ACM Transactions on Computational Logic

Published by Association for Computing Machinery
Print ISSN: 1529-3785
Automata on infinite words are used for specification and verification of nonterminating programs. Different types of automata induce different levels of expressive power, of succinctness, and of complexity. Alternating automata have both existential and universal branching modes and are particularly suitable for specification of programs. In a weak alternating automaton, the state space is partitioned into partially ordered sets, and the automaton can proceed from a certain set only to smaller sets. Reasoning about weak alternating automata is easier than reasoning about alternating automata with no restricted structure. Known translations of alternating automata to weak alternating automata involve determinization, and therefore involve a double-exponential blow-up. In this paper we describe a quadratic translation, which circumvents the need for determinization, of Buchi (1962) and co-Buchi alternating automata to weak alternating automata. Beyond the independent interest of such a translation, it gives rise to a simple complementation algorithm for nondeterministic Buchi automata
LF is a dependent type theory in which many other formal systems can be conveniently embedded. However, correct use of LF relies on nontrivial metatheoretic developments such as proofs of correctness of decision procedures for LF's judgments. Although detailed informal proofs of these properties have been published, they have not been formally verified in a theorem prover. We have formalized these properties within Isabelle/HOL using the nominal datatype package, closely following a recent article by Harper and Pfenning. In the process, we identified and resolved a gap in one of the proofs and a small number of minor lacunae in others. Besides its intrinsic interest, our formalization provides a foundation for studying the adequacy of LF encodings, the correctness of Twelf-style metatheoretic reasoning, and the metatheory of extensions to LF.
Logical frameworks based on intuitionistic or linear logics with higher-type quantification have been successfully used to give high-level, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that evaluation preserves types. One challenge in developing a framework for such reasoning is that higher-order abstract syntax (HOAS), an elegant and declarative treatment of object-level abstraction and substitution, is difficult to treat in proofs involving induction. In this paper, we present a meta-logic that can be used to reason about judgments coded using HOAS; this meta-logic is an extension of a simple intuitionistic logic that admits higher-order quantification over simply typed lambda-terms (key ingredients for HOAS) as well as induction and a notion of definition. We explore the difficulties of formal meta-theoretic analysis of HOAS encodings by considering encodings of intuitionistic and linear logics, and formally derive the admissibility of cut for important subsets of these logics. We then propose an approach to avoid the apparent tradeoff between the benefits of higher-order abstract syntax and the ability to analyze the resulting encodings. We illustrate this approach through examples involving the simple functional and imperative programming languages PCF and PCF:=. We formally derive such properties as unicity of typing, subject reduction, determinacy of evaluation, and the equivalence of transition semantics and natural semantics presentations of evaluation.
Abstraction and Concretization.  
Abstraction and Concretization for the initial state for the example.  
Reachability after 1 iteration for the example.  
Predicate abstraction provides a powerful tool for verifying properties of infinite-state systems using a combination of a decision procedure for a subset of first-order logic and symbolic methods originally developed for finite-state model checking. We consider models containing first-order state variables, where the system state includes mutable functions and predicates. Such a model can describe systems containing arbitrarily large memories, buffers, and arrays of identical processes. We describe a form of predicate abstraction that constructs a formula over a set of universally quantified variables to describe invariant properties of the first-order state variables. We provide a formal justification of the soundness of our approach and describe how it has been used to verify several hardware and software designs, including a directory-based cache coherence protocol.
We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logic with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ).
We propose a type inference algorithm for lambda terms in elementary affine logic (EAL). The algorithm decorates the syntax tree of a simple typed lambda term and collects a set of linear constraints. The result is a parametric elementary type that can be instantiated with any solution of the set of collected constraints. We point out that the typeability of lambda terms in EAL has a practical counterpart, since it is possible to reduce any EAL-typeable lambda terms with the Lamping's abstract algorithm obtaining a substantial increase of performances. We show how to apply the same techniques to obtain decorations of intuitionistic proofs into linear logic proofs.
This paper is aimed at providing a uniform framework for reasoning about beliefs of multiple agents and their fusion. In the first part of the paper, we develop logics for reasoning about cautiously merged beliefs of agents with different degrees of reliability. The logics are obtained by combining the multi-agent epistemic logic and multi-sources reasoning systems. Every ordering for the reliability of the agents is represented by a modal operator, so we can reason with the merged results under different situations. The fusion is cautious in the sense that if an agent's belief is in conflict with those of higher priorities, then his belief is completely discarded from the merged result. We consider two strategies for the cautious merging of beliefs. In the first one, if inconsistency occurs at some level, then all beliefs at the lower levels are discarded simultaneously, so it is called level cutting strategy. For the second one, only the level at which the inconsistency occurs is skipped, so it is called level skipping strategy. The formal semantics and axiomatic systems for these two strategies are presented. In the second part, we extend the logics both syntactically and semantically to cover some more sophisticated belief fusion and revision operators. While most existing approaches treat belief fusion operators as meta-level constructs, these operators are directly incorporated into our object logic language. Thus it is possible to reason not only with the merged results but also about the fusion process in our logics. The relationship of our extended logics with the conditional logics of belief revision is also discussed.
