A decision procedure and complete axiomatization for projection temporal logic☆
Introduction
Projection Temporal Logic (PTL) [1], [2] is an interval based Temporal Logic (TL) by introducing a new projection construct, , and supporting both finite and infinite time. Further, Propositional Projection Temporal Logic (PPTL) [3], the propositional subset of PTL, has the expressiveness power of the full regular expressions [4]. Moreover, the Modeling, Simulation and Verification Language (MSVL) [5], [6], [7], [8], [9], an executable subset of PTL with framing technique, supports a rich set of data types and many useful programming statements (e.g., frame, function call, if, while, and parallel ), and has ability to model, simulate and verify concurrent and reactive systems within a same logical system [10].
To verify the properties of concurrent and reactive systems, basically, two formal verification approaches, model checking [11] and theorem proving [12], are widely used in practice. The advantage of model checking is that the verification can be done automatically with a model checker. However, it suffers from the state explosion problem and thus does not fit to verify data intensive applications since the treatment of the data usually produces infinite state spaces [13]. With theorem proving approach, to verify whether or not a system S satisfies a property P is to prove whether or not is a theorem within the proof system. The advantage is that theorem proving avoids the state explosion problem and can verify both finite-state and infinite-state systems, including the data intensive applications. However, within the verification process, lots of assertions need to be inserted in the context of the program modeling the system, and the use of theorem prover requires considerable expertise to guide and assist the verification process. The decision problem of PPTL has been solved [3], [14], [15], based on which a model checker [16] is developed to verify the typical hardware and software systems such as process scheduling and handover protocol with success [17], [18], [19], [20]. However, to reason about concurrent systems based on the theorem proving approach, a proof system of PTL is required. Therefore, in this paper, we are motivated to formalize a complete axiomatization for PTL.
To this end, PTL is restricted to a finite domain, and the completeness argument is shown by means of a decision procedure based on a labeled normal form graph (LNFG), which in turn is constructed according to a labeled normal form (LNF), i.e., the traditional normal form (NF) possibly with some special marks, for PTL formulas. The techniques of NF and LNFG are first introduced in [1] and [21] respectively, and are further formalized in [3], [14] to check the satisfiability of PPTL formulas. In this paper, we extend the technique to first order projection temporal logic by proposing an LNF and the algorithms to construct and simplify LNFG for quantifier free PTL (QFPTL) formulas. The result shows that for any quantifier free formula P, P is unsatisfied if and only if the simplified LNFG of P is empty. Further, any quantified PTL formula is proved to be equivalent to a QFPTL formula and the related transformation algorithm is also given. Thus, the satisfiability of any PTL formula can also be checked.
The rest of the paper is organized as follows. Section 2 precisely presents the syntax, semantics and abbreviations of the underlying logic. Section 3 gives the axioms, inference rules of the proof system, and some useful theorems. In Section 4, the normal form and normal form graph (NFG) are defined and some important properties of NFG are discussed. Section 5 gives the definitions of LNF and LNFG, and an algorithm Lnfg for constructing LNFG. In Section 6, a decision procedure for checking the satisfiability of QFPTL formulas is described and two examples are given to illustrate how the decision procedure works. Section 7 formalizes an algorithm Q2Qftpl for transforming a quantified PTL formula into its equivalent QFPTL formula, and a decision procedure CheckPtl for checking the satisfiability of any PTL formula. In Section 8, the completeness of the axiomatic system is proved. In Section 9, the related work is addressed. Finally, conclusions are drawn in Section 10.
Section snippets
Projection temporal logic
In this section, we first introduce the syntax and semantics of projection temporal logic, and then give the definitions of some useful PTL formulas.
An axiomatization for PTL
The axioms and inference rules of our axiomatic system are given in Table 5 and Table 6 respectively, part of which are derived from some existing proof systems for QPTL [22] and PPTL [23]. Let P and Q be any PTL formulas, for convenience, sometimes we denote by and by .
A formula P deduced from the axiom system is called a PTL , denoted by ⊢P. The completeness proof of the axiomatic system uses many theorems and derived inference rules, part of which are listed in
Normal form and normal form graph
The technique of normal form (NF) [1], [2], [3] and normal form graph (NFG) [3] are useful to check the satisfiability of propositional projection temporal logic formulas. In the following, we extend the techniques to first order projection temporal logic.
Labeled normal form and labeled normal form graph
As depicted above, the NFG of a PTL formula P describes the models of P for it is constructed according to the normal form. However, this is not always true for formulas containing chop construct. For instance, formula is equivalent to , but there exists an infinite path in its NFG as shown in Fig. 2. Obviously, the NFG of formula is isomorphic to that of formula .
To solve the problem, for a chop formula , we need to make a deep analysis on the
A decision procedure for QFPTL formula
Definition 6.1 In the LNFG of a labeled PTL formula , a strongly connected subgraph is called an F decomposition cycle (F cycle for short) if there exists a DCI such that all the nodes are labeled with the same DCI . Further, if there does not exist such a DCI , the decomposition cycle is called an acceptable decomposition cycle (acceptable cycle for short).
The LNFG of a QFPTL formula R enjoys an important property that each F decomposition cycle in the LNFG
Transformation of QPTL formula into QFPTL formula
In the previous section, a decision procedure for QFPTL has been presented. As for a QPTL formula, our basic idea is to transform it into an equivalent QFPTL formula. Thus, the satisfiability of the PTL formula can be identified by checking the corresponding QFPTL formula with the decision procedure given above.
Decision procedure and completeness proof
For any PTL formula P, we can employ algorithm Q2Qfptl in Section 7 to transform it into an equivalent QFPTL formula . Thus, the satisfiability of formula P can be identified by checking that of formula with algorithm CheckQfptl in Section 6. Following this idea, algorithm CheckPTL is formalized in Table 14. Based on the transformation process of QPTL formula to QFPTL formula and the decidability of QFPTL formula, the completeness proof of our axiomatic system is concluded in the following
Related work
In the past three decades, a number of axiomatizations for temporal logics have been proposed to verify properties of concurrent and reactive systems with success. However, the mainstream of theoretical studies is focused on propositional case [24], [27], [28], [29], [30], [31], [32] for the corresponding propositional temporal logics are decidable and can be fully axiomatizable. Unfortunately, except for restricted fragments, full first order temporal logics and their derivations are not
Conclusion
In this paper, we present a complete proof system for the projection temporal logic supporting both finite and infinite time under the condition of finite data domain. The completeness argument about the proof system is proceeded by a labeled normal form graph based decision procedure. In the near future, as case study, we will further apply our proof system to verify properties of protocols, software and hardware systems. In particular, we are interested in modeling and verifying of composite
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This research is supported by NSFC Grant Nos. 61732013, 61420106004, Scientific Research Foundation of Education Department of Shaanxi Province (No. 11JK1037), and Industrial Research Project of Shaanxi Province (No. 2017GY-076).