60 research outputs found
On Low Treewidth Approximations of Conjunctive Queries
We recently initiated the study of approximations of conjunctive queries within classes that admit tractable query evaluation (with respect to combined complexity). Those include classes of acyclic, bounded treewidth, or bounded hypertreewidth queries. Such approximations are always guaranteed to exist. However, while for acyclic and bounded hypertreewidth queries we have shown a number of examples of interesting approximations, for queries of bounded treewidth the study had been restricted to queries over graphs, where such approximations usually trivialize. In this note we show that for relations of arity greater than two, the notion of low treewidth approximations is a rich one, as many queries possess them. In fact we look at approximations of queries of maximum possible treewidth by queries of minimum possible treewidth (i.e., one), and show that even in this case the structure of approximations remain rather rich as long as input relations are not binary
First-Order and Temporal Logics for Nested Words
Nested words are a structured model of execution paths in procedural
programs, reflecting their call and return nesting structure. Finite nested
words also capture the structure of parse trees and other tree-structured data,
such as XML. We provide new temporal logics for finite and infinite nested
words, which are natural extensions of LTL, and prove that these logics are
first-order expressively-complete. One of them is based on adding a "within"
modality, evaluating a formula on a subword, to a logic CaRet previously
studied in the context of verifying properties of recursive state machines
(RSMs). The other logic, NWTL, is based on the notion of a summary path that
uses both the linear and nesting structures. For NWTL we show that
satisfiability is EXPTIME-complete, and that model-checking can be done in time
polynomial in the size of the RSM model and exponential in the size of the NWTL
formula (and is also EXPTIME-complete). Finally, we prove that first-order
logic over nested words has the three-variable property, and we present a
temporal logic for nested words which is complete for the two-variable fragment
of first-order.Comment: revised and corrected version of Mar 03, 201
Logical Languages Accepted by Transformer Encoders with Hard Attention
We contribute to the study of formal languages that can be recognized by
transformer encoders. We focus on two self-attention mechanisms: (1) UHAT
(Unique Hard Attention Transformers) and (2) AHAT (Average Hard Attention
Transformers). UHAT encoders are known to recognize only languages inside the
circuit complexity class , i.e., accepted by a family of poly-sized
and depth-bounded boolean circuits with unbounded fan-ins. On the other hand,
AHAT encoders can recognize languages outside ), but their
expressive power still lies within the bigger circuit complexity class , i.e., -circuits extended by majority gates. We first show a
negative result that there is an -language that cannot be
recognized by an UHAT encoder. On the positive side, we show that UHAT encoders
can recognize a rich fragment of -languages, namely, all languages
definable in first-order logic with arbitrary unary numerical predicates. This
logic, includes, for example, all regular languages from . We then
show that AHAT encoders can recognize all languages of our logic even when we
enrich it with counting terms. We apply these results to derive new results on
the expressive power of UHAT and AHAT up to permutation of letters (a.k.a.
Parikh images)
A Symbolic Language for Interpreting Decision Trees
The recent development of formal explainable AI has disputed the folklore
claim that "decision trees are readily interpretable models", showing different
interpretability queries that are computationally hard on decision trees, as
well as proposing different methods to deal with them in practice. Nonetheless,
no single explainability query or score works as a "silver bullet" that is
appropriate for every context and end-user. This naturally suggests the
possibility of "interpretability languages" in which a wide variety of queries
can be expressed, giving control to the end-user to tailor queries to their
particular needs. In this context, our work presents ExplainDT, a symbolic
language for interpreting decision trees. ExplainDT is rooted in a carefully
constructed fragment of first-ordered logic that we call StratiFOILed.
StratiFOILed balances expressiveness and complexity of evaluation, allowing for
the computation of many post-hoc explanations--both local (e.g., abductive and
contrastive explanations) and global ones (e.g., feature relevancy)--while
remaining in the Boolean Hierarchy over NP. Furthermore, StratiFOILed queries
can be written as a Boolean combination of NP-problems, thus allowing us to
evaluate them in practice with a constant number of calls to a SAT solver. On
the theoretical side, our main contribution is an in-depth analysis of the
expressiveness and complexity of StratiFOILed, while on the practical side, we
provide an optimized implementation for encoding StratiFOILed queries as
propositional formulas, together with an experimental study on its efficiency
Regular Languages of Nested Words: Fixed Points, Automata, and Synchronization
Nested words provide a natural model of runs of programs with recursive procedure calls. The usual connection between monadic second-order logic (MSO) and automata extends from words to nested words and gives us a natural notion of regular languages of nested words. In this paper we look at some well-known aspects of regular languagesâtheir characterization via fixed points, deterministic and alternating automata for them, and synchronization for defining regular relationsâand extend them to nested words. We show that mu-calculus is as expressive as MSO over finite and infinite nested words, and the equivalence holds, more generally, for mu-calculus with past modalities eval-uated in arbitrary positions in a word, not only in the first position. We introduce the notion of alternating automata for nested words, show that they are as expressive as the usual automata, and also prove that Muller automata can be determinized (unlike in the case of visibly pushdown languages). Finally we look at synchronization over nested words. We show that the usual letter-to-letter synchronization is completely incompatible with nested words (in the sense that even the weakest form of it leads to an undecidable formalism) and present an alternative form of synchronization that gives us decidable notions of regular relations
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