多文字可满足SAT问题的相变点上界

可满足（SAT）问题是指：是否存在一组布尔变元赋值,使得随机合取范式（CNF）公式中每个子句至少有1个文字为真。多文字可满足SAT问题是指：是否存在一组布尔变元赋值,使得随机CNF公式中每个子句至少有2个文字为真。此问题仍然是一个NP难问题。定义约束密度α为CNF公式子句数与变元数之比,对该问题的相变点上界α*进行了研究。如果α>α*,则多文字可满足SAT问题高概率不可满足。通过一阶矩一个简单的推断,可以证明α*=-ln 2/ln(1-(k+1)/2k),当k=3时,α*=1。利用Kirousis等人的局部最大值技术,提升了多文字可满足3-SAT问题的相变点上界α*=0.7193。最后,选择了大量数据进行实验验证,结果表明,理论结果与实验结果相吻合。     

Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical D^P-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999).Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails.     

一个正则NP-完全问题及其不可近似性

通过一个适当的归约变换,可以将一个CNF (conjunctive normal form)公式变换为另一个具有某种特殊结构或性质的公式,使两者具有相同的可满足性.带有正则结构的CNF公式的因子图在图论中具有某些良好的性质和结果,可以用于研究公式的可满足性和计算复杂性.极小不可满足公式具有一个临界特征,公式本身不可满足,从原始公式中删去任意一个子句后得到的公式可满足.借助此临界特性,给出了一个从3-CNF公式到正则(3,4)-CNF公式的多项式归约转换.这里,正则(3,4)-CNF公式是指公式中每个子句的长度恰为3,每个变元出现的次数恰为4.因此,正则(3,4)-SAT问题是一个NP-完全问题,并且MAX(3,4)-SAT是不可近似问题.     

k-LSAT (k≥3)是NP-完全的

合取范式(conjunctive normal form,简称CNF)公式F是线性公式,如果F中任意两个不同子句至多有一个公共变元.如果F中的任意两个不同子句恰好含有一个公共变元,则称F是严格线性的.所有的严格线性公式均是可满足的,而对于线性公式类LCNF,对应的判定问题LSAT仍然是NP-完全的.LCNF_≥k是子句长度大于或等于k的CNF公式子类,判定问题LSA(≥k)的NP-完全性与LCNF(≥k)中是否含有不可满足公式密切相关.即LSAT_≥k的NP-完全性取决于LCNF_≥k是否含有不可满足公式.S.Porschen等人用超图和拉丁方的方法构造了LCNF_≥3和LCNF_≥4中的不可满足公式,并提出公开问题:对于k≥5,LCNF_≥k是否含有不可满足公式?将极小不可满足公式应用于公式的归约,引入了一个简单的一般构造方法.证明了对于k≥3,k-LCNF含有不可满足公式,从而证明了一个更强的结果:对于k≥3,k-LSAT是NP-完全的.     

Sequent forms of Herbrand theorem and their applications

New sequent forms* of the famous Herbrand theorem are proved for first-order classical logic without equality. These forms use the original notion of an admissible substitution and a certain modification of the Herbrand universe, which is constructed from constants, special variables, and functional symbols occurring only in the signature of an initial theory. Other well-known forms of the Herbrand theorem are obtained as special cases of the sequent ones. Besides, the sequent forms give an approach to the construction and theoretical investigation of computer-oriented calculi for efficient logical inference search in the signature of an initial theory. In a comparably simple way, they provide us with some technique for proving the completeness and soundness of the calculi. *A part of this investigation was performed during a visit to the University of Liverpool supported by the grant NAL/00841/G given by the Nuffield foundation.     

