Explicit Versus Symbolic Algorithms for Solving ALFP Constraints

In this paper, we investigate the model checking (MC) problem for Halpern and Shoham's interval temporal logic HS. In the last years, interval temporal logic MC has received an increasing attention as a viable alternative to the traditional (point-based) temporal logic MC, which can be recovered as a special case. Most results have been obtained under the homogeneity assumption, that constrains a proposition letter to hold over an interval if and only if it holds over each component state. Recently, Lomuscio and Michaliszyn proposed a way to relax such an assumption by exploiting regular expressions to define the behaviour of proposition letters over intervals in terms of their component states. When homogeneity is assumed, the exact complexity of MC is a difficult open question for full HS and for its two syntactically maximal fragments AA'BB'E' and AA'EB'E'. In this paper, we provide an asymptotically optimal bound to the complexity of these two fragments under the more expressive semantic variant based on regular expressions by showing that their MC problem is AEXP_pol-complete, where AEXP_pol denotes the complexity class of problems decided by exponential-time bounded alternating Turing Machines making a polynomially bounded number of alternations.

The paper proposes a new technique to mitigate the state explosion in model checking. The technique is called a divide & conquer approach to leads-to model checking. As indicated by the name, the technique is dedicated to leads-to properties. It is known that many important systems requirements can be expressed as leads-to properties, thus it is worth focusing on leads-to properties. The technique divides an original leads-to model checking problem into multiple smaller model checking problems and tackles each smaller one. We prove a theorem that the multiple smaller model checking problems are equivalent to the original leads-to model checking problem. We conduct two case studies demonstrating the power of the proposed technique.

Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the program's constants. We define a fixed point logic (FPL) extension of Clark's completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called (loosely) guarded programs. Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in the decidable (loosely) guarded fixed point logic (μ(L)GF). Moreover, we reduce normal closed ASP to loosely guarded OASP, enabling a characterization of an answer set semantics by μLGF formulas. Finally, we relate guarded OASP to Datalog LITE, thus linking an answer set semantics to a semantics based on fixed point models of extended stratified Datalog programs. From this correspondence, we deduce 2-EXPTIME-completeness of satisfiability checking w.r.t. (loosely) guarded programs.

The two central decision problems that arise during the design of safety critical systems are the satisfiability and the model checking problem. While model checking can only be applied after implementing the system, satisfiability checking answers the question whether a system that satisfies the specification exists. Model checking is traditionally considered to be the simpler problem --- for branching-time and fixed point logics such as CTL, CTL*, ATL, and the classical and alternating time μ-calculus, the complexity of satisfiability checking is considerably higher than the model checking complexity. We show that ATL* is a notable exception of this rule: Both ATL* model checking and ATL* satisfiability checking are 2EXPTIME-complete.

We study the complexity of the model checking problem, for fixed models A, over certain fragments $\mathcal{L}$ of first-order logic, obtained by restricting which of the quantifiers and boolean connectives we permit. These are sometimes known as the expression complexities of $\mathcal{L}$. We obtain various full and partial complexity classification theorems for these logics $\mathcal{L}$ as each ranges over models A, in the spirit of the dichotomy conjecture for the Constraint Satisfaction Problem --- which itself may be seen as the model checking problem for existential conjunctive positive first-order logic.

The regular spanners (characterised by vset-automata) are closed under the algebraic operations of union, join and projection, and have desirable algorithmic properties. The core spanners (introduced by Fagin, Kimelfeld, Reiss, and Vansummeren (PODS 2013, JACM 2015) as a formalisation of the core functionality of the query language AQL used in IBM's SystemT) additionally need string-equality selections and it has been shown by Freydenberger and Holldack (ICDT 2016, Theory of Computing Systems 2018) that this leads to high complexity and even undecidability of the typical problems in static analysis and query evaluation. We propose an alternative approach to core spanners: by incorporating the string-equality selections directly into the regular language that represents the underlying regular spanner (instead of treating it as an algebraic operation on the table extracted by the regular spanner), we obtain a fragment of core spanners that, while having slightly weaker expressive power than the full class of core spanners, arguably still covers the intuitive applications of string-equality selections for information extraction and has much better upper complexity bounds of the typical problems in static analysis and query evaluation.

