First-order Dynamic Logic Research Articles

A dynamic logic with branching modalities

We propose a dynamic logic DLb called “dynamic logic with branching modalities”, which extends the temporal dynamic logic DLT with a “branching modality” for specifying safety properties of regular programs with tests (simply “regular programs”). Compared to the trace modality of DLT for while programs that do not abort, branching modality of DLb does not exclude aborting traces introduced by regular programs, thus is able to capture a type of safety properties which are important for systems with failure behaviors. Moreover, it is congruent to the compositionality of regular programs so that the proof system naturally extended from that of DLT is proved to be complete for DLb. In this paper, we build the theory of DLb on both propositional and first-ordered levels, defining two logics: propositional DLb (PDLb) and first-ordered DLb (FODLb). PDLb forms the theoretical basis of DLb while FODLb is useful for practical verification. We propose the proof systems for PDLb and FODLb, and analyze their decidability, soundness and (relative) completeness in a formal way, through comparing their expressiveness and deduction capabilities with propositional dynamic logic (PDL) and first-order dynamic logic (FODL) respectively. We show that FODLb is actually an extension of DLT, and illustrate the motivations of using the branching modality through an example.

A dynamic logic for verification of synchronous models based on theorem proving

Synchronous model is a type of formal models for modelling and specifying reactive systems. It has a great advantage over other real-time models that its modelling paradigm supports a deterministic concurrent behaviour of systems. Various approaches have been utilized for verification of synchronous models based on different techniques, such as model checking, SAT/SMT sovling, term rewriting, type inference and so on. In this paper, we propose a verification approach for synchronous models based on compositional reasoning and term rewriting. Specifically, we initially propose a variation of dynamic logic, called synchronous dynamic logic (SDL). SDL extends the regular program model of first-order dynamic logic (FODL) with necessary primitives to capture the notion of synchrony and synchronous communication between parallel programs, and enriches FODL formulas with temporal dynamic logical formulas to specify safety properties -- a type of properties mainly concerned in reactive systems. To rightly capture the synchronous communications, we define a constructive semantics for the program model of SDL. We build a sound and relatively complete proof system for SDL. Compared to previous verification approaches, SDL provides a divide and conquer way to analyze and verify synchronous models based on compositional reasoning of the syntactic structure of the programs of SDL. To illustrate the usefulness of SDL, we apply SDL to specify and verify a small example in the synchronous model SyncChart, which shows the potential of SDL to be used in practice.

A clock-based dynamic logic for the verification of CCSL specifications in synchronous systems

The Clock Constraint Specification Language (CCSL) is a clock-based specification language for real-time embedded systems. With logical clocks defined as first-class citizens, CCSL provides a natural way for describing clock constraints in synchronous systems — a classical model of concurrency for real-time embedded systems. In this paper, we propose a clock-based dynamic logic called CCSL Dynamic Logic (CDL) for the verification of CCSL specifications in synchronous systems. It extends the first-order dynamic logic with a synchronous execution mechanism in its program model and with CCSL primitives as terms in its logical formulae. We build a sound and relatively complete proof system for CDL to support the verification. Compared with previous approaches for verifying CCSL specifications, which are based on model checking and SMT checking techniques, our approach, which is based on theorem-proving, offers a unified verification framework in which both bounded and unbounded CCSL specifications can be verified. Technically, with the proof system of CDL, a complex CDL formula can be semi-automatically transformed into a set of quantifier-free, arithmetical first-order logic (QF-AFOL) formulae which can be checked by an SMT solver in an efficient way. As a case study, we analyze a simple synchronous system throughout the paper to illustrate how CDL works. We analyze and prove the soundness and completeness of the proof system for CDL. Currently, CDL is partially mechanized in Coq. • Dynamic Logic to deal with Logical Clocked Systems. • Proof system for CCSL and synchronous systems. • Rely on a decidable subset theory for efficient use of SMT solvers. • Proof of soundness and relative completeness.

Transition semantics: the dynamics of dependence logic

We examine the relationship between dependence logic and game logics. A variant of dynamic game logic, called Transition Logic, is developed, and we show that its relationship with dependence logic is comparable to the one between first-order logic and dynamic game logic discussed by van Benthem. This suggests a new perspective on the interpretation of dependence logic formulas, in terms of assertions about reachability in games of imperfect information against Nature. We then capitalize on this intuition by developing expressively equivalent variants of dependence logic in which this interpretation is taken to the foreground.

