Temporal Formulas Research Articles

Decidability for a temporal logic used in discrete-event system analysis

The type of plant considered is one that can be modeled by a nondeterministic finite-state machine P. The regulator is a deterministic finite state machine R. The closed-loop system is formed by connecting P and R in a regulator configuration. Formulas in a propositional temporal language are used to describe the behavior of the closed-loop system. It is shown that there is a mechanical procedure which, for a given P and R, and a temporal formula Psi, will determine in a finite number of steps whether or not Psi must be true. This 'decidability' result could be proven using other known results on temporal logic. The proof given here shows that the behavior of the closed-loop system may safely be assumed to be ultimately periodic. The results are illustrated on two discrete-event system examples.

Transformation of dynamic integrity constraints into transaction specifications

Dynamic database behaviour can be specified by dynamic integrity constraints, which determine admissible sequences of database states, and by transaction specifications, which induce executable sequences. Constraints are expressed by formulae of temporal logic, whereas transactions are defined by pre⧸postconditions in predicate logic. This article presents concepts and rules for transforming dynamic constraints into transaction specifications in order to prepare integrity monitoring by transactions. At first, such transition graphs must be constructed from temporal formulae that have paths corresponding to admissible sequences. Then these graphs are utilized to refine and to simplify pre⧸postconditions systematically so that every executable state sequence will become admissible, too.

Denote CSP with temporal formulas

This is a sequel of [1] to give a temporal semantics to a full version of CSP, including hiding operator and nested parallelism. The semantic definition is of denotational style, and employs set of temporal formulas as denotations. The continuity is proved, when the hiding operator is restricted to channels only with finite possible messages.

Deriving Safety Monitors from Formal Specifications

Safety is not guaranteed by the proper behaviour of the target system itself - it also depends on the properties of the plant and the environment. The paper shows how a safety monitor which supervises the assumptions under which the target system is safe, can be derived from formal specifications. The essential means are finite transition graphs which can be constructed from temporal formula by utilizing an appropriate normal form.

VVSL: A language for structured VDM specifications

Abstract VVSL is a VDM specification language of the “British School” with modularisation constructs allowing sharing of hidden state variables and parameterisation constructs for structuring specifications, and with constructs for expressing temporal aspects of the concurrent execution of operations which interfere via state variables. The modularisation and parameterisation constructs have been inspired by the “kernel” design language COLD-K from the ESPRIT project 432: METEOR, and the constructs for expressing temporal aspects by various temporal logics based on linear and discrete time. VVSL is provided with a well-defined semantics by defining a translation to COLD-K extended with constructs which are required for translation of the VVSL constructs for expressing temporal aspects. In this paper, the syntax for the modularisation and parameterisation constructs of VVSL is outlined. Their meaning is informally described by giving an intuitive explanation and by outlining the translation to COLD-K. It is explained in some detail how sharing of hidden state variables is modelled. Examples of the use of the modularisation and parameterisation constructs are also given. These examples are based on a formal definition of the relational data model. With respect to the constructs for expressing temporal aspects, the ideas underlying the use of temporal formulae in VVSL are briefly outlined and a simple example is given.

Incompleteness of first-order temporal logic with until

The results presented in this paper concern the axiomatizability problem of first-order temporal logic with linear and discrete time. We show that the logic is incomplete, i.e., it cannot be provided with a finitistic and complete proof system. We show two incompleteness theorems. Although the first one is weaker (it assumes some first-order signature), we decided to present it, for its proof is much simpler and contains an interesting fact that finite sets are characterizable by means of temporal formulas. The second theorem shows that the logic is incomplete independently of any particular signature.

Monitoring dynamic integrity constraints based on temporal logic

Dynamic integrity constraints are used to specify admissible sequences of database states. We present algorithmic fundamentals of monitoring constraints expressed in temporal logic. The essential means are finite transition graphs which can be constructed from temporal formulae by utilizing an appropriate normalform. To ensure admissibility of a state sequence, the integrity monitor has to follow a corresponding path through the graph and to check certain nontemporal conditions in each state; these conditions are provided as edge labels. Thus, monitoring dynamic integrity is reduced to a controlled checking of static integrity. All errors in present database behaviour are detected as well as many inevitable future errors.

