Many-sorted First-order Logic Research Articles

New Techniques for Static Symmetry Breaking in Many-Sorted Finite Model Finding

Symmetry in finite model finding problems of many-sorted first-order logic (MSFOL) can be exploited to reduce the number of interpretations considered during search, thereby improving solver performance for tools such as the Alloy Analyzer. We present a framework to soundly compose static symmetry breaking schemes for many-sorted finite model finding. Then, we introduce and prove the correctness of three static symmetry breaking schemes for MSFOL: 1) one for functions with distinct sorts in the domain and range; 2) one for functions where the range sort appears in the domain; and 3) one for predicates. We provide a novel presentation of sort inference in the context of symmetry breaking that yields a new mathematical link between sorts and symmetries. We empirically investigate how our symmetry breaking approaches affect solving performance.

A decidable and expressive fragment of Many-Sorted First-Order Linear Temporal Logic

First-Order Linear Temporal Logic (FOLTL) and its Many-Sorted variant (MSFOLTL) are well-suited to specify infinite-state systems. However, the satisfiability of (MS)FOLTL is not even semi-decidable, thus preventing automated verification. In this paper, we exhibit various fragments of increasing scope that provide a pertinent basis for the abstract specification of infinite-state systems. We show that these fragments enjoy the Bounded Domain Property (any satisfiable (MS)FOLTL formula has a model with a finite, bounded FO domains), which provides a basis for complete, automated verification by reduction to LTL satisfiability. Finally, we present a simple case study illustrating the applicability and limitations of our results.

Fraïssé–Hintikka theorem in institutions

Abstract We generalize the characterization of elementary equivalence by Ehrenfeucht–Fraïssé games to arbitrary institutions whose sentences are finitary. These include many-sorted first-order logic, higher-order logic with types, as well as a number of other logics arising in connection to specification languages. The gain for the classical case is that the characterization is proved directly for all signatures, including infinite ones.

Encoding TLA+ into unsorted and many-sorted first-order logic

is a specification language designed for the verification of concurrent and distributed algorithms and systems. We present an encoding of a non-temporal fragment of into (unsorted) first-order logic and many-sorted first-order logic, the input languages of first-order automated theorem provers. The non-temporal subset of is based on untyped set theory and includes functions, arithmetic expressions, and Hilbert's choice operator. The translation, based on encoding techniques such as boolification, injection of unsorted expressions into sorted languages, term rewriting, and abstraction, is the core component of a back-end prover based on first-order theorem provers and smt solvers for the Proof System. • A TLA + Proof System back-end prover based on first-order provers and SMT solvers. • New techniques for encoding set theory, functions and Hilbert's operator into FOL. • Encoding of an untyped language (TLA + ) into many-sorted FOL. • Improved verification of safety properties of distributed systems in TLA + . • Reduce effort required to write interactive proofs in the TLA + Proof System.

Downward Löwenheim–Skolem Theorem and interpolation in logics with constructors

The present paper describes a method for proving Downward Lowenheim-Skolem Theorem within an arbitrary institution satisfying certain logic properties. In order to demonstrate the applicability of the present approach, the abstract results are instantiated to many-sorted first-order logic and preorder algebra. In addition to the first technique for proving Downward Lowenheim-Skolem Theorem, another one is developed, in the spirit of institution-independent model theory, which consists of borrowing the result from a simpler institution across an institution comorphism. As a result the Downward LowenheimSkolem Property is exported from first-order logic to partial algebras, and from higher-order logic with intensional Henkin semantics to higher-order logic with extensional Henkin semantics. The second method successfully extends the domain of application of Downward Lowenheim-Skolem Theorem to other non-conventional logical systems for which the first technique may fail. One major application of Downward Lowenheim-Skolem Theorem is interpolation in constructor-based logics with universally quantified sentences. The interpolation property is established by borrowing it from a base institution for its constructor-based variant across an institution morphism. This result is important as interpolation for constructor-based first-order logics is still an open problem.

First-order logic system $\mathbf{IMTL_{Q}^{*}}$ and triple I method in fuzzy reasoning with linguistic quantifiers

Based on first-order fuzzy logic system $\mathbf{K}^{*}$ and Ying's framework for linguistic quantifiers modeled by Sugeno integrals, a many-sorted first-order logic system $\mathbf{IMTL_{Q}^{*}}$ is constructed. Applying the triple I method, some fuzzy reasoning forms with linguistic quantifiers are investigated.

Modular natural language interfaces to logic-based policy frameworks

We have developed a translation system that maps sentences of Attempto Controlled English to predicates of many-sorted first-order logic, which can be directly imported into a logic-based policy management framework. Our translation achieves broader coverage than prior work that uses ACE, by a novel application of modern compositional semantics. This translation also natively supports question answering.The system significantly features a modular architecture, enabling semi-automated porting to new policy domains. We initially developed the system for cognitive radio policies, then generalized and ported it to two other policy vocabularies. The system interoperates with policies written in the XACML language.

