An Improved Separation of Regular Resolution from Pool Resolution and Clause Learning

We prove that the graph tautology principles of Alekhnovich, Johannsen, Pitassi and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses and without degenerate resolution inferences. We also prove that these graph tautology principles can be refuted by polynomial size DPLL proofs with clause learning, even when restricted to greedy, unit-propagating DPLL search

Resolution Trees with Lemmas: Resolution Refinements that Characterize DLL Algorithms with Clause Learning

Resolution refinements called w-resolution trees with lemmas (WRTL) and with input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both WRTL and WRTI when there is no regularity condition. For regular proofs, an exponential separation between regular dag-like resolution and both regular WRTL and regular WRTI is given. It is proved that DLL proof search algorithms that use clause learning based on unit propagation can be polynomially simulated by regular WRTI. More generally, non-greedy DLL algorithms with learning by unit propagation are equivalent to regular WRTI. A general form of clause learning, called DLL-Learn, is defined that is equivalent to regular WRTL. A variable extension method is used to give simulations of resolution by regular WRTI, using a simplified form of proof trace extensions. DLL-Learn and non-greedy DLL algorithms with learning by unit propagation can use variable extensions to simulate general resolution without doing restarts. Finally, an exponential lower bound for WRTL where the lemmas are restricted to short clauses is shown

Towards Verifying Nonlinear Integer Arithmetic

We eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using CDCL SAT solvers, by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits. Such short proofs were conjectured not to exist. More precisely, we give n^{O(1)} size regular resolution proofs for arbitrary degree 2 identities on array, diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result

Limits of CDCL Learning via Merge Resolution

In their seminal work, Atserias et al. and independently Pipatsrisawat and Darwiche in 2009 showed that CDCL solvers can simulate resolution proofs with polynomial overhead. However, previous work does not address the tightness of the simulation, i.e., the question of how large this overhead needs to be. In this paper, we address this question by focusing on an important property of proofs generated by CDCL solvers that employ standard learning schemes, namely that the derivation of a learned clause has at least one inference where a literal appears in both premises (aka, a merge literal). Specifically, we show that proofs of this kind can simulate resolution proofs with at most a linear overhead, but there also exist formulas where such overhead is necessary or, more precisely, that there exist formulas with resolution proofs of linear length that require quadratic CDCL proofs

Limits of CDCL Learning via Merge Resolution

In their seminal work, Atserias et al. and independently Pipatsrisawat and Darwiche in 2009 showed that CDCL solvers can simulate resolution proofs with polynomial overhead. However, previous work does not address the tightness of the simulation, i.e., the question of how large this overhead needs to be. In this paper, we address this question by focusing on an important property of proofs generated by CDCL solvers that employ standard learning schemes, namely that the derivation of a learned clause has at least one inference where a literal appears in both premises (aka, a merge literal). Specifically, we show that proofs of this kind can simulate resolution proofs with at most a linear overhead, but there also exist formulas where such overhead is necessary or, more precisely, that there exist formulas with resolution proofs of linear length that require quadratic CDCL proofs

Geographical proximity and circulation of knowledge through inter-firm cooperation

The production of scientific and technological innovations has become essential for many firms, but the latter are seldom in possession of all the knowledge needed for this activity because of the increasing complexity of knowledge bases or because R&D departments are too small. As they do not possess internally all the skills they need, firms wishing to innovate have recourse to external sources, such as cooperation with other firms or public organizations of research. However, acquiring external knowledge is not sufficient; one must also be able to use it in a specific process of production, to transform it into organizational routines, because it is important not only to integrate this knowledge, but ideally to use it to produce new knowledge. This process of creation, re-creation or imitation of new resources not only necessitates several technical and organizational adaptations, but also requires frequent relations of cooperation and partnership. The integration of new knowledge cannot be done in one go, but progressively during the course of the innovation projects, which implies that relations be sustained for a period of time. But the interests of the participants to this interactive process, as well as their opinions concerning technical issues sometimes vary or diverge. This is why co-operations are also sources of tensions and conflicts that jeopardize the adaptation of knowledge produced somewhere else to the context of the firm or even completely hinder the innovation process. In this paper, we try to provide some answers to the following question: What is the role played by geographical and organized proximities in the context of these external acquisitions of knowledge? In other words, can they help reduce the intensity of conflicts and thus facilitate the interactive process of innovations? First, we present shortcomings of innovation theory and works on spillovers claiming the importance of geographical proximity for circulation of knowledge without considering organizational prerequisites to reach this impact. Having explained the relevance of permanent as well as temporary geographical proximity, we will then turn to a discussion of conflicts between cooperators within innovation processes from a theoretical as well as an empirical perspective. The empirical study is based on a case study of French biotechnology firms and will serve to prove our hypothesis that temporary geographical proximity play an important role in preventing and resolving conflicts between innovators.