Agents are small programs that autonomously take actions based on changes in their environment or ``state.'' Over the last few years, there have been an increasing number of efforts to build agents that can interact and/or collaborate with other agents. In one of these efforts, Eiter, Subrahmanian amd Pick (AIJ, 108(1-2), pages 179-255) have shown how agents may be built on top of legacy code. However, their framework assumes that agent states are completely determined, and there is no uncertainty in an agent's state. Thus, their framework allows an agent developer to specify how his agents will react when the agent is 100% sure about what is true/false in the world state. In this paper, we propose the concept of a \emph{probabilistic agent program} and show how, given an arbitrary program written in any imperative language, we may build a declarative ``probabilistic'' agent program on top of it which supports decision making in the presence of uncertainty. We provide two alternative semantics for probabilistic agent programs. We show that the second semantics, though more epistemically appealing, is more complex to compute. We provide sound and complete algorithms to compute the semantics of \emph{positive} agent programs.
Query equivalence is investigated for disjunctive aggregate queries with negated subgoals, constants and comparisons. A full characterization of equivalence is given for the aggregation functions count, max, sum, prod, toptwo and parity. A related problem is that of determining, for a given natural number N, whether two given queries are equivalent over all databases with at most N constants. We call this problem bounded equivalence. A complete characterization of decidability of bounded equivalence is given. In particular, it is shown that this problem is decidable for all the above aggregation functions as well as for count distinct and average. For quasilinear queries (i.e., queries where predicates that occur positively are not repeated) it is shown that equivalence can be decided in polynomial time for the aggregation functions count, max, sum, parity, prod, toptwo and average. A similar result holds for count distinct provided that a few additional conditions hold. The results are couched in terms of abstract characteristics of aggregation functions, and new proof techniques are used. Finally, the results above also imply that equivalence, under bag-set semantics, is decidable for non-aggregate queries with negation.
Answer set programming (ASP) is a logic programming paradigm that can be used to solve complex combinatorial search problems. Aggregates are an ASP construct that plays an important role in many applications. Defining a satisfactory semantics of aggregates turned out to be a difficult problem, and in this paper we propose a new approach, based on an analogy between aggregates and propositional connectives. First, we extend the definition of an answer set/stable model to cover arbitrary propositional theories; then we define aggregates on top of them both as primitive constructs and as abbreviations for formulas. Our definition of an aggregate combines expressiveness and simplicity, and it inherits many theorems about programs with nested expressions, such as theorems about strong equivalence and splitting.
Bellantoni and Cook have given a function-algebra characterization of the polynomial-time computable functions via an unbounded recursion scheme which is called safe recursion. Inspired by their work, we characterize the exponential-time computable functions with the use of a safe variant of nested recursion.
Sequential propositional logic deviates from ordinary propositional logic by taking into account that during the sequential evaluation of a propositional statement,atomic propositions may yield different Boolean values at repeated occurrences. We introduce `free valuations' to capture this dynamics of a propositional statement's environment. The resulting logic is phrased as an equationally specified algebra rather than in the form of proof rules, and is named `proposition algebra'. It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The four axioms for free valuation congruence are then combined with other axioms in order define a few more valuation congruences that gradually identify more propositional statements, up to static valuation congruence (which is the setting of conventional propositional logic). Proposition algebra is developed in a fashion similar to the process algebra ACP and the program algebra PGA, via an algebraic specification which has a meaningful initial algebra for which a range of coarser congruences are considered important as well. In addition infinite objects (that is propositional statements, processes and programs respectively) are dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts about infinite ones while reducing all facts about infinite objects to an infinity of facts about finite ones in return.
We introduce an algebra qCCS of pure quantum processes in which no classical data is involved, communications by moving quantum states physically are allowed, and computations is modeled by super-operators. An operational semantics of qCCS is presented in terms of (non-probabilistic) labeled transition systems. Strong bisimulation between processes modeled in qCCS is defined, and its fundamental algebraic properties are established, including uniqueness of the solutions of recursive equations. To model sequential computation in qCCS, a reduction relation between processes is defined. By combining reduction relation and strong bisimulation we introduce the notion of strong reduction-bisimulation, which is a device for observing interaction of computation and communication in quantum systems. Finally, a notion of strong approximate bisimulation (equivalently, strong bisimulation distance) and its reduction counterpart are introduced. It is proved that both approximate bisimilarity and approximate reduction-bisimilarity are preserved by various constructors of quantum processes. This provides us with a formal tool for observing robustness of quantum processes against inaccuracy in the implementation of its elementary gates.
