lim+, delta+, and Non-Permutability of beta-Steps

Using a human-oriented formal example proof of the (lim+) theorem, i.e. that the sum of limits is the limit of the sum, which is of value for reference on its own, we exhibit a non-permutability of beta-steps and delta+-steps (according to Smullyan's classification), which is not visible with non-liberalized delta-rules and not serious with further liberalized delta-rules, such as the delta++-rule. Besides a careful presentation of the search for a proof of (lim+) with several pedagogical intentions, the main subject is to explain why the order of beta-steps plays such a practically important role in some calculi.Comment: ii + 36 page

Lower Complexity Bounds for Lifted Inference

One of the big challenges in the development of probabilistic relational (or probabilistic logical) modeling and learning frameworks is the design of inference techniques that operate on the level of the abstract model representation language, rather than on the level of ground, propositional instances of the model. Numerous approaches for such "lifted inference" techniques have been proposed. While it has been demonstrated that these techniques will lead to significantly more efficient inference on some specific models, there are only very recent and still quite restricted results that show the feasibility of lifted inference on certain syntactically defined classes of models. Lower complexity bounds that imply some limitations for the feasibility of lifted inference on more expressive model classes were established early on in (Jaeger 2000). However, it is not immediate that these results also apply to the type of modeling languages that currently receive the most attention, i.e., weighted, quantifier-free formulas. In this paper we extend these earlier results, and show that under the assumption that NETIME =/= ETIME, there is no polynomial lifted inference algorithm for knowledge bases of weighted, quantifier- and function-free formulas. Further strengthening earlier results, this is also shown to hold for approximate inference, and for knowledge bases not containing the equality predicate.Comment: To appear in Theory and Practice of Logic Programming (TPLP

A semantics for open normal defaults via a modified preferential approach

We present a new approach for handling open normal defaults that makes it possible 1. to derive existentially quantified formulae from other existentially quantified formulae by default, 2. to derive universally quantified formulae by default, and 3. to treat cardinality formulae analogously to other formulae. This was not the case for previous approaches. Reiter uses Skolemization in his treatment of open defaults to achieve the first goal, but this has the unpleasant side-effect that logically equivalent facts may lead to different default consequences. In addition, Reiter\u27s approach does not comply with our second requirement. Lifschitz\u27s main motivation for his approach was to satisfy this second demand. However, to achieve this goal he has to violate the third requirement, and the first condition is also not observed. Differing from these two previous approaches, we will not view open defaults as schemata for certain instantiated defaults. Instead they will be used to define a preference relation on models. But unlike the usual approaches to preferential semantics we shall not always take the minimal models to construct our semantics. Due to this new treatment of preference relations the resulting nonmonotonic consequence operator has nice proof-theoretic properties such as cumulativity

Quantification and Predication in Mandarin Chinese: A Case Study of Dou

In the more recent generalized quantifier theory, \u27every\u27 is defined as a relation between two sets such that the first set is a subset of the second set (Cooper (1987), van Benthem (1986)). We argue in this dissertation that the formal definition of \u27every\u27 ought to reflect our intuition that this quantifier is always associated with a pairing. For instance, \u27Every student left\u27 means that for every student there is an event (Davidson (1966), Kroch (1974), Mourelatos (1978), Bach (1986)) such that the student left in that event. We propose that the formal translation of EVERY be augmented by relating its two arguments via a skolem function. A skolem function links two variables by making the choice of a value for one variable depend on the choice of a value for the other. This definition of EVERY, after which \u27every\u27 and its Chinese counterpart \u27mei\u27 can be modeled, can help us explain the co-occurrence pattern between \u27mei\u27 and the adverb \u27dou\u27. It was observed in S.-Z. Huang (1995a) that \u27mei\u27 requires either \u27dou\u27, or an indefinite phrase, or a reflexive in its scope. Under the skolemized definition of EVERY, this is explainable: The skolem function needs a variable in the scope of EVERY. We stipulate that only morphologically/lexically licensed variables are available for quantification (of this kind). \u27Dou\u27 occurs with \u27mei\u27 because \u27dou\u27 can license the event variable for skolemization. This function of \u27dou\u27 is performed by the tense operator in English, while Chinese, lacking tense, resorts to \u27dou\u27. \u27Dou\u27, we will argue, is a sum operator on the event variable. Thus, \u27dou VPs\u27 always assert plural events, which predicts that the distribution of \u27dou\u27 may or may not involve universal quantification. Among other things, our account explains scope ambiguity in Chinese, the optionality of \u27dou\u27, and the interchangeability, in a number of contexts, between \u27dou\u27 and conjunction/additive words for VPs such as \u27ye\u27 also, and , \u27you\u27 also, again , and \u27hai\u27 also, still

On Counterexample Guided Quantifier Instantiation for Synthesis in CVC4

We introduce the first program synthesis engine implemented inside an SMT solver. We present an approach that extracts solution functions from unsatisfiability proofs of the negated form of synthesis conjectures. We also discuss novel counterexample-guided techniques for quantifier instantiation that we use to make finding such proofs practically feasible. A particularly important class of specifications are single-invocation properties, for which we present a dedicated algorithm. To support syntax restrictions on generated solutions, our approach can transform a solution found without restrictions into the desired syntactic form. As an alternative, we show how to use evaluation function axioms to embed syntactic restrictions into constraints over algebraic datatypes, and then use an algebraic datatype decision procedure to drive synthesis. Our experimental evaluation on syntax-guided synthesis benchmarks shows that our implementation in the CVC4 SMT solver is competitive with state-of-the-art tools for synthesis