Nominalistic Logic (NL) is a new presentation of Paul Gilmore's Intensional Type Theory (ITT) as a sequent calculus together with a succinct nominalization axiom (N) that permits names of predicates as individuals in certain cases. The logic has a flexible comprehension axiom, but no extensionality axiom and no infinity axiom, although axiom N is the key to the derivation of Peano's postulates for the natural numbers.
This article responds to a critique of higher-order abstract syntax appearing in Logic Column 14, ``Nominal Logic and Abstract Syntax'', cs.LO/0511025.
Formalizing syntactic proofs of properties of logics, programming languages, security protocols, and other formal systems is a significant challenge, in large part because of the obligation to handle name-binding correctly. We present an approach called nominal abstract syntax that has attracted considerable interest since its introduction approximately six years ago. After an overview of other approaches, we describe nominal abstract syntax and nominal logic, a logic for reasoning about nominal abstract syntax. We also discuss applications of nominal techniques to programming, automated reaso…
Separation logics are a family of extensions of Hoare logic for reasoning about programs that mutate memory. These logics are "abstract" because they are independent of any particular concrete memory model. Their assertion languages, called propositional abstract separation logics, extend the logic of (Boolean) Bunched Implications (BBI) in various ways. We develop a modular proof theory for various propositional abstract separation logics using cut-free labelled sequent calculi. We first extend the cut-fee labelled sequent calculus for BBI of Hou et al to handle Calcagno et al's original logi…
We show that a proof in multiplicative linear logic can be represented as a decorated surface, such that two proofs are logically equivalent just when their surfaces are geometrically equivalent. This is an extended abstract for arXiv:1601.05372.
The paper presents an extension of temporal epistemic logic with operators that quantify over strategies. The language also provides a natural way to represent what agents would know were they to be aware of the strategies being used by other agents. Some examples are presented to motivate the framework, and relationships to several variants of alternating temporal epistemic logic are discussed. The computational complexity of model checking the logic is also characterized.
We study the notion of stratification, as used in subsystems of linear logic with low complexity bounds on the cut-elimination procedure (the so-called light logics), from an abstract point of view, introducing a logical system in which stratification is handled by a separate modality. This modality, which is a generalization of the paragraph modality of Girard's light linear logic, arises from a general categorical construction applicable to all models of linear logic. We thus learn that stratification may be formulated independently of exponential modalities; when it is forced to be connecte…
We develop a logic which enables reasoning about single steps of non-deterministic parallel Abstract State Machines (ASMs). Our logic builds upon the unifying logic introduced by Nanchen and Stärk for reasoning about hierarchical (parallel) ASMs. Our main contribution to this regard is the handling of non-determinism (both bounded and unbounded) within the logical formalism. Moreover, we do this without sacrificing the completeness of the logic for statements about single steps of non-deterministic parallel ASMs, such as invariants of rules, consistency conditions for rules, or step-by-step eq…
Complete Heyting Algebra Semantics for an Intuitionistic Version of Matching Logic (Extended Abstract)
Research Center for Logic, Optimization and Security (LOS), University of Bucharest（布加勒斯特大学逻辑、优化与安全研究中心）；Institute for Logic and Data Science, Bucharest（布加勒斯特逻辑与数据科学研究所）
We continue work of our earlier paper (Lewitzka and Brunner: Minimally generated abstract logics, Logica Universalis 3(2), 2009), where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the cate…
Existing work on theorem proving for the assertion language of separation logic (SL) either focuses on abstract semantics which are not readily available in most applications of program verification, or on concrete models for which completeness is not possible. An important element in concrete SL is the points-to predicate which denotes a singleton heap. SL with the points-to predicate has been shown to be non-recursively enumerable. In this paper, we develop a first-order SL, called FOASL, with an abstracted version of the points-to predicate. We prove that FOASL is sound and complete with re…
Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of models. To cope with the lack of explicit set of states in institutions, t…
Process behaviour is often defined either in terms of the tests they satisfy, or in terms of the logical properties they enjoy. Here we compare these two approaches, using extensional testing in the style of DeNicola, Hennessy, and a recursive version of the property logic HML. We first characterise subsets of this property logic which can be captured by tests. Then we show that those subsets of the property logic capture precisely the power of tests.
