Local Backbones Ronald de Haan 1 , Iyad Kanj 2 , Stefan Szeider 1 1 Technische Universit¨ at Wien 2 DePaul University SAT 2013
Backbones in propositional theories A backbone of a propositional theory is a variable that has the same truth value in each satisfying assignment. ◮ i.e., x ∈ Var( ϕ ) is a backbone of a CNF formula ϕ if ϕ | = x or ϕ | = ¬ x . Identifying backbones allows us to simplify the theory. Unfortunately, deciding whether a variable is a backbone is coNP-complete. Our approach: ◮ Relax and localize the notion of a backbone. ◮ It is reasonable that some variables are enforced locally (local backbones). ◮ Main theoretical tool: parameterized complexity theory. 1/16
Overview What are local backbones? Do local backbones occur? Parameterized complexity results Iterative local backbones
What are local backbones? Definition ( k -backbones). A k-backbone of a CNF formula ϕ is a variable x ∈ Var( ϕ ) such that for some ϕ ′ ⊆ ϕ with | ϕ ′ | ≤ k it holds that ϕ ′ | = x or ϕ ′ | = ¬ x . Example: x 2 is a 2-backbone of ϕ ( ¬ x 2 is implied by a subset of size 2). ϕ = {{ x 1 , ¬ x 2 } , {¬ x 1 , ¬ x 2 } , { x 2 , x 3 , x 4 } , { x 2 , ¬ x 3 , x 4 } , {¬ x 4 , x 5 }} ◮ Every k -backbone of ϕ is a backbone of ϕ . ◮ 1-backbones correspond to unit clauses. 2/16
Where were we? What are local backbones? Do local backbones occur? Parameterized complexity results Iterative local backbones
Distribution of local backbones 100 100 percentage of backbones that 80 80 are k -backbones 60 60 40 40 20 logistics 20 ssa7552 0 0 0 20 40 60 80 100 0 20 40 60 80 100 k k 100 100 percentage of backbones that 80 80 are k -backbones 60 60 40 40 ii32 20 20 random bmc-ibm 0 0 0 20 40 60 80 100 0 20 40 60 80 100 k k 3/16
Parameterized complexity theory Parameterized complexity theory investigates how to algorithmically exploit structure in problem instances. ◮ Takes into account a parameter k of the input, besides the input size n . If k is a constant, then finding k -backbones can be done in polynomial time. ◮ Brute force search in roughly n k time (XP). ◮ For k = 3 , 4 , . . . this is already not so practical. We would like to solve the problem in f ( k ) · n c time, for some function f and some constant c : fixed-parameter tractability (FPT). k k n n instead of 4/16
Parameterized complexity theory To give evidence that some problems are not FPT, there exist fixed-parameter intractability classes: FPT ⊆ W[1] ⊆ W[2] ⊆ · · · ⊆ W[P] The classes W[t] are based on the question whether certain Boolean circuits are satisfiable with k input nodes set to true. These classes are not fixed-parameter tractable unless the Exponential Time Hypothesis (ETH) fails. ◮ ETH: 3SAT cannot be solved in subexponential time. 5/16
Where were we? What are local backbones? Do local backbones occur? Parameterized complexity results Iterative local backbones
The parameterized decision problem We consider the following parameterized decision problems, for propositional languages C . L OCAL -B ACKBONE [ C ] a CNF formula ϕ ∈ C , a variable x ∈ Var( ϕ ), Instance: and an integer k ≥ 1 . Parameter: k. Question: Is x a k-backbone of ϕ ? 6/16
Formulas with bounded variable occurrence Fix an integer d ≥ 1. We let VO d denote the class of CNF formulas in which each variable occurs at most d times. Theorem. L OCAL -B ACKBONE ( VO d ) is FPT. Proof (idea). Bounded search tree. Search for a subset ϕ ′ ⊆ ϕ witnessing ϕ ′ | = ℓ for some ℓ ∈ { x , ¬ x } with a bounded search tree. Start with some clause c containing x . For each variable y in the current set ϕ ′ , guess a (non-empty) subset of clauses containing y . ◮ bounded number of branches, since y occurs in at most d clauses The depth of the search tree is at most k , since | ϕ ′ | ≤ k . 7/16
Formulas with bounded variable occurrence Fix an integer d ≥ 1. We let VO d denote the class of CNF formulas in which each variable occurs at most d times. Theorem. L OCAL -B ACKBONE ( VO d ) is FPT. Proof (idea). Bounded search tree. Search for a subset ϕ ′ ⊆ ϕ witnessing ϕ ′ | = ℓ for some ℓ ∈ { x , ¬ x } with a bounded search tree. Start with some clause c containing x . For each variable y in the current set ϕ ′ , guess a (non-empty) subset of clauses containing y . ◮ bounded number of branches, since y occurs in at most d clauses The depth of the search tree is at most k , since | ϕ ′ | ≤ k . 7/16