We reconstruct Peleg's concurrent dynamic logic in the context of modal Kleene algebras. We explore the algebraic structure of its multirelational semantics and develop an abstract axiomatisation of concurrent dynamic algebras from that basis. In this axiomatisation, sequential composition is not associative. It interacts with concurrent composition through a weak distributivity law. The modal operators of concurrent dynamic algebra are obtained from abstract axioms for domain and antidomain operators; the Kleene star is modelled as a least fixpoint. Algebraic variants of Peleg's axioms are shown to be valid in these algebras and their soundness is proved relative to the multirelational model. Additional results include iteration principles for the Kleene star and a refutation of variants of Segerberg's axiom in the multirelational setting. The most important results have been verified formally with Isabelle/HOL.
It is well known that, under certain conditions, it is possible to split logic programs under stable model semantics, i.e. to divide such a program into a number of different "levels", such that the models of the entire program can be constructed by incrementally constructing models for each level. Similar results exist for other non-monotonic formalisms, such as auto-epistemic logic and default logic. In this work, we present a general, algebraicsplitting theory for logics with a fixpoint semantics. Together with the framework of approximation theory, a general fixpoint theory for arbitrary operators, this gives us a uniform and powerful way of deriving splitting results for each logic with a fixpoint semantics. We demonstrate the usefulness of these results, by generalizing existing results for logic programming, auto-epistemic logic and default logic.
Let D denote an infinite alphabet -- a set that consists of infinitely many symbols. A word w = a0b0a1b1 ⋯ anbn of even length over D can be viewed as a directed graph Gw whose vertices are the symbols that appear in w, and the edges are (a0, b0), (a1, b1), ..., (an, bn). For a positive integer m, define a language Rm such that a word w = a0b0 ⋯ anbn ∈ Rm if and only if there is a path in the graph Gw of length ≤ m from the vertex a0 to the vertex bn. We establish the following hierarchy theorem for pebble automata over infinite alphabet. For every positive integer k, (i) there exists a k-pebble automaton that accepts the language R2k − 1; (ii) there is no k-pebble automaton that accepts the language R2k + 1 − 2. Using this fact, we establish the following main results in this article: (a) a strict hierarchy of the pebble automata languages based on the number of pebbles; (b) the separation of monadic second order logic from the pebble automata languages; (c) the separation of one-way deterministic register automata languages from pebble automata languages.
A notion of alternating timed automata is proposed. It is shown that such automata with only one clock have decidable emptiness problem over finite words. This gives a new class of timed languages which is closed under boolean operations and which has an effective presentation. We prove that the complexity of the emptiness problem for alternating timed automata with one clock is non-primitive recursive. The proof gives also the same lower bound for the universality problem for nondeterministic timed automata with one clock. We investigate extension of the model with epsilon-transitions and prove that emptiness is undecidable. Over infinite words, we show undecidability of the universality problem.
A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. Safety one-way alternating automata with one register on infinite data words are considered, their nonemptiness is shown EXPSPACE-complete, and their inclusion decidable but not primitive recursive. The same complexity bounds are obtained for satisfiability and refinement, respectively, for the safety fragment of linear temporal logic with freeze quantification. Dropping the safety restriction, adding past temporal operators, or adding one more register, each causes undecidability. Comment: 23 pages
The final structure F for the formula ϑ = 1(p U q ∨ q) ∧ [[2]](♦p ∧ ¬q)
We develop a sound, complete and practically implementable tableaux-based decision method for constructive satisfiability testing and model synthesis in the fragment ATL+ of the full Alternating time temporal logic ATL star. The method extends in an essential way a previously developed tableaux-based decision method for ATL and works in 2EXPTIME, which is the optimal worst case complexity of the satisfiability problem for ATL+ . We also discuss how suitable parametrizations and syntactic restrictions on the class of input ATL+ formulae can reduce the complexity of the satisfiability problem.
A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element ("datum") from an infinite set, where the latter can only be compared for equality. The article considers alternating automata on data trees that can move downward and rightward, and have one register for storing data. The main results are that nonemptiness over finite data trees is decidable but not primitive recursive, and that nonemptiness of safety automata is decidable but not elementary. The proofs use nondeterministic tree automata with faulty counters. Allowing upward moves, leftward moves, or two registers, each causes undecidability. As corollaries, decidability is obtained for two data-sensitive fragments of the XPath query language. Comment: 23 pages
We present proof methods for the logics of nonmonotonic reasoning defined by Kraus, Lehmann and Magidor (KLM). We intro- duce tableaux calculi (called T ST) for all KLM logics. We provide deci- sion procedures for KLM logics based on these calculi, and we study their complexity. Finally, we describe KLMLean 2.0, a theorem prover imple- menting the calculi T ST inspired by the "lean" methodology. KLMLean 2.0 is implemented in SICStus Prolog and it also comprises a graphical interface written in Java1.
