Quantitative separation logic (QSL) is an extension of separation logic (SL) for the verification of probabilistic pointer programs. In QSL, formulae evaluate to real numbers instead of truth values, e.g., the probability of memory-safe termination in a given symbolic heap. As with \SL, one of the key problems when reasoning with QSL is \emph{entailment}: does a formula f entail another formula g?
We give a generic reduction from entailment checking in QSL to entailment checking in SL. This allows to leverage the large body of SL research for the automated verification of probabilistic pointer programs. We analyze the complexity of our approach and demonstrate its applicability. In particular, we obtain the first decidability results for the verification of such programs by applying our reduction to a quantitative extension of the well-known symbolic-heap fragment of separation logic.
The entailment between separation logic formulae with inductive predicates, also known as symbolic heaps, has been shown to be decidable for a large class of inductive definitions. Recently, a 2-EXPTIME algorithm was proposed and an EXPTIME-hard bound was established; however no precise lower bound is known. In this paper, we show that deciding entailment between predicate atoms is 2-EXPTIME-hard. The proof is based on a reduction from the membership problem for exponential-space bounded alternating Turing machines.
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 above 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-ex…
We present a sequent-based deductive system for automatically proving entailments in separation logic by using mathematical induction. Our technique, called mutual explicit induction proof, is an instance of Noetherian induction. Specifically, we propose a novel induction principle on a well-founded relation of separation logic model and follow the explicit induction methods to implement this principle as inference rules, so that it can be easily integrated into a deductive system. We also support mutual induction, a natural feature of implicit induction, where the goal entailment and other en…
This paper presents an approach to lemma synthesis to support advanced inductive entailment procedures based on separation logic. We first propose a mechanism where lemmas are automatically proven and systematically applied. The lemmas may include universal guard and/or unknown predicate. While the former is critical for expressivity, the latter is essential for supporting relationships between multiple predicates. We further introduce lemma synthesis to support (i) automated inductive reasoning together with frame inference and (ii) theorem exploration. For (i) we automatically discover and p…
The symbolic-heap fragment of separation logic has been actively developed and advocated for verifying the memory-safety property of computer programs. At present, one of its biggest challenges is to effectively prove entailments containing inductive heap predicates. These entailments are usually proof obligations generated when verifying programs that manipulate complex data structures like linked lists, trees, or graphs. To assist in proving such entailments, this paper introduces a lemma synthesis framework, which automatically discovers lemmas to serve as eureka steps in the proofs. Mathem…
ISTI - CNR（意大利国家研究委员会信息科学与技术研究所）；CAIR - University of Cape Town（开普敦大学CAIR）
Lehmann and Magidor's rational closure is acknowledged as a landmark in the field of non-monotonic logics and it has also been re-formulated in the context of Description Logics (DLs). We show here how to model a rational form of entailment for expressive DLs, such as SROIQ, providing a novel reasoning procedure that compiles a non-monotone DL knowledge base into a description logic program (dl-program).
Separation Logic (SL) with inductive definitions is a natural formalism for specifying complex recursive data structures, used in compositional verification of programs manipulating such structures. The key ingredient of any automated verification procedure based on SL is the decidability of the entailment problem. In this work, we reduce the entailment problem for a non-trivial subset of SL describing trees (and beyond) to the language inclusion of tree automata (TA). Our reduction provides tight complexity bounds for the problem and shows that entailment in our fragment is EXPTIME-complete. …
In this paper we consider the class of truth-functional many-valued logics with a finite set of truth-values. The main result of this paper is the development of a new \emph{binary} sequent calculi (each sequent is a pair of formulae) for many valued logic with a finite set of truth values, and of Kripke-like semantics for it that is both sound and complete. We did not use the logic entailment based on matrix with a strict subset of designated truth values, but a different new kind of semantics based on the generalization of the classic 2-valued truth-invariance entailment. In order to define …
The entailment problem $φ\models ψ$ in Separation Logic \cite{IshtiaqOHearn01,Reynolds02}, between separated conjunctions of equational ($x \iseq y$ and $x \not\iseq y$), spatial ($x \mapsto (y_1,\ldots,y_\rank)$) and predicate ($p(x_1,\ldots,x_n)$) atoms, interpreted by a finite set of inductive rules, is undecidable in general. Certain restrictions on the set of inductive definitions lead to decidable classes of entailment problems. Currently, there are two such decidable classes, based on two restrictions, called \emph{establishment} \cite{IosifRogalewiczSimacek13,KatelaanMathejaZuleger19,P…
Technische Universität Dresden（德累斯顿工业大学）；University of Wrocław（弗罗茨瓦夫大学）
We present a unified approach to (both finite and unrestricted) worst-case optimal entailment of (unions of) conjunctive queries (U)CQs in the wide class of "locally-forward" description logics. The main technique that we employ is a generalisation of Lutz's spoiler technique, originally developed for CQ entailment in ALCHQ. Our result closes numerous gaps present in the literature, most notably implying ExpTime-completeness of (U)CQ-querying for any superlogic of ALC contained in ALCHbregQ, and, as we believe, is abstract enough to be employed as a black-box in many new scenarios.