Abstract interpretation, Hoare logic, and incorrectness (or reverse Hoare) logic are powerful techniques for static analysis of computer programs. All of them have been successfully extended to the quantum setting, but largely developed in parallel. In this paper, we examine the relationship between these techniques in the context of verifying quantum while-programs, where the abstract domain and the set of assertions for quantum states are well-structured. In particular, we show that any complete quantum abstract interpretation induces a quantum Hoare logic and a quantum incorrectness logic, …
Combining higher-order abstract syntax and (co)induction in a logical framework is well known to be problematic. Previous work described the implementation of a tool called Hybrid, within Isabelle HOL, which aims to address many of these difficulties. It allows object logics to be represented using higher-order abstract syntax, and reasoned about using tactical theorem proving and principles of (co)induction. In this paper we describe how to use it in a multi-level reasoning fashion, similar in spirit to other meta-logics such as Twelf. By explicitly referencing provability in a middle layer c…
In database theory, the term $\textit{database transformation}$ was used to refer to a unifying treatment for computable queries and updates. Recently, it was shown that non-deterministic database transformations can be captured exactly by a variant of ASMs, the so-called Database Abstract State Machines (DB-ASMs). In this article we present a logic for DB-ASMs, extending the logic of Nanchen and Stärk for ASMs. In particular, we develop a rigorous proof system for the logic for DB-ASMs, which is proven to be sound and complete. The most difficult challenge to be handled by the extension is a …
We add strong negation $N$ to classical logic and interpret the attack relation of "$x$ attacks $y$" in argumentation as $(x\to Ny)$. We write a corresponding object level (using $N$ only) classical theory for each argumentation network and show that the classical models of this theory correspond exactly to the complete extensions of the argumentation network. We show by example how this approach simplifies the study of abstract argumentation networks. We compare with other translations of abstract argumentation networks into logic, such as classical predicate logic or modal logics, or logic p…
Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free regular category FRg(T) on a set T. From this point of view, regular theories are certain monoidal 2-functors from a suitable 2-category of contexts -- the 2-category of relations in FRg(T) -- to that of posets. Such functors assign to each context the set of formulas in that context, ordered by entailment. We refer to such a 2-functor as a regular calculus…
We present an approach for representing abstract argumentation frameworks based on an encoding into classical higher-order logic. This provides a uniform framework for computer-assisted assessment of abstract argumentation frameworks using interactive and automated reasoning tools. This enables the formal analysis and verification of meta-theoretical properties as well as the flexible generation of extensions and labellings with respect to well-known argumentation semantics.