An approach to the formal analysis of license interoperability

With recent advances in information and telecommunications technologies, a large range of digital content is distributed over the Internet. Whereas diverse licenses are provided to protect the content legally and have the advantage of offering authors many choices, the obstruction of smooth content distribution may occur if the relationships between licenses are not revealed because of differences between the restrictions imposed by each license. To activate digital content distribution, license interoperability must be revealed. In this paper, we propose a framework for formally examining license interoperability by using many-sorted first-order logic. We formalize five actual licenses and examine their interoperability to prove the effectiveness of our proposed framework. The results show that the framework reveals the relationships between licenses.

MDG-SAT: an automated methodology for efficient safety checking

Multiway decision graph (MDG) is a canonical representation of a subset of many-sorted first-order logic. It generalises the logic of equality with abstract types and uninterpreted function symbols. The area of satisfiability (SAT) has been the subject of intensive research in recent years, with significant theoretical and practical contributions. In this paper, we propose a new design verification tool integrating MDG and SAT, to check the safety of a design by invariant checking. Using MDG to encode the set of states provides a powerful mean of abstraction. We use a SAT solver to search for paths of reachable states violating the property under certain encoding constraints. In addition, we introduce an automated conversion-verification methodology to convert a directed formula (DF) into a conjunctive normal form (CNF) formula that can be fed to a SAT solver. The formal verification of this conversion is conducted within the HOL theorem prover. Finally, we present experimental results and a case study to show the correctness and the efficiency of our proposed methodology.

Unsorted Functional Translations

In this article we first show how the functional and the optimized functional translation from modal logic to many-sorted first-order logic can be naturally extended to the hybrid language H(@,↓). The translation is correct not only when reasoning over the class of all models, but for any first-order definable class. We then show that sorts can be safely removed (i.e., without affecting the satisfiability status of the formula) for frame classes that can be defined in the basic modal language, and show a counterexample for a frame class defined using nominals.

Automated Search for Impossibility Theorems in Social Choice Theory: Ranking Sets of Objects

We present a method for using standard techniques from satisfiability checking to automatically verify and discover theorems in an area of economic theory known as ranking sets of objects. The key question in this area, which has important applications in social choice theory and decision making under uncertainty, is how to extend an agent's preferences over a number of objects to a preference relation over nonempty sets of such objects. Certain combinations of seemingly natural principles for this kind of preference extension can result in logical inconsistencies, which has led to a number of important impossibility theorems. We first prove a general result that shows that for a wide range of such principles, characterised by their syntactic form when expressed in a many-sorted first-order logic, any impossibility exhibited at a fixed (small) domain size will necessarily extend to the general case. We then show how to formulate candidates for impossibility theorems at a fixed domain size in propositional logic, which in turn enables us to automatically search for (general) impossibility theorems using a SAT solver. When applied to a space of 20 principles for preference extension familiar from the literature, this method yields a total of 84 impossibility theorems, including both known and nontrivial new results.

NuMDG: a new tool for multiway decision graphs construction

Multiway Decision Graphs (MDGs) are a canonical representation of a subset of many-sorted first-order logic. This subset generalizes the logic of equality with abstract types and uninterpreted function symbols. The distinction between abstract and concrete sorts mirrors the hardware distinction between data path and control. Here we consider ways to improve MDGs construction. Efficiency is achieved through the use of the Generalized-If-Then-Else (GITE) commonly operator in Binary Decision Diagram packages. Consequently, we review the main algorithms used for MDGs verification techniques. In particular, Relational Product and Pruning by Subsumption are algorithms defined uniformly through this single GITE operator which will lead to a more efficient implementation. Moreover, we provide their correctness proof. This work can be viewed as a way to accommodate the ROBBD algorithms to the realm of abstract sorts and un interpreted functions. The new tool, called NuMDG, accepts an extended SMV language, supporting abstract data sorts. Finally, we present experimental results demonstrating the efficiency of the NuMDG tool and evaluating its performance using a set of benchmarks from the SMV package.

Decidable fragments of many-sorted logic

Many natural specifications use types. We investigate the decidability of fragments of many-sorted first-order logic. We identified some decidable fragments and illustrated their usefulness by formalizing specifications considered in the literature. Often the intended interpretations of specifications are finite. We prove that the formulas in these fragments are valid iff they are valid over the finite structures. We extend these results to logics that allow a restricted form of transitive closure. We tried to extend the classical classification of the quantifier prefixes into decidable/undecidable classes to the many-sorted logic. However, our results indicate that a naive extension fails and more subtle classification is needed.

An Institution-Independent Proof of the Robinson Consistency Theorem

We prove an institutional version of A. Robinson’s Consistency Theorem. This result is then appliedto the institution of many-sorted first-order predicate logic and to two of its variations, infinitary and partial, obtaining very general syntactic criteria sufficient for a signature square in order to satisfy the Robinson consistency and Craig interpolation properties.