There are numerous extensions of RDF that support temporal reasoning, reasoning about pedigree, reasoning about uncertainty, and so on. In this paper, we present Annotated RDF (or aRDF for short) in which RDF triples are annotated by members of a partially ordered set (with bottom element) that can be selected in any way desired by the user. We present a formal declarative semantics (model theory) for annotated RDF and develop algorithms to check consistency of aRDF theories and to answer queries to aRDF theories. We show that annotated RDF captures versions of all the forms of reasoning mentioned above within a single unified framework. We develop a prototype aRDF implementation and show that our algorithms work very fast indeed – in fact, in just a matter of seconds for theories with over 100,000 nodes.
In recent research on non-monotonic logic programming, repeatedly strong equivalence of logic programs P and Q has been considered, which holds if the programs P union R and Q union R have the same answer sets for any other program R. This property strengthens equivalence of P and Q with respect to answer sets (which is the particular case for R is the empty set), and has its applications in program optimization, verification, and modular logic programming. In this paper, we consider more liberal notions of strong equivalence, in which the actual form of R may be syntactically restricted. On the one hand, we consider uniform equivalence, where R is a set of facts rather than a set of rules. This notion, which is well known in the area of deductive databases, is particularly useful for assessing whether programs P and Q are equivalent as components of a logic program which is modularly structured. On the other hand, we consider relativized notions of equivalence, where R ranges over rules over a fixed alphabet, and thus generalize our results to relativized notions of strong and uniform equivalence. For all these notions, we consider disjunctive logic programs in the propositional (ground) case, as well as some restricted classes, provide semantical characterizations and analyze the computational complexity. Our results, which naturally extend to answer set semantics for programs with strong negation, complement the results on strong equivalence of logic programs and pave the way for optimizations in answer set solvers as a tool for input-based problem solving.
Example Structure t ⇒ AFc 
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, for the first time, a characterization of an answer set semantics by μLGF formulas. We further extend the open answer set semantics for programs with generalized literals. Such generalized programs (gPs) have interesting properties, for example, the ability to express infinity axioms. We restrict the syntax of gPs such that both rules and generalized literals are guarded . Via a translation to guarded fixed-point logic, we deduce 2-EXPTIME-completeness of satisfiability checking in such guarded gPs (GgPs). Bound GgPs are restricted GgPs with EXPTIME-complete satisfiability checking, but still sufficiently expressive to optimally simulate computation tree logic (CTL). We translate Datalog lite programs to GgPs, establishing equivalence of GgPs under an open answer set semantics, alternation-free μGF, and Datalog LITE.
We consider logic-based argumentation in which an argument is a pair (Fi,al), where the support Fi is a minimal consistent set of formulae taken from a given knowledge base (usually denoted by De) that entails the claim al (a formula). We study the complexity of three central problems in argumentation: the existence of a support Fi ss De, the validity of a support and the relevance problem (given psi is there a support Fi such that psi ss Fi?). When arguments are given in the full language of propositional logic these problems are computationally costly tasks, the validity problem is DP-complete, the others are SigP2-complete. We study these problems in Schaefer's famous framework where the considered propositional formulae are in generalized conjunctive normal form. This means that formulae are conjunctions of constraints build upon a fixed finite set of Boolean relations Ga (the constraint language). We show that according to the properties of this language Ga, deciding whether there exists a support for a claim in a given knowledge base is either polynomial, NP-complete, coNP-complete or SigP2-complete. We present a dichotomous classification, P or DP-complete, for the verification problem and a trichotomous classification for the relevance problem into either polynomial, NP-complete, or SigP2-complete. These last two classifications are obtained by means of algebraic tools.
Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by this approach. In this paper, we prove an upper bound on the the number of states of the minimal deterministic automaton for a Presburger arithmetic formula. This bound depends on the length of the formula and the quantifiers occurring in the formula. The upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that our bound is tight, even for nondeterministic automata. Moreover, we provide optimal automata constructions for linear equations and inequations.
This paper gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers. Precisely, this means that for every first-order formula phi(y,z_1,...,z_k) over the signature {<,+} there is a first-order formula psi(x,z_1,...,z_k) which expresses over the structure <Nat,<,+> (respectively, over initial segments of this structure) that the variable x is interpreted exactly by the number of possible interpretations of the variable y for which the formula phi(y,z_1,...,z_k) is satisfied. Applying this theorem, we obtain an easy proof of Ruhl's result that reachability (and similarly, connectivity) in finite graphs is not expressible in first-order logic with unary counting quantifiers and addition. Furthermore, the above result on Presburger arithmetic helps to show the failure of a particular version of the Crane Beach conjecture.
Relevant clones and their inclusion structure; the shading indicates the complexity of EXP(B).  
Autoepistemic logic extends propositional logic by the modal operator L. A formula that is preceded by an L is said to be "believed". The logic was introduced by Moore 1985 for modeling an ideally rational agent's behavior and reasoning about his own beliefs. In this paper we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of counting the number of stable expansions of a given knowledge base. To the best of our knowledge this is the first paper analyzing the counting problem for autoepistemic logic.