In the present paper, we propose Abstract Algebraic Logic (AAL) as a general logical framework for Judgment Aggregation. Our main contribution is a generalization of Herzberg's algebraic approach to characterization results in on judgment aggregation and propositional-attitude aggregation, characterizing certain Arrovian classes of aggregators as Boolean algebra and MV-algebra homomorphisms, respectively. The characterization result of the present paper applies to agendas of formulas of an arbitrary selfextensional logic. This notion comes from AAL, and encompasses a vast class of logics, of w…
University of Nebraska Omaha（内布拉斯加大学奥马哈分校）；IRIT（图卢兹计算机科学研究所）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）
In this work, we show that both logic programming and abstract argumentation frameworks can be interpreted in terms of Nelson's constructive logic N4. We do so by formalizing, in this logic, two principles that we call non-contradictory inference and strengthened closed world assumption: the first states that no belief can be held based on contradictory evidence while the latter forces both unknown and contradictory evidence to be regarded as false. Using these principles, both logic programming and abstract argumentation frameworks are translated into constructive logic in a modular way and u…
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,…
Type-preserving (or typed) compilation uses typing derivations to certify correctness properties of compilation. We have designed and implemented a type-preserving compiler for a simply-typed dialect of Prolog we call T-Prolog. The crux of our approach is a new certifying abstract machine which we call the Typed Warren Abstract Machine (TWAM). The TWAM has a dependent type system strong enough to specify the semantics of a logic program in the logical framework LF. We present a soundness metatheorem which constitutes a partial correctness guarantee: well-typed programs implement the logic prog…
The problem of forward abstract interpretation of {\em normal} logic programs has not been formally addressed in the literature although negation as failure is dealt with through the built-in predicate ! in the way it is implemented in Prolog. This paper proposes a solution to this problem by deriving two generic fixed-point abstract semantics $F^b and $F^\diamond for forward abstract interpretation of {\em normal} logic programs. $F^b$ is intended for inferring data descriptions for edges in the program graph where an edge denotes the possibility that the control of execution transfers from i…
Logical systems with classical negation and means for sentential or propositional self-reference involve, in some way, paradoxical statements such as the liar. However, the paradox disappears if one replaces classical by an appropriate non-classical negation such as a paraconsistent one (no paradox arises if the liar is both true and false). We consider a non-Fregean logic which is a revised and extended version (Lewitzka 2012) of Epsilon-T-Logic originally introduced by (Straeter 1992) as a logic with a total truth predicate and propositional quantifiers. Self-reference is achieved by means o…
Computer Science Institute Leipzig University（莱比锡大学计算机科学研究所）
We analyse the expressiveness of the two-valued semantics of abstract argumentation frameworks, normal logic programs and abstract dialectical frameworks. By expressiveness we mean the ability to encode a desired set of two-valued interpretations over a given propositional signature using only atoms from that signature. While the computational complexity of the two-valued model existence problem for all these languages is (almost) the same, we show that the languages form a neat hierarchy with respect to their expressiveness.
An Encoding of Abstract Dialectical Frameworks into Higher-Order Logic
University of Luxembourg, Department of Computer Science（卢森堡大学计算机科学系）；Universität Greifswald, Institut für Mathematik und Informatik（格赖夫斯瓦尔德大学数学与信息学研究所）
We introduce and study logic programs whose clauses are built out of monotone constraint atoms. We show that the operational concept of the one-step provability operator generalizes to programs with monotone constraint atoms, but the generalization involves nondeterminism. Our main results demonstrate that our formalism is a common generalization of (1) normal logic programming with its semantics of models, supported models and stable models, (2) logic programming with weight atoms (lparse programs) with the semantics of stable models, as defined by Niemela, Simons and Soininen, and (3) of dis…
Abstract interpretation is a well-established technique for performing static analyses of logic programs. However, choosing the abstract domain, widening, fixpoint, etc. that provides the best precision-cost trade-off remains an open problem. This is in a good part because of the challenges involved in measuring and comparing the precision of different analyses. We propose a new approach for measuring such precision, based on defining distances in abstract domains and extending them to distances between whole analyses of a given program, thus allowing comparing precision across different analy…
In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge. Building on Stalnaker's core insights, and using frameworks developed by Bjorndahl and Baltag et al., we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker's system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the princip…
Abstract separation logics are a family of extensions of Hoare logic for reasoning about programs that manipulate resources such as memory locations. These logics are "abstract" because they are independent of any particular concrete resource model. Their assertion languages, called propositional abstract separation logics (PASLs), extend the logic of (Boolean) Bunched Implications (BBI) in various ways. In particular, these logics contain the connectives $*$ and $-\!*$, denoting the composition and extension of resources respectively. This added expressive power comes at a price since the res…
Bialgebrae provide an abstract framework encompassing the semantics of different kinds of computational models. In this paper we propose a bialgebraic approach to the semantics of logic programming. Our methodology is to study logic programs as reactive systems and exploit abstract techniques developed in that setting. First we use saturation to model the operational semantics of logic programs as coalgebrae on presheaves. Then, we make explicit the underlying algebraic structure by using bialgebrae on presheaves. The resulting semantics turns out to be compositional with respect to conjunctio…
The existence of a coalition strategy to achieve a goal does not necessarily mean that the coalition has enough information to know how to follow the strategy. Neither does it mean that the coalition knows that such a strategy exists. The paper studies an interplay between the distributed knowledge, coalition strategies, and coalition "know-how" strategies. The main technical result is a sound and complete trimodal logical system that describes the properties of this interplay.
