Resolution Refutation Systems

1982 ◽  
pp. 161-191
Author(s):  
Nils J. Nilsson
Keyword(s):  
Refutation Systems ◽  
Resolution Refutation

RCA-Based Detection Methods for Resolution Refutation

2004 ◽  
pp. 32-36 ◽  
Author(s):  
In-Hee Lee ◽  
Ji Yoon Park ◽  
Young-Gyu Chai ◽  
Byoung-Tak Zhang
Keyword(s):  
Detection Methods ◽  
Resolution Refutation

scholarly journals DNA Implementation of Theorem Proving with Resolution Refutation in Propositional Logic

2003 ◽  
pp. 156-167 ◽  
Author(s):  
In-Hee Lee ◽  
Ji-Yoon Park ◽  
Hae-Man Jang ◽  
Young-Gyu Chai ◽  
Byoung-Tak Zhang
Keyword(s):  
Theorem Proving ◽  
Propositional Logic ◽  
Resolution Refutation

NAE-resolution: A new resolution refutation technique to prove not-all-equal unsatisfiability

2020 ◽  
Vol 30 (7) ◽  
pp. 736-751
Author(s):  
Hans Kleine Büning ◽  
P. Wojciechowski ◽  
K. Subramani
Keyword(s):  
Normal Form ◽  
Conjunctive Normal Form ◽  
Principal Result ◽  
Satisfying Assignment ◽  
Boolean Formulas ◽  
Complete Procedure ◽  
Np Complete ◽  
Resolution Refutation ◽  
Original Formula ◽  
Cnf Formulas

AbstractIn this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

scholarly journals Hybrid Deduction-Refutation Systems for FDE-Based Logics

2021 ◽  
Vol 18 (6) ◽  
pp. 599-615
Author(s):  
Eoin Moore
Keyword(s):  
Hybrid Systems ◽  
Hybrid System ◽  
Classical Logic ◽  
First Degree Entailment ◽  
Refutation Systems

Hybrid deduction-refuation systems are presented for four first-degree entailment based logics. The hybrid systems are shown to be deductively and refutationally sound with respect to their logics. The proofs of completeness are presented in a uniform way. The paper builds on work by Goranko, who presented a deductively and refutationally sound and complete hybrid system for classical logic.

scholarly journals Friends or Foes? On Planning as Satisfiability and Abstract CNF Encodings

2009 ◽  
Vol 36 ◽  
pp. 415-469 ◽  
Author(s):  
C. Domshlak ◽  
J. Hoffmann ◽  
A. Sabharwal
Keyword(s):  
Point Of View ◽  
Theoretical Point ◽  
Optimal Plan ◽  
Interesting Phenomenon ◽  
State Spaces ◽  
Planning Task ◽  
Resolution Refutation ◽  
International Planning ◽  
Almost All ◽  
Planning Graph

Planning as satisfiability, as implemented in, for instance, the SATPLAN tool, is a highly competitive method for finding parallel step-optimal plans. A bottleneck in this approach is to *prove the absence* of plans of a certain length. Specifically, if the optimal plan has N steps, then it is typically very costly to prove that there is no plan of length N-1. We pursue the idea of leading this proof within solution length preserving abstractions (over-approximations) of the original planning task. This is promising because the abstraction may have a much smaller state space; related methods are highly successful in model checking. In particular, we design a novel abstraction technique based on which one can, in several widely used planning benchmarks, construct abstractions that have exponentially smaller state spaces while preserving the length of an optimal plan. Surprisingly, the idea turns out to appear quite hopeless in the context of planning as satisfiability. Evaluating our idea empirically, we run experiments on almost all benchmarks of the international planning competitions up to IPC 2004, and find that even hand-made abstractions do not tend to improve the performance of SATPLAN. Exploring these findings from a theoretical point of view, we identify an interesting phenomenon that may cause this behavior. We compare various planning-graph based CNF encodings F of the original planning task with the CNF encodings F_abs of the abstracted planning task. We prove that, in many cases, the shortest resolution refutation for F_abs can never be shorter than that for F. This suggests a fundamental weakness of the approach, and motivates further investigation of the interplay between declarative transition-systems, over-approximating abstractions, and SAT encodings.

scholarly journals Non-clausal Redundancy Properties

2021 ◽  
pp. 252-272
Author(s):  
Lee A. Barnett ◽  
Armin Biere
Keyword(s):  
State Of The Art ◽  
Binary Decision Diagrams ◽  
Gaussian Elimination ◽  
Decision Diagrams ◽  
Binary Decision ◽  
Refutation Systems ◽  
Unit Propagation

AbstractState-of-the-art refutation systems for SAT are largely based on the derivation of clauses meeting some redundancy criteria, ensuring their addition to a formula does not alter its satisfiability. However, there are strong propositional reasoning techniques whose inferences are not easily expressed in such systems. This paper extends the redundancy framework beyond clauses to characterize redundancy for Boolean constraints in general. We show this characterization can be instantiated to develop efficiently checkable refutation systems using redundancy properties for Binary Decision Diagrams (BDDs). Using a form of reverse unit propagation over conjunctions of BDDs, these systems capture, for instance, Gaussian elimination reasoning over XOR constraints encoded in a formula, without the need for clausal translations or extension variables. Notably, these systems generalize those based on the strong Propagation Redundancy (PR) property, without an increase in complexity.

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