The model D(2, 2), described in Definition 8.3. B(2, 2), as in Definition 8.1, is the submodel on the right-hand side of the dotted line and C(2) is the submodel on its left. Arrows indicate f D , while is the transitive, reflexive closure of the relation represented by together with the ellipse on the left, which represents ≈. Points represented by a circle satisfy p 1 , by a triangle, p 2 .
The relation ∼ 1 on B(2, 2). Points inside the large lower-left triangle are ∼ 1-related if and only if they satisfy the same propositional variable, which in this figure occurs when they have the same shape. This triangle would grow if we were to consider ∼ 0 , or shrink if we were to consider ∼ 2 .
The model D(1, 3). Notation is similar to that for Figure 2, but this time we have only partially indicated the relations; points marked with a square now satisfy p 3 .
Dynamic topological logic (DTL) is a polymodal logic designed for reasoning about dynamic topological systems. These are pairs 〈 X, f〉, where X is a topological space and f:X → X is continuous. DTL uses a language L which combines the topological S4 modality &square; with temporal operators from linear temporal logic. Recently, we gave a sound and complete axiomatization DTL&ast; for an extension of the logic to the language L&ast;, where ◊ is allowed to act on finite sets of formulas and is interpreted as a tangled closure operator. No complete axiomatization is known in the language L, although one proof system, which we shall call KM, was conjectured to be complete by Kremer and Mints. In this article, we show that given any language L' such that L ⊆ L' ⊆ L&ast;, the set of valid formulas of L' is not finitely axiomatizable. It follows, in particular, that KM is incomplete.
Coalgebras provide a uniform framework to study dynamical systems, including several types of automata. In this paper, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalised powerset construction that determinises coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor $FT$, where $T$ is a monad describing the branching of the systems (e.g. non-determinism, weights, probability etc.), has as a quotient the rational fixpoint of the "determinised" type functor $\bar F$, a lifting of $F$ to the category of $T$-algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain non-deterministic automata, where we recover Rabinovich's sound and complete calculus for language equivalence.
We show that constant-depth Frege systems with counting axioms modulo $m$ polynomially simulate Nullstellensatz refutations modulo $m$. Central to this is a new definition of reducibility from formulas to systems of polynomials with the property that, for most previously studied translations of formulas to systems of polynomials, a formula reduces to its translation. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new, small refutations for certain CNFs by constant-depth Frege systems with counting axioms.
Over the last two decades, propositional satisfiability (SAT) has become one of the most successful and widely applied techniques for the solution of NP-complete problems. The aim of this paper is to investigate theoretically how Sat can be utilized for the efficient solution of problems that are harder than NP or co-NP. In particular, we consider the fundamental reasoning problems in propositional disjunctive answer set programming (ASP), Brave Reasoning and Skeptical Reasoning, which ask whether a given atom is contained in at least one or in all answer sets, respectively. Both problems are located at the second level of the Polynomial Hierarchy and thus assumed to be harder than NP or co-NP. One cannot transform these two reasoning problems into SAT in polynomial time, unless the Polynomial Hierarchy collapses. We show that certain structural aspects of disjunctive logic programs can be utilized to break through this complexity barrier, using new techniques from Parameterized Complexity. In particular, we exhibit transformations from Brave and Skeptical Reasoning to SAT that run in time O(2^k n^2) where k is a structural parameter of the instance and n the input size. In other words, the reduction is fixed-parameter tractable for parameter k. As the parameter k we take the size of a smallest backdoor with respect to the class of normal (i.e., disjunction-free) programs. Such a backdoor is a set of atoms that when deleted makes the program normal. In consequence, the combinatorial explosion, which is expected when transforming a problem from the second level of the Polynomial Hierarchy to the first level, can now be confined to the parameter k, while the running time of the reduction is polynomial in the input size n, where the order of the polynomial is independent of k.
Using the left merge and communication merge from ACP, we present an equational base (i.e., a ground-complete and $\omega$-complete set of valid equations) for the fragment of CCS without recursion, restriction and relabelling. Our equational base is finite if the set of actions is finite.
