130  has either one or two children.  FIG. 3  depicts the “stripped pivot proof” version of the original proof shown in  FIG. 2 . The stripped pivot proof is a stripped proof where each internal node  110 ,  120 ,  130  with two children is annotated by a literal pivot. 
         [0025]      FIG. 4  depicts the proof after reduction, where one leg representing a redundant resolution leading to leaf node  131  is removed according to the preferred embodiment.  FIG. 5  depicts the reconstructed proof after it is reconstructed without the removed leaf  131 .  FIG. 6  depicts the final proof after further manipulation to simplify the DAG by substituting node  151  in place of redundant resolutions represented by nodes  130 ,  131  and  141 . 
         [0026]    The following pseudo code provides a preferred implementation of the claimed method. 
       Notation: 
       [0027]      
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 res (C L ,C R ) is the resolution clause of C L  and C R   
               
               
                 piv (C L , C R ) is the pivot lit, it is assumed that piv (C L , C R ) is included in 
               
               
                 C L  and − piv (C L , C R ) is included in C R . 
               
               
                   
               
             
          
         
       
     
       Definition Proof 
       [0028]    A proof is a directed acyclic graph (DAG) P with a single root, P.root, where each node represents a clause n.C. The single root represents the clause C Conclusion . For every node n the following holds: 
         [0000]                                    1.   n is a leaf of the graph.       2.   n has only one child, n.L. In this case n.C is equal to n.L.C.       3.   The node has exactly two children representing the clauses CL and           CR. In this case n.c=res(n.L.C, n.R.C)                    
Definition (Stripped Proof) A stripped proof is a directed acyclic graph (DAG) with a single root where each leaf node represents a clause. Every non leaf node has either one or two children.
 
Defenition (Stripped Pivot Proof). A stripped pivot proof is a stripped proof where each internal node n with 2 children is annotated by a literal pivot n. piv.
 
Algorithm stripp
 
         [0029]    Input: A proof P 
         [0030]    Output: A striped proof P′ 
         [0000]                                                1.   for each node n in P           2.   add a node n’ to P’           3.   for each node n in P           4.   if n is a leaf of P           5.   n’.C := n.C           6.   else if n has two children           7.   n’.piv = piv (n.L.C, n.R.C)                        
Comment: For each Proof P Stripping (P) is a stripped Proof
 
Algorithm reconstruct
 
         [0031]    Input: A stripped pivot proof P 
         [0032]    Output: P holds a proof 
         [0033]    mark all nodes of P as unvisited 
         [0034]    recursiveReconsturct (P, root (P)) 
         [0000]    Algorithm recursiveReconsturct 
         [0035]    Input: Stripped Proof P and node n 
         [0036]    Output: P such that the sub proof starting at n is a proof of n.c. 
         [0000]                                                1.   if n visited           2.   return           3.   mark n as visited           4.   if n is a leaf           5.   return           6.   if n has a single child n.L           7.   n.C = n.L.C           8.   else           9.   recursiveReconsturct (P,n.L)           10.   recursiveReconsturct (P,n.R)           11.   if piv is in N.C.L and −piv is in N.C.R           12.   n.C = res( n.L.C,n.R.C)           13.   else if (piv is in n.L.c and not in n.R.C)           14.   n.C = n.R.C           15.   n.L = nill           16.   else if (−piv is in n.R.c and not in n.L.C)           17.   n.C = n.L.C           18.   n.R = nill           19.   else           20.   heuristically choose side from L or R           21.   n.C = n.side.C           22.   n.otherSide = nill           23.   return                        
Comment for each Proof P reconstruct (stripp (P))==P
 
Definition clause partial order
 
         [0037]    C 1 ≧C 2  iff all lits of C 1  are included in C 2   
         [0000]    Definition Proof partial order 
         [0000]                                            P1 ≧ P2 iff            1.  leafs of P1    l eafs of P2            2.  P1.root ≧ P2.root                        
Algorithm: doublePivotReduction
 
input: a stripped proof P
 
output: a (“stronger”, see comment below) stripped proof P
 
         [0000]                                                                    1.   doublePivotRecReduction (P.root,{ })           2.   return P            doublePivotRecReduction (n, removableLits)                1.   if n.visited           2.   return           3.   n.visited=true           4.   if leaf           5.   return           6.   if (n has more then one parent)           7.   removableLits = { }           8.   if (piv in n.L.c and −piv in n.R.c )           9.   doublePivotRecReduction(nL,removableLits U {−piv})           10.   doublePivotRecReduction(nR,removableLits U {piv})           11.   else if (piv in n.L.c and −piv not in n.R.c U {−piv})           12.   n.L = nill           13.   doublePivotRecReduction(n1,removableLits);           14.   else if (piv not in n.L.c and −piv in n.R.c )           15.   n.R = nill           16.   doublePivotRecReduction(nr,removableLits U {piv});           17.   else           18.   choose huristicly a side from L and R           19.   n.sideNotChosen = nill           20.   doublePivotRecReduction(n.side,removableLits);                        
comment: reconstruct (doublePivotReduction (stripp (P))==P
 
         [0038]    While the preferred embodiment to the invention has been described, it will be understood that those skilled in the art, both now and in the future, may make various improvements and enhancements which fall within the scope of the claims which follow. These claims should be construed to maintain the proper protection for the invention first described.