In 1969, Per Lindstrom proved his celebrated theorem characterising the first-order logic and established criteria for the first-order definability of formal theories for discrete structures. K. J. Barwise, S. Shelah, J. Vaananen and others extended Lindstrom's characterizability program to classes of infinitary logic systems, including a recent paper by M. Dzamonja and J. Vaananen on Karp's chain logic, which satisfies interpolation, undefinability of well-order, and is maximal in the class of logic systems with these properties. The novelty of the chain logic is in its new definition of sati…
Epistemic logics typically talk about knowledge of individual agents or groups of explicitly listed agents. Often, however, one wishes to express knowledge of groups of agents specified by a given property, as in `it is common knowledge among economists'. We introduce such a logic of common knowledge, which we term abstract-group epistemic logic (AGEL). That is, AGEL features a common knowledge operator for groups of agents given by concepts in a separate agent logic that we keep generic, with one possible agent logic being ALC. We show that AGEL is EXPTIME-complete, with the lower bound estab…
The use of formal analysis tools on models or source code often requires the availability of auxiliary invariants about the studied system. Abstract interpretation is currently one of the best approaches to discover useful invariants, especially numerical ones. However, its application is limited by two orthogonal issues: (i) developing an abstract interpretation is often non-trivial; each transfer function of the system has to be represented at the abstract level, depending on the abstract domain used; (ii) with precise but costly abstract domains, the information computed by the abstract int…
Local Completeness Logic (LCL) has been put forward as a program logic for proving both the correctness and incorrectness of program specifications. LCL is an abstract logic, parameterized by an abstract domain that allows combining over- and under-approximations of program behaviors. It turns out that LCL instantiated to the trivial singleton abstraction boils down to O'Hearn incorrectness logic, which allows us to prove the presence of program bugs. It has been recently proved that suitable extensions of Kleene algebra with tests (KAT) allow representing both O'Hearn incorrectness and Hoare …
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 rea…
In this paper we show that reversible analysis of logic languages by abstract interpretation can be performed without loss of precision by systematically refining abstract domains. The idea is to include semantic structures into abstract domains in such a way that the refined abstract domain becomes rich enough to allow approximate bottom-up and top-down semantics to agree. These domains are known as condensing abstract domains. Substantially, an abstract domain is condensing if goal-driven and goal-independent analyses agree, namely no loss of precision is introduced by approximating queries …
We present Hypersequent Classical Processes (HCP), a revised interpretation of the "Proofs as Processes" correspondence between linear logic and the π-calculus initially proposed by Abramsky [1994], and later developed by Bellin and Scott [1994], Caires and Pfenning [2010], and Wadler [2014], among others. HCP mends the discrepancies between linear logic and the syntax and observable semantics of parallel composition in the π-calculus, by conservatively extending linear logic to hyperenvironments (collections of environments, inspired by the hypersequents by Avron [1991]). Separation of enviro…
We identify multirole logic as a new form of logic in which conjunction/disjunction is interpreted as an ultrafilter on the power set of some underlying set (of roles) and the notion of negation is generalized to endomorphisms on this underlying set. We formalize both multirole logic (MRL) and linear multirole logic (LMRL) as natural generalizations of classical logic (CL) and classical linear logic (CLL), respectively, and also present a filter-based interpretation for intuitionism in multirole logic. Among various meta-properties established for MRL and LMRL, we obtain one named multiparty c…
Recently there has been an increasing interest in frameworks extending Dung's abstract Argumentation Framework (AF). Popular extensions include bipolar AFs and AFs with recursive attacks and necessary supports. Although the relationships between AF semantics and Partial Stable Models (PSMs) of logic programs has been deeply investigated, this is not the case for more general frameworks extending AF. In this paper we explore the relationships between AF-based frameworks and PSMs. We show that every AF-based framework $Δ$ can be translated into a logic program $P_Δ$ so that the extensions prescr…