Hybrid MKNF knowledge bases are one of the most prominent tightly integrated combinations of open-world ontology languages with closed-world (nonmonotonic) rule paradigms. Based on the logic of minimal knowledge and negation as failure (MKNF), the definition of Hybrid MKNF is parametric on the description logic (DL) underlying the ontology language, in the sense that nonmonotonic rules can extend any decidable DL language. Two related semantics have been defined for Hybrid MKNF: one that is based on the Stable Model Semantics for logic programs and one on the Well-Founded Semantics (WFS). Under WFS, the definition of Hybrid MKNF relies on a bottom-up computation that has polynomial data complexity whenever the DL language is tractable. Here we define a general query-driven procedure for Hybrid MKNF that is sound with respect to the stable model-based semantics, and sound and complete with respect to its WFS variant. This procedure is able to answer a slightly restricted form of conjunctive queries, and is based on tabled rule evaluation extended with an external oracle that captures reasoning within the ontology. Such an (abstract) oracle receives as input a query along with knowledge already derived, and replies with a (possibly empty) set of atoms, defined in the rules, whose truth would suffice to prove the initial query. With appropriate assumptions on the complexity of the abstract oracle, the general procedure maintains the data complexity of the WFS for Hybrid MKNF knowledge bases. To illustrate this approach, we provide a concrete oracle for EL⁺, a fragment of the lightweight DL EL⁺⁺. Such an oracle has practical use, as EL⁺⁺ is the language underlying OWL 2 EL, which is part of the W3C recommendations for the Semantic Web, and is tractable for reasoning tasks such as subsumption. We show that query-driven Hybrid MKNF preserves polynomial data complexity when using the EL⁺ oracle and WFS.
We generalize the QSQR evaluation method to give a set-oriented depth-first evaluation method for Horn knowledge bases. The resulting procedure closely simulates SLD-resolution (to take advantages of the goal-directed approach) and highly exploits set-at-a-time tabling. Our generalized QSQR evaluation procedure is sound, complete, and tight. It does not use adornments and annotations. To deal with function symbols, our procedure uses iterative deepening search which iteratively increases term depth bound for atoms occurring in the computation. When the term depth bound is fixed, our evaluation procedure runs in polynomial time in the size of extensional relations.
First-order temporal logics are notorious for their bad computational behavior. It is known that even the two-variable monadic fragment is highly undecidable over various linear timelines, and over branching time even one-variable fragments might be undecidable. However, there have been several attempts at finding well-behaved fragments of first-order temporal logics and related temporal description logics, mostly either by restricting the available quantifier patterns or by considering sub-Boolean languages. Here we analyze seemingly “mild” extensions of decidable one-variable fragments with counting capabilities, interpreted in models with constant, decreasing, and expanding first-order domains. We show that over most classes of linear orders, these logics are (sometimes highly) undecidable, even without constant and function symbols, and with the sole temporal operator “eventually.” We establish connections with bimodal logics over 2D product structures having linear and “difference” (inequality) component relations and prove our results in this bimodal setting. We show a general result saying that satisfiability over many classes of bimodal models with commuting “unbounded” linear and difference relations is undecidable. As a byproduct, we also obtain new examples of finitely axiomatizable but Kripke incomplete bimodal logics. Our results generalize similar lower bounds on bimodal logics over products of two linear relations, and our proof methods are quite different from the known proofs of these results. Unlike previous proofs that first “diagonally encode” an infinite grid and then use reductions of tiling or Turing machine problems, here we make direct use of the grid-like structure of product frames and obtain lower-complexity bounds by reductions of counter (Minsky) machine problems. Representing counter machine runs apparently requires less control over neighboring grid points than tilings or Turing machine runs, and so this technique is possibly more versatile, even if one component of the underlying product structures is “close to” being the universal relation.
With the previous notions of bisimulation presented in literature, to check if two quantum processes are bisimilar, we have to instantiate the free quantum variables of them with arbitrary quantum states, and verify the bisimilarity of resultant configurations. This makes checking bisimilarity infeasible from an algorithmic point of view because quantum states constitute a continuum. In this paper, we introduce a symbolic operational semantics for quantum processes directly at the quantum operation level, which allows us to describe the bisimulation between quantum processes without resorting to quantum states. We show that the symbolic bisimulation defined here is equivalent to the open bisimulation for quantum processes in the previous work, when strong bisimulations are considered. An algorithm for checking symbolic ground bisimilarity is presented. We also give a modal logical characterisation for quantum bisimilarity based on an extension of Hennessy-Milner logic to quantum processes.
We specify the operational semantics and bisimulation relations for the finite pi-calculus within a logic that contains the nabla quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The nabla quantifier helps with the delicate issues surrounding the scope of variables within pi-calculus expressions and their executions (proofs). We shall illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no side-conditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arises from familar logic distinctions; the interplay between the three quantifiers (for all, exists, and nabla) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for one-step transitions, bisimulation, and satisfaction in modal logic.
Cryptographic protocols with single blind copying were defined and modeled by Comon and Cortier using the new class $\mathcal C$ of first order clauses. They showed its satisfiability problem to be in 3-DEXPTIME. We improve this result by showing that satisfiability for this class is NEXPTIME-complete, using new resolution techniques. We show satisfiability to be DEXPTIME-complete if clauses are Horn, which is what is required for modeling cryptographic protocols. While translation to Horn clauses only gives a DEXPTIME upper bound for the secrecy problem for these protocols, we further show that this secrecy problem is actually DEXPTIME-complete.
