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Resolution lower bounds for the weak pigeonhole principle

Ran Raz
2004 Journal of the ACM  
One corollary is that a certain propositional formulation of the statement N P 6 P = p o l ydoes not have short Resolution proofs.  ...  The Resolution rule says that if C and D are two clauses and x i is a variable then any assignment that satisfies both of the clauses, C _ x i and D _ : x i , also  ...  N P 6 P = p o l ydoes not have short Resolution proofs. The Pigeon Hole Principle (PHP) is one of the most widely studied tautologies in propositional proof theory.  ... 
doi:10.1145/972639.972640 fatcat:o4wjhfe7cndozfiso4gn4k6uua

On Finding Short Resolution Refutations and Small Unsatisfiable Subsets [chapter]

Michael R. Fellows, Stefan Szeider, Graham Wrightson
2004 Lecture Notes in Computer Science  
We consider the parameterized problems of whether a given set of clauses can be refuted within k resolution steps, and whether a given set of clauses contains an unsatisfiable subset of size at most k.  ...  Our results remain true if restricted to 3-SAT instances and/or to various restricted versions of resolution including tree-like resolution, input resolution, and read-once resolution.  ...  clauses labeling its predecessors; (2) R contains exactly one sink; the sink is labeled with the empty clause.  ... 
doi:10.1007/978-3-540-28639-4_20 fatcat:2xzwc2sbnjgonjhrukgnucwxai

On finding short resolution refutations and small unsatisfiable subsets

Michael R. Fellows, Stefan Szeider, Graham Wrightson
2006 Theoretical Computer Science  
We consider the parameterized problems of whether a given set of clauses can be refuted within k resolution steps, and whether a given set of clauses contains an unsatisfiable subset of size at most k.  ...  Our results remain true if restricted to 3-SAT instances and/or to various restricted versions of resolution including tree-like resolution, input resolution, and read-once resolution.  ...  clauses labeling its predecessors; (2) R contains exactly one sink; the sink is labeled with the empty clause.  ... 
doi:10.1016/j.tcs.2005.10.005 fatcat:ncfeoii32bcc7fmlp356njhkxu

Locality and Hard SAT-Instances

Klas Markström, Daniel Le Berre, Laurent Simon
2006 Journal on Satisfiability, Boolean Modeling and Computation  
In this note we construct a family of SAT-instance based on Eulerian graphs which are aimed at being hard for resolution based SAT-solvers.  ...  We discuss some experiments made with instances of this type and how a solver can try to avoid at least some of the pitfalls presented by these instances.  ...  In connection with this we will also give a short discussion of the importance of local dense subformulae for the performance of resolution based solvers.  ... 
doi:10.3233/sat190024 fatcat:y3z7fqup3zalba22qtraboyiuu

Resolution lower bounds for the weak pigeonhole principle

Ran Raz
2002 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing - STOC '02  
One corollary is that a certain propositional formulation of the statement N P 6 P = p o l ydoes not have short Resolution proofs.  ...  The Resolution rule says that if C and D are two clauses and x i is a variable then any assignment that satisfies both of the clauses, C _ x i and D _ : x i , also  ...  N P 6 P = p o l ydoes not have short Resolution proofs. The Pigeon Hole Principle (PHP) is one of the most widely studied tautologies in propositional proof theory.  ... 
doi:10.1145/509984.509987 fatcat:7pu7sukc5vcsrhoua54tsakile

Resolution lower bounds for the weak pigeonhole principle

Ran Raz
2002 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing - STOC '02  
One corollary is that a certain propositional formulation of the statement N P 6 P = p o l ydoes not have short Resolution proofs.  ...  The Resolution rule says that if C and D are two clauses and x i is a variable then any assignment that satisfies both of the clauses, C _ x i and D _ : x i , also  ...  N P 6 P = p o l ydoes not have short Resolution proofs. The Pigeon Hole Principle (PHP) is one of the most widely studied tautologies in propositional proof theory.  ... 
doi:10.1145/509907.509987 dblp:conf/stoc/Raz02a fatcat:rr4ys2lxu5gw5lpe75c3dyu2gi

Boundary Points and Resolution [chapter]

Eugene Goldberg
2009 Lecture Notes in Computer Science  
A lit(x)-boundary point mandates a resolution on variable x. Adding the resolvent of this resolution to F eliminates this boundary point.  ...  Any resolution proof has to eventually eliminate all boundary points of F. Hence one can study resolution proofs from the viewpoint of boundary point elimination.  ...  In the experiments we compared these short proofs with ones generated by the SAT-solver DMRP-SAT [11] . We performed the experiments on a Pentium-4 PC with clock frequency of 3 GHz.  ... 
doi:10.1007/978-3-642-02777-2_16 fatcat:4exoqvstpbgchfkbpus7jbpuju

Boundary Points and Resolution [chapter]