We define a class of Separation Logic formulae, whose entailment problem: given formulae $ϕ, ψ_1, \ldots, ψ_n$, is every model of $ϕ$ a model of some $ψ_i$? is 2EXPTIME-complete. The formulae in this class are existentially quantified separating conjunctions involving predicate atoms, interpreted by the least sets of store-heap structures that satisfy a set of inductive rules, which is also part of the input to the entailment problem. Previous work consider established sets of rules, meaning that every existentially quantified variable in a rule must eventually be bound to an allocated locatio…
An efficient entailment proof system is essential to compositional verification using separation logic. Unfortunately, existing decision procedures are either inexpressive or inefficient. For example, Smallfoot is an efficient procedure but only works with hardwired lists and trees. Other procedures that can support general inductive predicates run exponentially in time as their proof search requires back-tracking to deal with a disjunction in the consequent. This paper presents a decision procedure to derive cyclic entailment proofs for general inductive predicates in polynomial time. Our pro…
The Resource Description Framework (RDF) is a Semantic Web standard that provides a data language, simply called RDF, as well as a lightweight ontology language, called RDF Schema. We investigate embeddings of RDF in logic and show how standard logic programming and description logic technology can be used for reasoning with RDF. We subsequently consider extensions of RDF with datatype support, considering D entailment, defined in the RDF semantics specification, and D* entailment, a semantic weakening of D entailment, introduced by ter Horst. We use the embeddings and properties of the logics…
Two results are presented concerning the entailment problem in Separation Logic with inductively defined predicate symbols and theory reasoning. First, we show that the entailment problem is undecidable for rules with bounded tree-width, if theory reasoning is considered. The result holds for a wide class of theories, even with a very low expressive power. For instance it applies to the natural numbers with the successor function, or with the usual order. Second, we show that every entailment problem can be reduced to an entailment problem containing no equality (neither in the formulas nor in…
We prove the decidability of Ticket Entailment. Raised by Anderson and Belnap within the framework of relevance logic, this question is equivalent to the question of the decidability of type inhabitation in simply-typed combinatory logic with the partial basis BB'IW. We solve the equivalent problem of type inhabitation for the restriction of simply-typed lambda-calculus to hereditarily right-maximal terms.
We extend the epistemic logic with De Morgan negation by Fagin et al. (Artif. Intell. 79, 203-240, 1995) by adding operators for universal and common knowledge in a group of agents, and with a formalization of information update using a generalized version of the left division connective of the non-associative Lambek calculus. We provide sound and complete axiomatizations of the basic logic with the group operators and the basic logic with group operators and updates. Both logics are shown to be decidable.
Cardiff University（卡迪夫大学）；University of Luxembourg（卢森堡大学）；University of Cape Town（开普敦大学）；CAIR（南非人工智能研究中心）；Univ. Artois（阿尔图瓦大学）；CNRS（法国国家科学研究中心）
Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator capturing the most typical (alias normal or conventional) situations in which a given sentence holds. The semantics of PTL is in terms of ranked models as studied in the well-known KLM approach to preferential reasoning and therefore KLM-style rational consequence relations can be embedded in PTL. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for P…
Towards Non-Monotonic Entailment in Propositional Defeasible Standpoint Logic
University of Cape Town and CAIR, South Africa（开普敦大学和CAIR，南非）；Université Sorbonne Paris Nord, Inserm, Sorbonne Université, Limics, 93017 Bobigny, France（巴黎-索邦大学，Inserm，索邦大学，Limics，法国93017博比尼）；ISTI-CNR, Pisa, Italy（意大利比萨ISTI-CNR）
We present a logic for reasoning with if-then formulas which involve constants for rational truth degrees from the unit interval. We introduce graded semantic and syntactic entailment of formulas. We prove the logic is complete in Pavelka style and depending on the choice of structure of truth degrees, the logic is a decidable fragment of the Rational Pavelka logic (RPL) or the Rational Product Logic (RΠL). We also present a characterization of the entailment based on least models and study related closure structures.