We analyse the problem of solving Boolean equation systems through the use of structure graphs. The latter are obtained through an elegant set of Plotkin-style deduction rules. Our main contribution is that we show that equation systems with bisimilar structure graphs have the same solution. We show that our work conservatively extends earlier work, conducted by Keiren and Willemse, in which dependency graphs were used to analyse a subclass of Boolean equation systems, viz., equation systems in standard recursive form. We illustrate our approach by a small example, demonstrating the effect of simplifying an equation system through minimisation of its structure graph.
Proof-search (the basis of logic programming) with multiplicative inference rules, such as linear logic's R and '8L, is problematic because of the required non-deterministic splitting of resources. Similarly, searching with additive rules such as &L and R requires a non-deterministic choice between two formulae. Many strategies which resolve such non-determinism, either locally or globally, are available. We present a characterization of a range of strategies for distributing and selecting resources in linear sequent calculus proof-search via a sequent calculus annotated with Boolean constraints. Strategies are characterized by calculations of solutions of sets of Boolean equations generated by searches. Our characterization encompasses lazy (or local), eager (or global) and intermediate (mixed local and global) strategies.
A relational structure is d -degree-bounded, for some integer d , if each element of the domain belongs to at most d tuples. In this paper, we revisit the complexity of the evaluation problem of not necessarily Boolean first-order ( FO ) queries over d -degree-bounded structures. Query evaluation is considered here as a dynamical process. We prove that any FO query on d -degree-bounded structures belongs to the complexity class constant-Delay lin , that is, can be computed by an algorithm that has two separate parts: it has a precomputation step of time linear in the size of the structure and then, it outputs all solutions (i.e., tuples that satisfy the formula) one by one with a constant delay (i.e., depending on the size of the formula only) between each. Seen as a global process, this implies that queries on d -degree-bounded structures can be evaluated in total time f (|φ|).(| S | + |φ( S )|) and space g (|φ|).| S | where S is the structure, φ is the formula, φ( S ) is the result of the query and f , g are some fixed functions. Among other things, our results generalize a result of Seese on the data complexity of the model-checking problem for d -degree-bounded structures. Besides, the originality of our approach compared to related results is that it does not rely on the Hanf's model-theoretic technique and is simple and informative since it essentially rests on a quantifier elimination method.
Tree decomposition T of schema (R, F ) in Example 2.1 
Induced substructures T ′ s and ¯ T ′ s of the tree decomposition T w.r.t. the node s. 
Bounded treewidth and Monadic Second Order (MSO) logic have proved to be key concepts in establishing fixed-parameter tractability results. Indeed, by Courcelle's Theorem we know: Any property of finite structures, which is expressible by an MSO sentence, can be decided in linear time (data complexity) if the structures have bounded treewidth. In principle, Courcelle's Theorem can be applied directly to construct concrete algorithms by transforming the MSO evaluation problem into a tree language recognition problem. The latter can then be solved via a finite tree automaton (FTA). However, this approach has turned out to be problematical, since even relatively simple MSO formulae may lead to a ``state explosion'' of the FTA. In this work we propose monadic datalog (i.e., datalog where all intentional predicate symbols are unary) as an alternative method to tackle this class of fixed-parameter tractable problems. We show that if some property of finite structures is expressible in MSO then this property can also be expressed by means of a monadic datalog program over the structure plus the tree decomposition. Moreover, we show that the resulting fragment of datalog can be evaluated in linear time (both w.r.t. the program size and w.r.t. the data size). This new approach is put to work by devising new algorithms for the 3-Colorability problem of graphs and for the PRIMALITY problem of relational schemas (i.e., testing if some attribute in a relational schema is part of a key). We also report on experimental results with a prototype implementation.
Given two structures $G$ and $H$ distinguishable in $\fo k$ (first-order logic with $k$ variables), let $A^k(G,H)$ denote the minimum alternation depth of a $\fo k$ formula distinguishing $G$ from $H$. Let $A^k(n)$ be the maximum value of $A^k(G,H)$ over $n$-element structures. We prove the strictness of the quantifier alternation hierarchy of $\fo 2$ in a strong quantitative form, namely $A^2(n)\ge n/8-2$, which is tight up to a constant factor. For each $k\ge2$, it holds that $A^k(n)>\log_{k+1}n-2$ even over colored trees, which is also tight up to a constant factor if $k\ge3$. For $k\ge 3$ the last lower bound holds also over uncolored trees, while the alternation hierarchy of $\fo 2$ collapses even over all uncolored graphs. We also show examples of colored graphs $G$ and $H$ on $n$ vertices that can be distinguished in $\fo 2$ much more succinctly if the alternation number is increased just by one: while in $\Sigma_{i}$ it is possible to distinguish $G$ from $H$ with bounded quantifier depth, in $\Pi_{i}$ this requires quantifier depth $\Omega(n^2)$. The quadratic lower bound is best possible here because, if $G$ and $H$ can be distinguished in $\fo k$ with $i$ quantifier alternations, this can be done with quantifier depth $n^{2k-2}$.