Formulas with bounded variable occurrence Fix an integer d ≥ 1. We let VO d denote the class of CNF formulas in which each variable occurs at most d times. Theorem. L OCAL -B ACKBONE ( VO d ) is FPT. Proof (idea). Bounded search tree. Example: ϕ = {{¬ x 1 , x 2 } , { x 2 , x 3 } , {¬ x 2 } , {¬ x 3 , x 4 } , {¬ x 3 , ¬ x 4 } , { x 4 , x 5 }} {{ x 2 , x 3 }} 7/16
Formulas with bounded variable occurrence Fix an integer d ≥ 1. We let VO d denote the class of CNF formulas in which each variable occurs at most d times. Theorem. L OCAL -B ACKBONE ( VO d ) is FPT. Proof (idea). Bounded search tree. Example: ϕ = {{¬ x 1 , x 2 } , { x 2 , x 3 } , {¬ x 2 } , {¬ x 3 , x 4 } , {¬ x 3 , ¬ x 4 } , { x 4 , x 5 }} {{ x 2 , x 3 }} {{ x 2 , x 3 } , {¬ x 1 , x 2 }} {{ x 2 , x 3 } , {¬ x 2 }} {{ x 2 , x 3 } , {¬ x 1 , x 2 } , {¬ x 2 }} • • • • • • • 7/16
Formulas with bounded variable occurrence Fix an integer d ≥ 1. We let VO d denote the class of CNF formulas in which each variable occurs at most d times. Theorem. L OCAL -B ACKBONE ( VO d ) is FPT. Proof (idea). Bounded search tree. Example: ϕ = {{¬ x 1 , x 2 } , { x 2 , x 3 } , {¬ x 2 } , {¬ x 3 , x 4 } , {¬ x 3 , ¬ x 4 } , { x 4 , x 5 }} {{ x 2 , x 3 }} {{ x 2 , x 3 } , {¬ x 1 , x 2 }} {{ x 2 , x 3 } , {¬ x 2 }} {{ x 2 , x 3 } , {¬ x 1 , x 2 } , {¬ x 2 }} • • • • • • • . . . . . . . . . . . . . . . {{ x 2 , x 3 } , {¬ x 2 }{ x 3 , ¬ x 4 } , {¬ x 3 , ¬ x 4 }} . . . • • • • • • • 7/16
Formulas with bounded variable occurrence Fix an integer d ≥ 1. We let VO d denote the class of CNF formulas in which each variable occurs at most d times. Theorem. L OCAL -B ACKBONE ( VO d ) is FPT. Proof (idea). Bounded search tree. Example: ϕ = {{¬ x 1 , x 2 } , { x 2 , x 3 } , {¬ x 2 } , {¬ x 3 , x 4 } , {¬ x 3 , ¬ x 4 } , { x 4 , x 5 }} branching ≤ 2 d {{ x 2 , x 3 }} {{ x 2 , x 3 } , {¬ x 1 , x 2 }} {{ x 2 , x 3 } , {¬ x 2 }} {{ x 2 , x 3 } , {¬ x 1 , x 2 } , {¬ x 2 }} • • • • • • • depth ≤ k . . . . . . . . . . . . . . . {{ x 2 , x 3 } , {¬ x 2 }{ x 3 , ¬ x 4 } , {¬ x 3 , ¬ x 4 }} . . . • • • • • • • 7/16
Local backbones of various propositional fragments Complexity of L OCAL -B ACKBONE [ C ] , for C ⊆ { D,N,K,H } : - H N K D NH KH DH NK DN DK NKH DNH DKH DNK DNKH D: no purely negative clauses N: no unit clauses K: clauses are Krom e.g., DH corresponds to definite Horn H: clauses are Horn 8/16
Local backbones of various propositional fragments Complexity of L OCAL -B ACKBONE [ C ] , for C ⊆ { D,N,K,H } : - H N K D NH KH DH NK DN DK NKH DNH DKH DNK W[1]-complete (NP-complete) FPT DNKH D: no purely negative clauses N: no unit clauses K: clauses are Krom e.g., DH corresponds to definite Horn H: clauses are Horn (All results hold also for the restriction to 3CNF.) 8/16
Local backbones of various propositional fragments Complexity of L OCAL -B ACKBONE [ C ] , for C ⊆ { D,N,K,H } : - SAT is NP-hard H N K D NH KH DH NK DN DK NKH DNH DKH DNK W[1]-complete (NP-complete) FPT DNKH D: no purely negative clauses N: no unit clauses K: clauses are Krom e.g., DH corresponds to definite Horn H: clauses are Horn (All results hold also for the restriction to 3CNF.) 8/16
Small Unsatisfiable Subsets Local backbones are closely related to small unsatisfiable subsets. ◮ useful for the repair of inconsistent knowledge bases. Originally considered in Fellows et al. (2006). S MALL -U NSATISFIABLE -S UBSET [ C ] a CNF formula ϕ ∈ C , and an integer k ≥ 1 . Instance: Parameter: k. Is there an unsatisfiable ϕ ′ ⊆ ϕ Question: with at most k clauses? Theorem. For any C , S MALL -U NSATISFIABLE -S UBSET [ C ] has the same parameterized complexity as L OCAL -B ACKBONE [ C ] . 9/16
Small Unsatisfiable Subsets Local backbones are closely related to small unsatisfiable subsets. ◮ useful for the repair of inconsistent knowledge bases. Originally considered in Fellows et al. (2006). S MALL -U NSATISFIABLE -S UBSET [ C ] a CNF formula ϕ ∈ C , and an integer k ≥ 1 . Instance: Parameter: k. Is there an unsatisfiable ϕ ′ ⊆ ϕ Question: with at most k clauses? Theorem. For any C , S MALL -U NSATISFIABLE -S UBSET [ C ] has the same parameterized complexity as L OCAL -B ACKBONE [ C ] . 9/16
Definite Horn Formulas Theorem. L OCAL -B ACKBONE [ DefHorn ] is W[1]-hard. Proof (idea). Reduction from M ULTICOLORED -C LIQUE (see below). 2 · · · · · · · · · · · p 1 , 2 · · · s p 2 , 3 · · 1 t · · p 1 , 3 · · · · · · · · · · · · 3 (A slight modification of the proof works for the case of NH.) 10/16
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