Universität zu Köln（科隆大学）；Fraunhofer-Institut FKIE（弗劳恩霍夫FKIE研究所）
We present the architecture and the evaluation of a new system for recognizing textual entailment (RTE). In RTE we want to identify automatically the type of a logical relation between two input texts. In particular, we are interested in proving the existence of an entailment between them. We conceive our system as a modular environment allowing for a high-coverage syntactic and semantic text analysis combined with logical inference. For the syntactic and semantic analysis we combine a deep semantic analysis with a shallow one supported by statistical models in order to increase the quality an…
We study logic for reasoning with if-then formulas describing dependencies between attributes of objects which are observed in consecutive points in time. We introduce semantic entailment of the formulas, show its fixed-point characterization, investigate closure properties of model classes, present an axiomatization and prove its completeness, and investigate alternative axiomatizations and normalized proofs. We investigate decidability and complexity issues of the logic and prove that the entailment problem is NP-hard and belongs to EXPSPACE. We show that by restricting to predictive formula…
We investigate array separation logic (ASL), a variant of symbolic-heap separation logic in which the data structures are either pointers or arrays, i.e., contiguous blocks of allocated memory. This logic provides a language for compositional memory safety proofs of imperative array programs. We focus on the biabduction problem for this logic, which has been established as the key to automatic specification inference at the industrial scale. We present an NP decision procedure for biabduction in ASL that produces solutions of reasonable quality, and we also show that the problem of finding a c…
Explanations for description logic (DL) entailments provide important support for the maintenance of large ontologies. The "justifications" usually employed for this purpose in ontology editors pinpoint the parts of the ontology responsible for a given entailment. Proofs for entailments make the intermediate reasoning steps explicit, and thus explain how a consequence can actually be derived. We present an interactive system for exploring description logic proofs, called Evonne, which visualizes proofs of consequences for ontologies written in expressive DLs. We describe the methods used for c…
In the present work I introduce a semantics based on the cognitive attitudes of acception and rejection entertained by a given society of agents for logics inspired on Dunn and Belnap's First Degree Entailment ($\mathbf{E}$). In contrast to the epistemic situations originally employed by $\mathbf{E}$, the cognitive attitudes do not coincide with truth-values and they seem more suitable to logics that intend to consider the informational content of propositions "said to be true" as well as of propositions "said to be false" as determinant of the notion of logical validity. After analyzing some …
University of Cape Town（开普敦大学）；CAIR, South Africa（南非人工智能研究中心）；Technische Universität Dresden（德累斯顿工业大学）；ScaDS.AI – Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig, Germany（德国德累斯顿/莱比锡可扩展数据分析与人工智能中心）
Logic-based approaches to AI have the advantage that their behaviour can in principle be explained by providing their users with proofs for the derived consequences. However, if such proofs get very large, then it may be hard to understand a consequence even if the individual derivation steps are easy to comprehend. This motivates our interest in finding small proofs for Description Logic (DL) entailments. Instead of concentrating on a specific DL and proof calculus for this DL, we introduce a general framework in which proofs are represented as labeled, directed hypergraphs, where each hypere…
We investigate the complexity consequences of adding pointer arithmetic to separation logic. Specifically, we study extensions of the points-to fragment of symbolic-heap separation logic with various forms of Presburger arithmetic constraints. Most significantly, we find that, even in the minimal case when we allow only conjunctions of simple "difference constraints" (x'\leq x+k) where k is an integer, polynomial-time decidability is already impossible: satisfiability becomes NP-complete, while quantifier-free entailment becomes coNP-complete and quantified entailment becomes P2-complete (P2 i…
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Oxford University Computing Laboratory（牛津大学计算实验室）
Description Logics are knowledge representation formalisms that provide, for example, the logical underpinning of the W3C OWL standards. Conjunctive queries, the standard query language in databases, have recently gained significant attention as an expressive formalism for querying Description Logic knowledge bases. Several different techniques for deciding conjunctive query entailment are available for a wide range of DLs. Nevertheless, the combination of nominals, inverse roles, and number restrictions in OWL 1 and OWL 2 DL causes unsolvable problems for the techniques hitherto available. We…
We give the first ExpTime (complexity-optimal) tableau decision procedure for checking satisfiability of a knowledge base in the description logic SHIQ when numbers are coded in unary. Our procedure is based on global state caching and integer linear feasibility checking.