We study the complexity of reachability problems on branching extensions of vector addition systems, which allows us to derive new non-elementary complexity bounds for fragments and variants of propositional linear logic. We show that provability in the multiplicative exponential fragment is Tower-hard already in the affine case---and hence non-elementary. We match this lower bound for the full propositional affine linear logic, proving its Tower-completeness. We also show that provability in propositional contractive linear logic is Ackermann-complete.
A concrete Kripke structure V and its existential abstraction Vα.
There is a growing interest in techniques for detecting whether a logic specification is satisfied too easily, or vacuously. For example, the specification “every request is eventually followed by an acknowledgment” is satisfied vacuously by a system that never generates any requests. Vacuous satisfaction misleads users of model-checking into thinking that a system is correct. It is a serious problem in practice. There are several existing definitions of vacuity. Originally, Beer et al. [1997] formalized vacuity as insensitivity to syntactic perturbation (syntactic vacuity). This formulation captures the intuition of “vacuity” when applied to a single occurrence of a subformula. Armoni et al. argued that vacuity must be robust; not affected by semantically invariant changes, such as extending a model with additional atomic propositions. They show that syntactic vacuity is not robust for subformulas of linear temporal logic, and propose an alternative definition; trace vacuity. In this article, we continue this line of research. We show that trace vacuity is not robust for branching time logic. We further refine the notion of vacuity so that it applies uniformly to linear and branching time logic and does not suffer from the common pitfalls of prior definitions. Our new definition, bisimulation vacuity, is a proper and nontrivial extension of both syntactic and trace vacuity. We discuss the complexity of detecting bisimulation vacuity, and identify several practically-relevant subsets of CTL* for which vacuity detection problem is reducible to model-checking. We believe that in most practical applications, bisimulation vacuity provides both the desired theoretical properties and is tractable computationally.
A fragment of second-order lambda calculus (System F) is defined that characterizes the elementary recursive functions. Type quantification is restricted to be non-interleaved and stratified, i.e., the types are assigned levels, and a quantified variable can only be instantiated by a type of smaller level, with a slightly liberalized treatment of the level zero.
Symbolic event recognition systems have been successfully applied to a variety of application domains, extracting useful information in the form of events, allowing experts or other systems to monitor and respond when significant events are recognised. In a typical event recognition application, however, these systems often have to deal with a significant amount of uncertainty. In this paper, we address the issue of uncertainty in logic-based event recognition by extending the Event Calculus with probabilistic reasoning. Markov Logic Networks are a natural candidate for our logic-based formalism. However, the temporal semantics of the Event Calculus introduce a number of challenges for the proposed model. We show how and under what assumptions we can overcome these problems. Additionally, we study how probabilistic modelling changes the behaviour of the formalism, affecting its key property, the inertia of fluents. Furthermore, we demonstrate the advantages of the probabilistic Event Calculus through examples and experiments in the domain of activity recognition, using a publicly available dataset for video surveillance.
Many problems in robust control and motion planning can be reduced to either find a sound approximation of the solution space determined by a set of nonlinear inequalities, or to the ``guaranteed tuning problem'' as defined by Jaulin and Walter, which amounts to finding a value for some tuning parameter such that a set of inequalities be verified for all the possible values of some perturbation vector. A classical approach to solve these problems, which satisfies the strong soundness requirement, involves some quantifier elimination procedure such as Collins' Cylindrical Algebraic Decomposition symbolic method. Sound numerical methods using interval arithmetic and local consistency enforcement to prune the search space are presented in this paper as much faster alternatives for both soundly solving systems of nonlinear inequalities, and addressing the guaranteed tuning problem whenever the perturbation vector has dimension one. The use of these methods in camera control is investigated, and experiments with the prototype of a declarative modeller to express camera motion using a cinematic language are reported and commented.
Well-known theorems of Hanf’s and Gaifman’s establishing locality of first-order definable properties have been used in many applications. These theorems were recently generalized to other logics, which led to new applications in descriptive complexity and database theory. However, a logical characterization of local properties that correspond to Hanf’s and Gaifman’s theorems, is still lacking. Such a characterization only exists for structures of bounded valence. In this paper, we give logical characterizations of local properties behind Hanf’s and Gaifman’s theorems. We first deal with an infinitary logic with counting terms and quantifiers, that is known to capture Hanf-locality on structures of bounded valence. We show that testing isomorphism of neighborhoods can be added to it without violating Hanf-locality, while increasing its expressive power. We then show that adding local second-order quantification to it captures precisely all Hanf-local properties. To capture Gaifman-locality, one must also add a (potentially infinite) case statement. We further show that the hierarchy based on the number of variants in the case statement is strict.
Top-cited authors
Thomas Eiter
Georg Gottlob
  • University of Oxford
Nicola Leone
  • Università della Calabria
Gerald Pfeifer
Wolfgang Faber
  • Alpen-Adria-Universität Klagenfurt