变量极小不可满足在模型检测中的应用

提出一个结合变量抽象和有界模型检测(BMC)的验证框架,用于证明反例不存在或输出存在反例.引入变量极小不可满足(VMU)的数学概念来驱动抽象精化的验证过程.一个VMU公式F的变量集合是保证其不可满足性的一个极小集合.严格证明了VMU驱动的精化满足抽象精化框架中的两个理想性质:有效性和极小性.虽然VMU的判定问题和极小不可满足(MU)一样难,即DP完全的,该案例研究表明,在变量抽象精化过程中,VMU比MU更为有效.     

k-LSAT(k≥3)是NP-完全的(英文)

合取范式(conjunctive normal form,简称CNF)公式F是线性公式,如果F中任意两个不同子句至多有一个公共变元.如果F中的任意两个不同子句恰好含有一个公共变元,则称F是严格线性的.所有的严格线性公式均是可满足的,而对于线性公式类LCNF,对应的判定问题LSAT仍然是NP-完全的.LCNF≥k是子句长度大于或等于k的CNF公式子类,判定问题LSAT≥k的NP-完全性与LCNF≥k中是否含有不可满足公式密切相关.即LSAT≥k的NP-完全性取决于LCNF≥k是否含有不可满足公式.S.Porschen等人用超图和拉丁方的方法构造了LCNF≥3和LCNF≥4中的不可满足公式,并提出公开问题:对于k≥5,LCNF≥k是否含有不可满足公式?将极小不可满足公式应用于公式的归约,引入了一个简单的一般构造方法.证明了对于k≥3,k-LCNF含有不可满足公式,从而证明了一个更强的结果:对于k≥3,k-LSAT是NP-完全的.     

The Complexity of Read-Once Resolution

We investigate the complexity of deciding whether a propositional formula has a read-once resolution proof. We give a new and general proof of Iwama–Miynano's theorem which states that the problem whether a formula has a read-once resolution proof is NP-complete. Moreover, we show for fixed k2 that the additional restriction that in each resolution step one of the parent clauses is a k-clause preserves the NP-completeness. If we demand that the formulas are minimal unsatisfiable and read-once refutable then the problem remains NP-complete. For the subclasses MU(k) of minimal unsatisfiable formulas we present a pol-time algorithm deciding whether a MU(k)-formula has a read-once resolution proof. Furthermore, we show that the problems whether a formula contains a MU(k)-subformula or a read-once refutable MU(k)-subformula are NP-complete.     

鸽巢公式的一些性质

由鸽巢原理定义的鸽巢公式PH nn+1是著名的消解难例之一,研究该公式的结构和性质有助于其他难例的构造.证明了PH nn+1是一个极小不可满足公式,根据其极小不可满足性,给出了最大可满足真值指派的两种标准形式,Haken关于PH nn+1的难解证明用到了其中一种标准形式.公式PH nn+1具有良好的子结构同构性质,如果DPLL算法中允许使用同构规则,则存在PH nn+1的反驳证明,其复杂性可以降至O(n3).     

Efficient Craig interpolation for linear Diophantine (dis)equations and linear modular equations

The use of Craig interpolants has enabled the development of powerful hardware and software model checking techniques. Efficient algorithms are known for computing interpolants in rational and real linear arithmetic. We focus on subsets of integer linear arithmetic. Our main results are polynomial time algorithms for obtaining interpolants for conjunctions of linear Diophantine equations, linear modular equations (linear congruences), and linear Diophantine disequations. We also present an interpolation result for conjunctions of mixed integer linear equations. We show the utility of the proposed interpolation algorithms for discovering modular/divisibility predicates in a counterexample guided abstraction refinement (CEGAR) framework. This has enabled verification of simple programs that cannot be checked using existing CEGAR based model checkers. This paper is an extended version of [14]. This research was sponsored by the Gigascale Systems Research Center (GSRC), Semiconductor Research Corporation (SRC), the National Science Foundation (NSF), the Office of Naval Research (ONR), the Naval Research Laboratory (NRL), the Defense Advanced Research Projects Agency (DARPA), the Army Research Office (ARO), and the General Motors Collaborative Research Lab at CMU. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of GSRC, SRC, NSF, ONR, NRL, DARPA, ARO, GM, or the U.S. government.