Deciding full branching time logic

In this paper the full branching time logic (CTL*) is studied. It has basic modalities consisting of a path quantifier, either A (“for all paths”) of E (“for some path”), followed by an arbitrary linear time assertion composed of unrestricted combinations of the usual linear temporal operators F (“sometime”), G (“always”), X (“nexttime”), and U (“until”). It is shown that the problem of determining if a CTL* formula is satisfiable in a structure generated by a binary relation is decidable in triple exponential time. The decision procedure exploits the special structure of the finite state ω-automata for linear temporal formulae which allows them to be determinized with only a single exponential blowup in size. Also the expressive power of tree automata is compared with that of CTL* augmented by quantified auxillary propositions.

Relationships of the Baja California Ground Snakes, Genus Sonora

Sonora bancroftae Klauber, 1943, and S. mosaueri Stickel, 1938, are placed in the synonymy of S. semiannulata Baird and Girard, 1853. The characters that have been used to separate these taxa (color pattern, numbers of ventrals and subcaudals, and dorsal scale row reduction pattern) are shown to be insufficient for that purpose. Three species of Sonora presently are recognized as occurring in Baja California, M6xico, and the adjacent United States. S. bancroftae Klauber, 1943 (type locality: 2 mi. E San Jorge, Baja California Norte), is found in northwestern Baja California, and S. mosaueri Stickel, 1938 (type locality: Comondfi, Baja California Sur), is found in the southern half of Baja California. S. semiannulata Baird and Girard, 1853 (type locality: Sonora, restricted by Stickel, 1943, to the vicinity of the Santa Rita Mountains of southern Arizona, United States) is found in the southwestern quadrant of the United States and northern mainland Mexico, but in Baja California occurs only in the northeastern part, northward from the vicinity of Santa Rosalia (Stickel, 1943; Stebbins, 1966). These species have not been reviewed since Stickel (1938, 1943). In the intervening years, several specimens have been collected that improve our understanding of the relationships of these ground snakes. As first revised by Stickel (1938), the Sonora semiannulata complex (all Sonora except S. aemula and S. michoacanensis) contained five species: S. taylori (South Texas), S. episcopa (Great Plains), S. mosaueri (Baja California), S. semiannulata (western United States), and S. miniata (Idaho south to Arizona and California). The features used to diagnose these phenetically very similar species were temporal formula (1+1 in S. taylori and 1+2 in all others), numbers of ventrals and subcaudals, numbers of dorsal scale rows, and color pattern. Although Stickel was aware of color pattern polymorphism in S. episcopa and in S. semiannulata in the eastern part of its range, he distinguished western S. semiannulata from the largely sympatric

The temporal semantics of concurrent programs

The formalism of Temporal Logic is suggested as an appropriate tool for formalizing the semantics of concurrent programs. A simple model of concurrent program is presented in which n processors are executing concurrent n disjoint programs under a shared memory environment. The semantics of such a program specifies the class of state sequences which are admissible as proper execution sequences under the program. The two main criteria which are required are 1. (a) Each state is obtained from its predecessor in the sequence by exactly one processor performing an atomic instruction in its process. 2. (b) Fair Scheduling: no processor which is infinitely often enabled will be indefinitely delayed. The basic elements of Temporal Logic are introduced in a particular logic framework DX. The usefulness of Temporal Logic notation in describing properties of concurrent programs is demonstrated. A construction is then given for assigning to a program P a temporal formula W( P) which is true on all proper execution sequences of P. In order to prove that a program P possesses a property R 1, one has only to prove the implications W( P)⊃ R. An example of such proof is given. It is then demonstrated that specification of the Temporal character of the program's behavior is absolutely essential for the unambiguous understanding of the meaning of programming constructs.