Dynamic Non-Commutative Logic

A first-order dynamic non-commutative logic (DN), which has no structural rules and has some program operators, is introduced as a Gentzen-type sequent calculus. Decidability, cut-elimination and completeness theorems are shown for DN or its fragments. DN is intended to represent not only program-based, resource-sensitive, ordered, sequence-based, but also hierarchical (tree-based) reasoning.

Institutions: abstract model theory for specification and programming

There is a population explosion among the logical systems used in computing science. Examples include first-order logic, equational logic, Horn-clause logic, higher-order logic, infinitary logic, dynamic logic, intuitionistic logic, order-sorted logic, and temporal logic; moreover, there is a tendency for each theorem prover to have its own idiosyncratic logical system. The concept of institution is introduced to formalize the informal notion of “logical system.” The major requirement is that there is a satisfaction relation between models and sentences that is consistent under change of notation. Institutions enable abstracting away from syntactic and semantic detail when working on language structure “in-the-large”; for example, we can define language features for building large logical system. This applies to both specification languages and programming languages. Institutions also have applications to such areas as database theory and the semantics of artificial and natural languages. A first main result of this paper says that any institution such that signatures (which define notation) can be glued together, also allows gluing together theories (which are just collections of sentences over a fixed signature). A second main result considers when theory structuring is preserved by institution morphisms. A third main result gives conditions under which it is sound to use a theorem prover for one institution on theories from another. A fourth main result shows how to extend institutions so that their theories may include, in addition to the original sentences, various kinds of constraint that are useful for defining abstract data types, including both “data” and “hierarchy” constraints. Further results show how to define institutions that allow sentences and constraints from two or more institutions. All our general results apply to such “duplex” and “multiplex” institutions.

Definability by deterministic and non-deterministic programs (with applications to first-order dynamic logic)

We make explicit a connection between the “unwind property” and first-order logics of programs. Using known results on the unwind property, we can then quickly compare various logics of programs. In Section 1, we give a sample of these comparative results, which are already known but established differently in this paper. In Sections 2 and 3, given an arbitrary deterministic regular program S (with or without parameterless recursive calls), we show how to construct a first-order structure where S will unwind. Based on this construction, we then prove that the logic of regular programs (with or without parameterless recursive calls) is more expressive than the logic of deterministic regular programs (with or without parameterless recursive calls, respectively).

Propositional dynamic logic with local assignment

We propose an extension of Propositional Dynamic Logic which allows a new kind of program terms—local assignments to propositional variables. They are very close to the known array assignments in Dynamic Logic beacuse they allow us to change the truth values of predicate variables. In this logic, many notions, like equivalence of programs, looping and finitely branching, are expressible on a propositional level. In fact, we show that the resulting logic is equivalent in expressive power to first-order logic augmented by a device to express transitive closures. In other words, it is (modulo extra predicate symbols) equivalent to first-order dynamic logic. Not suprisingly, therefore, the validity problem for this extension is Π 1 1-complete.

A probabilistic dynamic logic

A logic, PrDL, is presented, which enables formal reasoning about probabilistic programs or, alternatively, reasoning probabilistically about conventional programs. The syntax of PrDL derives from Pratt's first-order dynamic logic and the semantics extends Kozen's semantics of probabilistic programs. An axiom system for PrDL is presented and shown to be complete relative to an extension of first-order analysis. For discrete probabilities it is shown that first-order analysis actually suffices. Examples are presented, both of the expressive power of PrDL, and of a proof in the axiom system.

David Harel. First-order dynamic logic. Lecture notes in computer science, vol. 68. Springer-Verlag, Berlin, Heidelberg, and New York, 1979, X + 133 pp.

David Harel. First-order dynamic logic. Lecture notes in computer science, vol. 68. Springer-Verlag, Berlin, Heidelberg, and New York, 1979, X + 133 pp.

Arithmetical Completeness in First-Order Dynamic Logic for Concurrent Programs

We extend Harel's [4] arithmetical axiomatization P of regular first-order dynamic logic so as to include concurrent programs a///?, and then establish its arithmetical completeness.