We give examples of classes of rigid structures which are of unbounded rigidity but least fixed point (Partial fixed point) logic can express all Boolean PTIME (PSPACE) queries on these classes. This shows that definability of linear order in FO+LEP although sufficient for it to capture Boolean PTIME queries, is not necessary even on the classes of rigid structures. The situation however appears very different for nonzero-ary queries. Next, we turn to the study of fixed point logics on arbitrary classes of structures. We completely characterize the recursively enumerable classes of finite structures on which PFP captures all PSPACE queries of arbitrary arities. We also state in some alternative forms several natural necessary and some sufficient conditions for PFP to capture PSPACE queries on classes of finite structures. The conditions similar to the ones proposed above work for LFP and PTIME also in some special cases but to prove the same necessary conditions in general for LFP to capture PTIME seems harder and remains open.

As a recently proposed membrane computing model, cP systems are capable of solving computational hard and distributed problems. Although several membrane system variants were formally verified in previous research, none of their approaches was applicable to cP systems. To formally verify the safety and liveness properties of cP systems, we solve a famous NP-Complete problem—the subset sum problem—in cP systems and use the PAT3 and ProB model checkers to verify the cP solution. Our cP solution outperforms previous work in time complexity and uses fewer rules. To perform model checking in cP systems, we define several mapping rules from cP notation to formal verification languages CSP# and B. We show that we can use model checkers to effectively detect design errors in cP systems. This work is the first study on formal verification of cP systems, and it showed that cP models can be effectively transformed into model checking problems and verified automatically.

Model-Checking of Correctness Conditions for Concurrent Objects

Streaming of XML transformations is a challenging task and only very few\nsystems support streaming. Research approaches generally define custom\nfragments of XQuery and XPath that are amenable to streaming, and then design\ncustom algorithms for each fragment. These languages have several shortcomings.\nHere we take a more principles approach to the problem of streaming\nXQuery-based transformations. We start with an elegant transducer model for\nwhich many static analysis problems are well-understood: the Macro Forest\nTransducer (MFT). We show that a large fragment of XQuery can be translated\ninto MFTs --- indeed, a fragment of XQuery, that can express important features\nthat are missing from other XQuery stream engines, such as GCX: our fragment of\nXQuery supports XPath predicates and let-statements. We then rely on a\nstreaming execution engine for MFTs, one which uses a well-founded set of\noptimizations from functional programming, such as strictness analysis and\ndeforestation. Our prototype achieves time and memory efficiency comparable to\nthe fastest known engine for XQuery streaming, GCX. This is surprising because\nour engine relies on the OCaml built in garbage collector and does not use any\nspecialized buffer management, while GCX's efficiency is due to clever and\nexplicit buffer management.\n

In this chapter we show that properties expressible in many logics are almost surely true or almost surely false; that is, either they hold for almost all structures, or they fail for almost all structures. This phenomenon is known as the zero-one law. We prove it for FO, fixed point logics, and L ∞ω ω . We shall also see that the “almost everywhere” behavior of logics is drastically different from their “everywhere” behavior. For example, while satisfiability in the finite is undecidable, it is decidable if a sentence is true in almost all finite models.

A Restricted Second Order Logic for Finite Structures

The Ordered conjecture of Kolaitis and Vardi asks whether fixed-point logic differs from first-order logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive definability and infinitary logic on proficient classes of finite structures (those admitting an unbounded induction). In particular, for a class of finite structures, we introduce a compactness notion which yields a new proof of a ramified version of McColm's second conjecture. Furthermore, we show a connection between a model-theoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). We also elaborate on complexity-theoretic implications of this line of research.KeywordsLinear OrderOrder LogicRelation SymbolPreservation PropertyInductive DefinitionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Automata and fixed point logic: A coalgebraic perspective

We consider fixed point logics, i.e., extensions of first order predicate logic with operators defining fixed points. A number of such operators, generalizing inductive definitions, have been studied in the context of finite model theory, including nondeterministic and alternating operators. We review results established in finite model theory, and also consider the expressive power of the resulting logics on infinite structures. In particular, we establish the relationship between inflationary and nondeterministic fixed point logics and second order logic, and we consider questions related to the determinacy of games associated with alternating fixed points.