Proof compression

In proof theory, an area of mathematical logic, proof compression is the problem of algorithmically compressing formal proofs. The developed algorithms can be used to improve the proofs generated by automated theorem proving tools such as SAT solvers, SMT-solvers, first-order theorem provers and proof assistants.

Problem Representation

In propositional logic a resolution proof of a clause from a set of clauses C is a directed acyclic graph (DAG): the input nodes are axiom inferences (without premises) whose conclusions are elements of C, the resolvent nodes are resolution inferences, and the proof has a node with conclusion .[1]

The DAG contains an edge from a node to a node if and only if a premise of is the conclusion of . In this case, is a child of , and is a parent of . A node with no children is a root.

A proof compression algorithm will try to create a new DAG with fewer nodes that represents a valid proof of or, in some cases, a valid proof of a subset of .

A simple example

Let's take a resolution proof for the clause from the set of clauses

Here we can see:

  • and are input nodes.
  • The node has a pivot ,
    • left resolved literal
    • right resolved literal
  • conclusion is the clause
  • premises are the conclusion of nodes and (its parents)
  • The DAG would be
  • and are parents of
  • is a child of and
  • is a root of the proof

A (resolution) refutation of C is a resolution proof of from C. It is a common given a node , to refer to the clause or ’s clause meaning the conclusion clause of , and (sub)proof meaning the (sub)proof having as its only root.

In some works can be found an algebraic representation of resolution inferences. The resolvent of and with pivot can be denoted as . When the pivot is uniquely defined or irrelevant, we omit it and write simply . In this way, the set of clauses can be seen as an algebra with a commutative operator; and terms in the corresponding term algebra denote resolution proofs in a notation style that is more compact and more convenient for describing resolution proofs than the usual graph notation.

In our last example the notation of the DAG would be or simply

We can identify .

Compression algorithms

Algorithms for compression of sequent calculus proofs include cut introduction and cut elimination.

Algorithms for compression of propositional resolution proofs include RecycleUnits,[2] RecyclePivots,[2] RecyclePivotsWithIntersection,[1] LowerUnits,[1] LowerUnivalents,[3] Split,[4] Reduce&Reconstruct,[5] and Subsumption.

Notes

  1. Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via Partial Regularization. 23rd Conference on Automated Deduction, 2011.
  2. Bar-Ilan, O.; Fuhrmann, O.; Hoory, S. ; Shacham, O. ; Strichman, O. Linear-time Reductions of Resolution Proofs. Hardware and Software: Verification and Testing, p. 114–128, Springer, 2011.
  3. "Skeptik/Doc/Papers/LUniv at develop · Paradoxika/Skeptik · GitHub". GitHub. Archived from the original on 11 April 2013.
  4. Cotton, Scott. "Two Techniques for Minimizing Resolution Proofs". 13th International Conference on Theory and Applications of Satisfiability Testing, 2010.
  5. Simone, S.F. ; Brutomesso, R. ; Sharygina, N. "An Efficient and Flexible Approach to Resolution Proof Reduction". 6th Haifa Verification Conference, 2010.
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