Technische Universität Dresden（德累斯顿工业大学）；University of Wrocław（弗罗茨瓦夫大学）
In logic-based knowledge representation, query answering has essentially replaced mere satisfiability checking as the inferencing problem of primary interest. For knowledge bases in the basic description logic ALC, the computational complexity of conjunctive query (CQ) answering is well known to be ExpTime-complete and hence not harder than satisfiability. This does not change when the logic is extended by certain features (such as counting or role hierarchies), whereas adding others (inverses, nominals or transitivity together with role-hierarchies) turns CQ answering exponentially harder. We…
Logic-based approaches to AI have the advantage that their behavior can in principle be explained to a user. If, for instance, a Description Logic reasoner derives a consequence that triggers some action of the overall system, then one can explain such an entailment by presenting a proof of the consequence in an appropriate calculus. How comprehensible such a proof is depends not only on the employed calculus, but also on the properties of the particular proof, such as its overall size, its depth, the complexity of the employed sentences and proof steps, etc. For this reason, we want to determ…
We study generalized possibilistic computation tree logic model checking in this paper, which is an extension of possibilistic computation logic model checking introduced by Y.Li, Y.Li and Z.Ma (2014). The system is modeled by generalized possibilistic Kripke structures (GPKS, in short), and the verifying property is specified by a generalized possibilistic computation tree logic (GPoCTL, in short) formula. Based on generalized possibility measures and generalized necessity measures, the method of generalized possibilistic computation tree logic model checking is discussed, and the correspondi…
The molecular computing has been successfully employed to solve more and more complex computation problems. However, as an important complex problem, the model checking are still far from fully resolved under the circumstance of molecular computing, since it is still a lack of method. To address this issue, a model checking method is presented for checking the basic constructs in a given temporal logic using molecular computing. Through the design of the new encoding and calling this process, we get a molecule-based approach for checking all of the basic constructs of this logic.
We propose a graph-based extension of Boolean logic called Boolean Graph Logic (BGL). Construing formula trees as the cotrees of cographs, we may state semantic notions such as evaluation and entailment in purely graph-theoretic terms, whence we recover the definition of BGL. Naturally, it is conservative over usual Boolean logic. Our contributions are the following: (1) We give a natural semantics of BGL based on Boolean relations, i.e. it is a multivalued semantics, and show adequacy of this semantics for the corresponding notions of entailment. (2) We show that the complexity of evaluation …
Path checking, the special case of the model checking problem where the model under consideration is a single path, plays an important role in monitoring, testing, and verification. We prove that for linear-time temporal logic (LTL), path checking can be efficiently parallelized. In addition to the core logic, we consider the extensions of LTL with bounded-future (BLTL) and past-time (LTL+Past) operators. Even though both extensions improve the succinctness of the logic exponentially, path checking remains efficiently parallelizable: Our algorithm for LTL, LTL+Past, and BLTL+Past is in AC^1(lo…
Propositional and modal inclusion logic are formalisms that belong to the family of logics based on team semantics. This article investigates the model checking and validity problems of these logics. We identify complexity bounds for both problems, covering both lax and strict team semantics. By doing so, we come close to finalising the programme that ultimately aims to classify the complexities of the central reasoning problems for modal and propositional dependence, independence, and inclusion logics.