Constructor-based observational logic

This paper focuses on the integration of reachability and observability concepts within an algebraic, institution-based framework. In the first part of this work, we develop the essential ingredients that are needed to define the constructor-based observational logic institution, called COL, which takes into account both the generation- and observation-oriented aspects of software systems. The underlying paradigm of our approach is that the semantics of a specification should be as loose as possible to capture all its correct realizations. We also consider the “black box” semantics of a specification which is useful to study the behavioral properties a user can observe when he/she is experimenting with the system. In the second part of this work, we develop proof techniques for structured COL-specifications. For this purpose we introduce an institution encoding from the COL institution to the institution of many-sorted first-order logic with equality and sort-generation constraints. Using this institution encoding, we can then reduce proofs of consequences of structured specifications built over COL to proofs of consequences of structured specifications written in a simple subset of the algebraic specification language C asl. This means, in particular, that any inductive theorem prover, such as e.g. the Larch Prover or PVS, can be used to prove theorems over structured COL-specifications.

Guards, Bounds, and Generalized Semantics

Some initial motivations for the Guarded Fragment still seem of interest in carrying its program further. First, we stress the equivalence between two perspectives: (a) satisfiability on standard models for guarded first-order formulas, and (b) satisfiability on general assignment models for arbitrary first-order formulas. In particular, we give a new straightforward reduction from the former notion to the latter. We also show how a perspective shift to general assignment models provides a new look at the fixed-point extension LFP(FO) of first-order logic, making it decidable. Next, we relate guarded syntax to earlier quantifier restriction strategies for achieving effective axiomatizability in second-order logic -- pointing at analogies with `persistent' formulas, which are essentially in the Bounded Fragment of many-sorted first-order logic. Finally, we look at some further unexplored directions, including the systematic use of `quasi-models' as a semantics by itself.

Generalized interpolation in [formula omitted

In this paper we consider the partial many-sorted first-order logic and its extension to the subsorted partial many-sorted first-order logic that underly the Casl specification formalism. First we present counterexamples showing that the generalization of the Craig Interpolation Property does not hold for these logics in general (i.e., with respect to arbitrary signature morphisms). Then we formulate conditions under which the generalization of the Craig Interpolation Property holds for the first logic.

Specification of abstract dynamic-data types: A temporal logic approach

A concrete dynamic-data type is just a partial algebra with predicates such that for some of the sorts there is a special predicate defining a transition relation. An abstract dynamic-data type (ad-dt) is an isomorphism class of such algebras. To obtain specifications for ad-dt's, we propose a logic which combines many-sorted first-order logic with branching-time combinators. We consider both an initial and a loose semantics for our specifications and give sufficient conditions for the existence of the initial models. Then we discuss structured specifications and implementation.

Type reducing correspondences and well-orderings: Frege's and Zermelo's constructions re-examined

A key idea in both Frege's development of arithmetic in the Grundlagen [7] and Zermelo's 1904 proof [10] of the well-ordering theorem is that of a “type reducing” correspondence between second-level and first-level entities. In Frege's construction, the correspondence obtains between concept and number, in Zermelo's (through the axiom of choice), between set and member. In this paper, a formulation is given and a detailed investigation undertaken of a system ℱ of many-sorted first-order logic (first outlined in the Appendix to [6]) in which this notion of type reducing correspondence is accorded a central role and which enables Frege's and Zermelo's constructions to be presented in such a way as to reveal their essential similarity. By adapting Bourbaki's version of Zermelo's proof of the well-ordering theorem, we show that, within ℱ, any correspondence c between second-level entities (here called concepts) and first-level ones (here called objects) induces a well-ordering relation W (c) in a canonical manner. We shall see that, when c is the “Fregean” correspondence between concepts and cardinal numbers, W (c) is (the well-ordering of) the ordinal ω + 1, and when c is a “Zermelian” choice function on concepts, W (c) is a well-ordering of the universal concept embracing all objects.In ℱ an important role is played by the notion of extension of a concept. To each concept X we assume there is assigned an object e(X) in such a way that, for any concepts X, Y satisfying a certain predicate E, we have e (X) = e (Y) iff the same objects fall under X and Y.

Many-Sorted First-Order Logic Database Language

A database languages based on Many-Sorted First-Order Logic (MSFOL) have many advantages over one based on One-Sorted First-Order Logic (OSFOL). The advantages includes ease-of-expressiveness, efficiency, and the abstraction mechanism. Many database researchers have used OSFOL to view the Relational Data Model (RDM) ; however, no RDM has been modelled by MSFOL. This paper first gives a formal definition for MSFOL and then its advantages of expressiveness and of abstraction are illustrated. Two reduction algorithms which can transform an MSFOL-based language into\from an OSFOL-based language are given. The semantic equivalence between languages based on MSFOL and Typed OSFOL is also proved. Recent extensions of RDMs require aggregation, classification, and generalisation/specialisation mechanisms which MSFOL-based languages can provide, but OSFOL-based languages cannot.