Eugene Goldberg, Panagiotis Manolios
2010 Advanced Techniques in Logic Synthesis, Optimizations and Applications  
A lit(x)-boundary point mandates a resolution on variable x. Adding the resolvent of this resolution to F eliminates this boundary point.  ...  Any resolution proof has to eventually eliminate all boundary points of F. Hence one can study resolution proofs from the viewpoint of boundary point elimination.  ...  In the experiments we compared these short proofs with ones generated by the SAT-solver DMRP-SAT [11] . We performed the experiments on a Pentium-4 PC with clock frequency of 3 GHz.  ... 
doi:10.1007/978-1-4419-7518-8_7 fatcat:74xfabft3rb4jj4pcs4c4dyghm

Local Symmetries in Propositional Logic [chapter]

Noriko H. Arai, Alasdair Urquhart
2000 Lecture Notes in Computer Science  
Examples are exhibited in which resolution with the global symmetry rule has an exponential speed-up with respect to the cutting plane refutation system.  ...  In the context of resolution, the rule enables the shortening of refutations by using symmetries present in an initial set of clauses. These symmetries can be local or global.  ...  It follows that C j must be identical with one of the premisses C h or C i . If the variable resolved on belongs to V , then C j can be inferred from C h and C i by resolution.  ... 
doi:10.1007/10722086_3 fatcat:daicfvl26fazxidevd7eqojkvy

Augmenting the Power of (Partial) MaxSat Resolution with Extension

Javier Larrosa, Emma Rollon
2020 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
In this paper we augment the MaxSAT resolution proof system with an extension rule.  ...  The refutation power of SAT and MaxSAT resolution is challenged by problems like the soft and hard Pigeon Hole Problem PHP for which short refutations do not exist.  ...  There is no short refutation for PHP hard with standard resolution (Haken 1985) , therefore there is no short refutation with MaxSAT resolution.  ... 
doi:10.1609/aaai.v34i02.5516 fatcat:yc4zw7ogyzdrnmm37cly5b42tu

Lower Bounds for Width-Restricted Clause Learning on Small Width Formulas [chapter]

Eli Ben-Sasson, Jan Johannsen
2010 Lecture Notes in Computer Science  
It is shown that lower bounds on the runtime of width-restricted clause learning follow from resolution width lower bounds.  ...  This yields the first lower bounds on width-restricted clause learning for formulas in 3-CNF.  ...  On the other hand, from the short regular resolution refutations of Ord n , short regular refutations of E 3 (Ord n ) are obtained easily.  ... 
doi:10.1007/978-3-642-14186-7_4 fatcat:c4k72ngmqze3pa4hohv3lxitie

Finding the Hardest Formulas for Resolution

Tomáš Peitl, Stefan Szeider
2021 The Journal of Artificial Intelligence Research  
A CNF formula is harder than another CNF formula with the same number of clauses if it requires a longer resolution proof.  ...  In this paper we introduce resolution hardness numbers; they give for m=1,2,... the length of a shortest proof of a hardest formula on m clauses.  ...  Start with s = 1, and increment s by one while short s (F ) is unsatisfiable.  ... 
doi:10.1613/jair.1.12589 fatcat:nlriwgx4fvc4vlaujxxpusvf6e

Legislative Bill Drafting

Paul Mason
1926 California Law Review  
In joint resolutions, since they deal with relations with the federal government, it is better practice to include the name of the state in the resolving clause.  ...  AMENDMENTS TO THE CODES For the purpose of amendment, the Codes are treated as acts with short titles.  ... 
doi:10.2307/3475509 fatcat:6kn3vkntefcahawsbi6owfxd7q

Towards a Better Understanding of (Partial Weighted) MaxSAT Proof Systems [chapter]

Javier Larrosa, Emma Rollon
2020 Lecture Notes in Computer Science  
MaxSAT is a very popular language for discrete optimization with many domains of application.  ...  First, we give some basic definitions on completeness and show that refutational completeness makes compleness redundant, as it happens in SAT.  ...  If u is a SAT symmetric resolution, the inference step is a MaxSAT resolution generating one clause with weight f (u).  ... 
doi:10.1007/978-3-030-51825-7_16 fatcat:unfyounsxre5ve45pzfspfrmfm

QRAT Polynomially Simulates Merge Resolution [article]

Sravanthi Chede, Anil Shukla
2021 arXiv   pre-print
Each line of MRes consists of clauses with only existential literals, together with information of countermodels stored as merge maps. As a result, MRes has strategy extraction by design.  ...  Merge Resolution (MRes [Beyersdorff et al. J. Autom. Reason.'2021] ) is a refutational proof system for quantified Boolean formulas (QBF).  ...  A clause D is a sub-clause of C if every literal of D are also a literal of C. A unit clause is a clause with only one literal in it.  ... 
arXiv:2107.09320v1 fatcat:khn37rgxojaydolrvm4ofvpwqi
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