ACM Transactions on Computational Logic最新文献

We study the MaxSAT Resolution (MaxRes) rule in the context of certifying unsatisfiability. We show that it can be exponentially more powerful than tree-like resolution, and when augmented with weakening (the system MaxResW), p-simulates tree-like resolution. In devising a lower bound technique specific to MaxRes (and not merely inheriting lower bounds from Res), we define a new proof system called the SubCubeSums proof system. This system, which p-simulates MaxResW, can be viewed as a special case of the semi-algebraic Sherali–Adams proof system. In expressivity, it is the integral restriction of conical juntas studied in the contexts of communication complexity and extension complexity. We show that it is not simulated by Res. Using a proof technique qualitatively different from the lower bounds that MaxResW inherits from Res, we show that Tseitin contradictions on expander graphs are hard to refute in SubCubeSums. We also establish a lower bound technique via lifting: for formulas requiring large degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums.

Within knowledge representation in artificial intelligence, a first-order ontology is a theory in first-order logic that axiomatizes the concepts in some domain. Ontology verification is concerned with the relationship between the intended models of an ontology and the models of the axiomatization of the ontology. In particular, we want to characterize the models of an ontology up to isomorphism and determine whether or not these models are equivalent to the intended models of the ontology. Unfortunately, it can be quite difficult to characterize the models of an ontology up to isomorphism. In the first half of this article, we review the different metalogical relationships between first-order theories and identify which relationship is needed for ontology verification. In particular, we will demonstrate that the notion of logical synonymy is needed to specify a representation theorem for the class of models of one first-order ontology with respect to another. In the second half of the article, we discuss the notion of reducible theories and show we can specify representation theorems by which models are constructed by amalgamating models of the constituent ontologies.

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3k+4) is a complete isomorphism test for the class of all graphs of rank width at most k. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width.

It was known that isomorphism of graphs of rank width k is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time n^f(k) for a non-elementary function f. Our result yields an isomorphism test for graphs of rank width k running in time n^O(k). Another consequence of our result is the first polynomial-time canonisation algorithm for graphs of bounded rank width.

Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.

Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [1] and Zhuk [2], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way.

We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now that the original Chen Conjecture (see [3]) is known to be false [4].

Switchability was introduced by Chen [1] as a generalization of the already-known collapsibility [5]. There, an algebra 𝔸 :=({ 0,1,2};r) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.

We develop a doubly exponential decision procedure for the satisfiability problem of guarded separation logic—a novel fragment of separation logic featuring user-supplied inductive predicates, Boolean connectives, and separating connectives, including restricted (guarded) versions of negation, magic wand, and septraction. Moreover, we show that dropping the guards for any of the preceding connectives leads to an undecidable fragment.

We further apply our decision procedure to reason about entailments in the popular symbolic heap fragment of separation logic. In particular, we obtain a doubly exponential decision procedure for entailments between (quantifier-free) symbolic heaps with inductive predicate definitions of bounded treewidth (SL_btw)—one of the most expressive decidable fragments of separation logic. Together with the recently shown 2ExpTime-hardness for entailments in said fragment, we conclude that the entailment problem for SL_btw is 2ExpTime-complete—thereby closing a previously open complexity gap.

Optimized SAT solvers not only preprocess the clause set, they also transform it during solving as inprocessing. Some preprocessing techniques have been generalized to first-order logic with equality. In this article, we port inprocessing techniques to work with superposition, a leading first-order proof calculus, and we strengthen known preprocessing techniques. Specifically, we look into elimination of hidden literals, variables (predicates), and blocked clauses. Our evaluation using the Zipperposition prover confirms that the new techniques usefully supplement the existing superposition machinery.

We study the reachability problem for continuous one-counter automata, COCA for short. In such automata, transitions are guarded by upper- and lower-bound tests against the counter value. Additionally, the counter updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor between zero and one. Our three main results are as follows: we prove (1) that the reachability problem for COCA with global upper- and lower-bound tests is in NC²; (2) that, in general, the problem is decidable in polynomial time; and (3) that it is NP-complete for COCA with parametric counter updates and bound tests.