In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion s…
We study the computational complexity of model checking and satisfiability problems of polyadic modal logics extended with permutations and Boolean operators on accessibility relations. First, we show that the combined complexity of the model checking problem for the resulting logic is PTime-complete. Secondly, we show that the satisfiability problem of polyadic modal logic extended with negation on accessibility relations is ExpTime-complete. Finally, we show that the satisfiability problem of polyadic modal logic with permutations and Boolean operators on accessibility relations is ExpTime-c…
In this paper we provide an alternative semantics for Equilibrium Logic and its monotonic basis, the logic of Here-and-There (also known as Gödel's G3 logic) that relies on the idea of "denotation" of a formula, that is, a function that collects the set of models of that formula. Using the three-valued logic G3 as a starting point and an ordering relation (for which equilibrium/stable models are minimal elements) we provide several elementary operations for sets of interpretations. By analysing structural properties of the denotation of formulas, we show some expressiveness results for G3 such…
We give the first cut-free ExpTime (optimal) tableau decision procedure for checking satisfiability of a knowledge base in the description logic SHI, which extends the description logic ALC with transitive roles, inverse roles and role hierarchies.
Given a formula in a temporal logic such as LTL or MTL, a fundamental problem is the complexity of evaluating the formula on a given finite word. For LTL, the complexity of this task was recently shown to be in NC. In this paper, we present an NC algorithm for MTL, a quantitative (or metric) extension of LTL, and give an NCC algorithm for UTL, the unary fragment of LTL. At the time of writing, MTL is the most expressive logic with an NC path-checking algorithm, and UTL is the most expressive fragment of LTL with a more efficient path-checking algorithm than for full LTL (subject to standard co…
We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC1. As basic tool we use the connection between intuitionistic logic and Heyting algebra, and investigate its complexity theoretical aspects. For superintuitionistic logics with one variable, we obtain NC1-completeness for the model checking problem.
A proof procedure, in the spirit of the sequent calculus, is proposed to check the validity of entailments between Separation Logic formulas combining inductively defined predicates denoted structures of bounded tree width and theory reasoning. The calculus is sound and complete, in the sense that a sequent is valid iff it admits a (possibly infinite) proof tree. We show that the procedure terminates in the two following cases: (i) When the inductive rules that define the predicates occurring on the left-hand side of the entailment terminate, in which case the proof tree is always finite. (ii)…
Hybrid Spatiotemporal Logic for Automotive Applications: Modeling and Model-Checking
TU Wien（维也纳工业大学）；National Institute of Advanced Industrial Science and Technology (AIST)（国立产业技术综合研究所）；National Institute of Informatics (NII)（国立情报学研究所）
Separation Logic is a widely used formalism for describing dynamically allocated linked data structures, such as lists, trees, etc. The decidability status of various fragments of the logic constitutes a long standing open problem. Current results report on techniques to decide satisfiability and validity of entailments for Separation Logic(s) over lists (possibly with data). In this paper we establish a more general decidability result. We prove that any Separation Logic formula using rather general recursively defined predicates is decidable for satisfiability, and moreover, entailments betw…
We study several extensions of linear-time and computation-tree temporal logics with quantifiers that allow for counting how often certain properties hold. For most of these extensions, the model-checking problem is undecidable, but we show that decidability can be recovered by considering flat Kripke structures where each state belongs to at most one simple loop. Most decision procedures are based on results on (flat) counter systems where counters are used to implement the evaluation of counting operators.
We consider quantitative extensions of the alternating-time temporal logics ATL/ATLs called quantitative alternating-time temporal logics (QATL/QATLs) in which the value of a counter can be compared to constants using equality, inequality and modulo constraints. We interpret these logics in one-counter game models which are infinite duration games played on finite control graphs where each transition can increase or decrease the value of an unbounded counter. That is, the state-space of these games are, generally, infinite. We consider the model-checking problem of the logics QATL and QATLs on…
The Epistemic Halpern-Shoham logic (EHS) is a temporal-epistemic logic that combines the interval operators of the Halpern-Shoham logic with epistemic modalities. The semantics of EHS is based on interpreted systems whose labelling function is defined on the endpoints of intervals. We show that this definition can be generalised by allowing the labelling function to be based on the whole interval by means of regular expressions. We prove that all the positive results known for EHS, notably the attractive complexity of its model checking problem for some of its fragments, still hold for its gen…
We investigate the decidability of model-checking logics of time, knowledge and probability, with respect to two epistemic semantics: the clock and synchronous perfect recall semantics in partially observed discrete-time Markov chains. Decidability results are known for certain restricted logics with respect to these semantics, subject to a variety of restrictions that are either unexplained or involve a longstanding unsolved mathematical problem. We show that mild generalizations of the known decidable cases suffice to render the model checking problem definitively undecidable. In particular,…
