An exactly solvable random satisfiability problem

We introduce a new model for the generation of random satisfiability problems. It is an extension of the hyper-SAT model of Ricci-Tersenghi, Weigt and Zecchina, which is a variant of the famous K-SAT model: it is extended to q-state variables and relates to a different choice of the statistical ensemble. The model has an exactly solvable statistic: the critical exponents and scaling functions of the SAT/UNSAT transition are calculable at zero temperature, with no need of replicas, also with exact finite-size corrections. We also introduce an exact duality of the model, and show an analogy of thermodynamic properties with the Random Energy Model of disordered spin systems theory. Relations with Error-Correcting Codes are also discussed.Comment: 31 pages, 1 figur

Taming Numbers and Durations in the Model Checking Integrated Planning System

The Model Checking Integrated Planning System (MIPS) is a temporal least commitment heuristic search planner based on a flexible object-oriented workbench architecture. Its design clearly separates explicit and symbolic directed exploration algorithms from the set of on-line and off-line computed estimates and associated data structures. MIPS has shown distinguished performance in the last two international planning competitions. In the last event the description language was extended from pure propositional planning to include numerical state variables, action durations, and plan quality objective functions. Plans were no longer sequences of actions but time-stamped schedules. As a participant of the fully automated track of the competition, MIPS has proven to be a general system; in each track and every benchmark domain it efficiently computed plans of remarkable quality. This article introduces and analyzes the most important algorithmic novelties that were necessary to tackle the new layers of expressiveness in the benchmark problems and to achieve a high level of performance. The extensions include critical path analysis of sequentially generated plans to generate corresponding optimal parallel plans. The linear time algorithm to compute the parallel plan bypasses known NP hardness results for partial ordering by scheduling plans with respect to the set of actions and the imposed precedence relations. The efficiency of this algorithm also allows us to improve the exploration guidance: for each encountered planning state the corresponding approximate sequential plan is scheduled. One major strength of MIPS is its static analysis phase that grounds and simplifies parameterized predicates, functions and operators, that infers knowledge to minimize the state description length, and that detects domain object symmetries. The latter aspect is analyzed in detail. MIPS has been developed to serve as a complete and optimal state space planner, with admissible estimates, exploration engines and branching cuts. In the competition version, however, certain performance compromises had to be made, including floating point arithmetic, weighted heuristic search exploration according to an inadmissible estimate and parameterized optimization

Stable domination and independence in algebraically closed valued fields

We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stable domination coincides with a natural notion of `orthogonality to the value group'. We also investigate other notions of independence, and show that they all agree, and are well-behaved, for stably dominated types. One of these is used to show that every type extends to an invariant type; definable types are dense. Much of this work requires the use of imaginary elements. We also show existence of prime models over reasonable bases, possibly including imaginaries

Embedded Finite Models beyond Restricted Quantifier Collapse

We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as interpreted vocabulary over an infinite domain: denoted in the past as embedded finite model theory. We extend the analysis of "collapse results": the ability to eliminate first-order quantifiers over the infinite domain in favor of quantification over the finite structure. We investigate several weakenings of collapse, one allowing higher-order quantification over the finite structure, another allowing expansion of the theory. We also provide results comparing collapse for unary signatures with general signatures, and new analyses of collapse for natural decidable theories