Simplest random K-satisfiability problem

We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of $\gamma N$ random boolean constraints which are to be satisfied simultaneously by $N$ logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical/topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor errors and references correcte

Learn with SAT to Minimize B\"uchi Automata

We describe a minimization procedure for nondeterministic B\"uchi automata (NBA). For an automaton A another automaton A_min with the minimal number of states is learned with the help of a SAT-solver. This is done by successively computing automata A' that approximate A in the sense that they accept a given finite set of positive examples and reject a given finite set of negative examples. In the course of the procedure these example sets are successively increased. Thus, our method can be seen as an instance of a generic learning algorithm based on a "minimally adequate teacher" in the sense of Angluin. We use a SAT solver to find an NBA for given sets of positive and negative examples. We use complementation via construction of deterministic parity automata to check candidates computed in this manner for equivalence with A. Failure of equivalence yields new positive or negative examples. Our method proved successful on complete samplings of small automata and of quite some examples of bigger automata. We successfully ran the minimization on over ten thousand automata with mostly up to ten states, including the complements of all possible automata with two states and alphabet size three and discuss results and runtimes; single examples had over 100 states.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Polynomial-time algorithms for generation of prime implicants

AbstractA notion of a neighborhood cube of a term of a Boolean function represented in the canonical disjunctive normal form is introduced. A relation between neighborhood cubes and prime implicants of a Boolean function is established. Various aspects of the problem of prime implicants generation are identified and neighborhood cube-based algorithms for their solution are developed. The correctness of algorithms is proven and their time complexity is analyzed. It is shown that all presented algorithms are polynomial in the number of minterms occurring in the canonical disjunctive normal form representation of a Boolean function. A summary of the known approaches to the solution of the problem of the generation of prime implicants is also included

JSKETCH: Sketching for Java

Sketch-based synthesis, epitomized by the SKETCH tool, lets developers synthesize software starting from a partial program, also called a sketch or template. This paper presents JSKETCH, a tool that brings sketch-based synthesis to Java. JSKETCH's input is a partial Java program that may include holes, which are unknown constants, expression generators, which range over sets of expressions, and class generators, which are partial classes. JSKETCH then translates the synthesis problem into a SKETCH problem; this translation is complex because SKETCH is not object-oriented. Finally, JSKETCH synthesizes an executable Java program by interpreting the output of SKETCH.Comment: This research was supported in part by NSF CCF-1139021, CCF- 1139056, CCF-1161775, and the partnership between UMIACS and the Laboratory for Telecommunication Science

Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization

None of the available minimizers for 2-level hazard-free logic minimization can synthesize very large circuits. This limitation has forced researchers to resort to manual and automated circuit partitioning techniques. This paper introduces two new 2-level logic minimizers:ESPRESSO-HF, a heuristic method which is loosely based on ESPRESSO-II, and IMPYMIN, an exact method based on implicit data structures. Both minimizers can solve all currently available examples, which range up to 32 inputs and 33 outputs.These include examples that have never been solved before.For examples that can be solved by other minimizers our methods are several orders of magnitude faster. As by-products of these algorithms, we also present two additional results. First, we introduce a fast new algorithm to check if a hazard-free covering problem can feasibly be solved. Second, we introduce a novel formulation of the 2-level hazard-free logic minimization problem by capturing hazard-freedom constraints within a synchronous function by adding new variables

BOOM - A Heuristic Boolean Minimizer

This paper presents an algorithm for two-level Boolean minimization (BOOM) based on a new implicant generation paradigm. In contrast to all previous minimization methods, where the implicants are generated bottom-up, the proposed method uses a top-down approach. Thus, instead of increasing the dimensionality of implicants by omitting literals from their terms, the dimension of a term is gradually decreased by adding new literals. The method is advantageous especially for functions with many input variables (up to thousands) and with only few care terms defined, where other minimization tools are not applicable because of the long runtime. The method has been tested on several different kinds of problems and the results were compared with ESPRESSO

Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization

None of the available minimizers for 2-level hazard-free logic minimization can synthesize very large circuits. This limitation has forced researchers to resort to manual and automated circuit partitioning techniques. This paper introduces two new 2-level logic minimizers:ESPRESSO-HF, a heuristic method which is loosely based on ESPRESSO-II, and IMPYMIN, an exact method based on implicit data structures. Both minimizers can solve all currently available examples, which range up to 32 inputs and 33 outputs.These include examples that have never been solved before.For examples that can be solved by other minimizers our methods are several orders of magnitude faster. As by-products of these algorithms, we also present two additional results. First, we introduce a fast new algorithm to check if a hazard-free covering problem can feasibly be solved. Second, we introduce a novel formulation of the 2-level hazard-free logic minimization problem by capturing hazard-freedom constraints within a synchronous function by adding new variables