We introduce a novel variant of logical relations that maps types not merely to partial equivalence relations on values, as is commonly done, but rather to a proof-relevant generalisation thereof, namely setoids. The objects of a setoid establish that values inhabit semantic types, whilst its morphisms are understood as proofs of semantic equivalence. The transition to proof-relevance solves two well-known problems caused by the use of existential quantification over future worlds in traditional Kripke logical relations: failure of admissibility, and spurious functional dependencies. We illust…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）；Mohamed bin Zayed University of Artificial Intelligence (MBZUAI)（穆罕默德·本·扎耶德人工智能大学）
Abstract reasoning, i.e., inferring complicated patterns from given observations, is a central building block of artificial general intelligence. While humans find the answer by either eliminating wrong candidates or first constructing the answer, prior deep neural network (DNN)-based methods focus on the former discriminative approach. This paper aims to design a framework for the latter approach and bridge the gap between artificial and human intelligence. To this end, we propose logic-guided generation (LoGe), a novel generative DNN framework that reduces abstract reasoning as an optimizati…
Weighted monadic second-order logic is a weighted extension of monadic second-order logic that captures exactly the behaviour of weighted automata. Its semantics is parameterized with respect to a semiring on which the values that weighted formulas output are evaluated. Gastin and Monmege (2018) gave abstract semantics for a version of weighted monadic second-order logic to give a more general and modular proof of the equivalence of the logic with weighted automata. We focus on the abstract semantics of the logic and we give a complete axiomatization both for the full logic and for a fragment …
This paper deals with descriptive complexity of picture languages of any dimension by syntactical fragments of existential second-order logic. - We uniformly generalize to any dimension the characterization by Giammarresi et al. \cite{GRST96} of the class of \emph{recognizable} picture languages in existential monadic second-order logic. - We state several logical characterizations of the class of picture languages recognized in linear time on nondeterministic cellular automata of any dimension. They are the first machine-independent characterizations of complexity classes of cellular automata…
In separation logic program analyses, tractability is generally achieved by restricting invariants to a finite abstract domain. As this domain cannot vary, loss of information can cause failure even when verification is possible in the underlying logic. In this paper, we propose a CEGAR-like method for detecting spurious failures and avoiding them by refining the abstract domain. Our approach is geared towards discovering existential properties, e.g. "list contains value x". To diagnose failures, we use abduction, a technique for inferring command preconditions. Our method works backwards from…
In this article we show that hybrid type-logical grammars are a fragment of first-order linear logic. This embedding result has several important consequences: it not only provides a simple new proof theory for the calculus, thereby clarifying the proof-theoretic foundations of hybrid type-logical grammars, but, since the translation is simple and direct, it also provides several new parsing strategies for hybrid type-logical grammars. Second, NP-completeness of hybrid type-logical grammars follows immediately. The main embedding result also sheds new light on problems with lambda grammars/abs…
Abstract Dialectical Frameworks (ADFs) are argumentation frameworks where each node is associated with an acceptance condition. This allows us to model different types of dependencies as supports and attacks. Previous studies provided a translation from Normal Logic Programs (NLPs) to ADFs and proved the stable models semantics for a normal logic program has an equivalent semantics to that of the corresponding ADF. However, these studies failed in identifying a semantics for ADFs equivalent to a three-valued semantics (as partial stable models and well-founded models) for NLPs. In this work, w…
