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...f the verification environment AProVE [6] and the Tseitin implementation of SAT4J [7]. For the top matrix, satisfiability for the threshold value kmin = 23 was shown using e.g. the SAT solver MiniSAT =-=[3]-=-, as reported in [5]. The unsatisfiability proof for kmin − 1 = 22 was only achieved later by CryptoMiniSAT [8] (cf. [4]). The corresponding value of kmin for the bottom matrix is currently unknown. I...

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... prenex normal form that can be used as input for a SAT solver (by dropping explicit existential quantification, encoding cardinality constraints using [5, 1], and performing Tseitin’s transformation =-=[14]-=-). δ3(ℓ) = ∧ ⎛ ⎝fℓ,i → ∧ ⎞ (ψ(j, i) ↔ aℓ,j) ⎠ ∧ exactly1(fℓ,1, . . . , fℓ,k) 1≤i≤k 1≤j≤n δ = ∃b1,1. . . . ∃bk,n.∃c1,1. . . ....∃ck,k.∃f1,1. . . . ∃fm,k. β1 ∧ ∧ 1≤ℓ≤m δ3(ℓ) For the implementation of δ ...

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...mplementations it is beneficial to represent propositional formulae not as trees, but as directed acyclic graphs with sharing of common subformulae. This technique is also known as structural hashing =-=[6]-=-. We perform standard Boolean simplifications (e.g., ϕ ∧ 1 = ϕ), we share Boolean junctor applications modulo commutativity and idempotence (where applicable), and we use varyadic ∧ and ∨. In contrast...

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.... . , fℓ,k) 1≤i≤k 1≤j≤n δ = ∃b1,1. . . . ∃bk,n.∃c1,1. . . ....∃ck,k.∃f1,1. . . . ∃fm,k. β1 ∧ ∧ 1≤ℓ≤m δ3(ℓ) For the implementation of δ we used the SAT framework in the verification environment AProVE =-=[8]-=- and the Tseitin implementation from SAT4J [12]. 3.1 Size of the Encoding Given a decision problem with an m × n matrix and a natural number k (where w.l.o.g. m ≤ k holds since for m > k, we could jus...

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...efit from linearity of the operation ⊕ on GF(2), which means that the absolute positiveness criterion for polynomials [10] (a simple technique commonly used in automated termination provers, cf. e.g. =-=[7]-=-) is not only sound, but also complete. Essentially, the idea is that two linear forms compute the same function iff their coefficients are identical. In this way, we can now drop the inputs x1, . . ....

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...r program lengths 12 to 35. They were created using our implementation for [5], which makes use of the SAT framework of the verification environment AProVE [6] and the Tseitin implementation of SAT4J =-=[7]-=-. For the top matrix, satisfiability for the threshold value kmin = 23 was shown using e.g. the SAT solver MiniSAT [3], as reported in [5]. The unsatisfiability proof for kmin − 1 = 22 was only achiev...

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...sfiability for the threshold value kmin = 23 was shown using e.g. the SAT solver MiniSAT [3], as reported in [5]. The unsatisfiability proof for kmin − 1 = 22 was only achieved later by CryptoMiniSAT =-=[8]-=- (cf. [4]). The corresponding value of kmin for the bottom matrix is currently unknown. In [2], Boyar and Peralta report that they have found linear straight-line programs of 30 lines for the bottom m...

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... ∧ ψ(j, p) 1≤p<i We finally get an encoding δ in prenex normal form that can be used as input for a SAT solver (by dropping explicit existential quantification, encoding cardinality constraints using =-=[5, 1]-=-, and performing Tseitin’s transformation [14]). δ3(ℓ) = ∧ ⎛ ⎝fℓ,i → ∧ ⎞ (ψ(j, i) ↔ aℓ,j) ⎠ ∧ exactly1(fℓ,1, . . . , fℓ,k) 1≤i≤k 1≤j≤n δ = ∃b1,1. . . . ∃bk,n.∃c1,1. . . ....∃ck,k.∃f1,1. . . . ∃fm,k. β...

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.... . .,xn, our program should yield the correct result. Fortunately, we can now benefit from linearity of the operation ⊕ on GF(2), which means that the absolute positiveness criterion for polynomials =-=[10]-=- (a simple technique commonly used in automated termination provers, cf. e.g. [7]) is not only sound, but also complete. Essentially, the idea is that two linear forms compute the same function iff th...

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...In Proc. SAT’10, LNCS, 2010. Supported by the G.I.F. grant 966-116.6 and the Danish Natural Science Research Council.While there are heuristic methods for finding short straight-line linear programs =-=[4]-=- (see also [3] for the corresponding patent application), to the best of our knowledge, there is no feasible method for finding an optimal solution. In this paper, we present an approach based on redu...

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... ∧ ψ(j, p) 1≤p<i We finally get an encoding δ in prenex normal form that can be used as input for a SAT solver (by dropping explicit existential quantification, encoding cardinality constraints using =-=[5, 1]-=-, and performing Tseitin’s transformation [14]). δ3(ℓ) = ∧ ⎛ ⎝fℓ,i → ∧ ⎞ (ψ(j, i) ↔ aℓ,j) ⎠ ∧ exactly1(fℓ,1, . . . , fℓ,k) 1≤i≤k 1≤j≤n δ = ∃b1,1. . . . ∃bk,n.∃c1,1. . . ....∃ck,k.∃f1,1. . . . ∃fm,k. β...

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...n problem (“Is there a program of length k?”) to satisfiability of propositional logic. The reduction is performed in a way that every model found by the SAT solver represents a solution. Recent work =-=[11]-=- has shown that reductions to satisfiability problems are a promising approach for circuit synthesis. By restricting our attention to linear functions, we now obtain a polynomial-size encoding. The st...

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...ust” 1500 variables. In this section we discuss three different approaches based on unary SAT encodings, on Pseudo-Boolean satisfiability, and on symmetry breaking. 6.1 Unary encodings As remarked by =-=[9]-=-, encoding arithmetic in unary representation instead of the more common binary (CPU-like) representation can be very beneficial for the performance of modern conflict-driven SAT solvers on the result...

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...utes all yℓ. Note that here we are aiming at a (provably) optimal solution. This is opposed to allowing approximations with more lines than actually necessary, which is currently the state of the art =-=[2]-=-. As a step towards solving this optimization problem, first let us consider the corresponding decision problem: Given n variables x1, . . .,xn over GF(2), m linear forms yℓ = aℓ,1 · x1 ⊕ . . . ⊕ aℓ,n...

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...0, LNCS, 2010. Supported by the G.I.F. grant 966-116.6 and the Danish Natural Science Research Council.While there are heuristic methods for finding short straight-line linear programs [4] (see also =-=[3]-=- for the corresponding patent application), to the best of our knowledge, there is no feasible method for finding an optimal solution. In this paper, we present an approach based on reducing the assoc...