Elsevier

Theoretical Computer Science

Volume 819, 2 June 2020, Pages 50-84
Theoretical Computer Science

A decision procedure and complete axiomatization for projection temporal logic

https://doi.org/10.1016/j.tcs.2017.09.026Get rights and content

Abstract

To specify and verify the concurrent and reactive systems with the theorem proving approach, a complete axiomatization is formalized for first order projection temporal logic (PTL) with both finite and infinite time. To this end, PTL is restricted to a finite domain, and the syntax, semantics as well as the axiomatization of PTL are presented. Further, the techniques of labeled normal form and labeled normal form graph of PTL formulas are introduced respectively, with which a decision procedure for quantifier free PTL (QFPTL) formulas is given. Moreover, a generalized labeled normal form graph is defined and employed to transform a quantified PTL formula into its equivalent QFPTL formula. Finally, a decision procedure for PTL is formalized and the completeness of the axiomatic system is proved based on the decidability of PTL formulas.

Introduction

Projection Temporal Logic (PTL) [1], [2] is an interval based Temporal Logic (TL) by introducing a new projection construct, (P1,,Pm)prjQ, 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 SP 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 (PQ) by PQ and (PQ) by PQ.

A formula P deduced from the axiom system is called a PTL theorem, 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 (x<2);x=2 is equivalent to false, but there exists an infinite path in its NFG as shown in Fig. 2. Obviously, the NFG of formula (x<2);x=2 is isomorphic to that of formula (x<2).

To solve the problem, for a chop formula P;Q, 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 R˜, a strongly connected subgraph ΠC={R˜1,,R˜h} (h1) is called an F decomposition cycle (F cycle for short) if there exists a DCI [k] such that all the nodes R˜i (1ih) are labeled with the same DCI [k]. Further, if there does not exist such a DCI [k], the decomposition cycle ΠC 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 P. Thus, the satisfiability of formula P can be identified by checking that of formula P 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

References (45)

  • Z. Duan

    An Extended Interval Temporal Logic and a Framing Technique for Temporal Logic Programming

    (1996)
  • Z. Duan

    Temporal Logic and Temporal Logic Programming

    (2005)
  • Z. Duan et al.

    A decision procedure for propositional projection temporal logic with infinite models

    Acta Inform.

    (2008)
  • Z. Duan et al.

    A framed temporal logic programming language

    J. Comput. Sci. Tech.

    (2004)
  • X. Shu et al.

    Extending MSVL with semaphore

  • X. Wang et al.

    MSVL: a typed language for temporal logic programming

    Front. Comput. Sci.

    (2017)
  • Z. Duan et al.

    A unified model checking approach with projection temporal logic

  • E.M. Clarke et al.

    25 years of model checking

  • K. Kanso et al.

    A light-weight integration of automated and interactive theorem proving

    Math. Structures Comput. Sci.

    (2016)
  • K.L. McMillan

    Symbolic Model Checking

    (1993)
  • N. Zhang et al.

    Model checking concurrent systems with MSVL

    Sci. China Ser. F

    (2016)
  • Z. Duan

    Modeling and Analysis of Hybrid Systems

    (2004)
  • Cited by (6)

    • A sound and complete proof system for a unified temporal logic

      2020, Theoretical Computer Science
      Citation Excerpt :

      Moszkowski proposes an axiom system over finite intervals for PITL [23], and later extends the work to infinite intervals [24]. Besides, two proof systems are formalized respectively for PPTL [25] and PTL [26], which constitute a basis for utilizing PTL to verify computer systems with the theorem proving approach [27]. This paper is an extended version of our conference paper [29], and the extension is made mainly in the following aspects.

    • Efficient decision procedure for propositional projection temporal logic

      2020, Theoretical Computer Science
      Citation Excerpt :

      In order to avoid an excessive number of parentheses, the precedence rules are used as shown in Table 1. The techniques of normal form (NF) and normal form graph (NFG) [19,21] are the bases of the decision procedure of PPTL formulas. In this section, we present an efficient method for computing NFs and constructing NFGs of PPTL formulas.

    • A JPSL Based Model Checking Approach for Java Programs

      2023, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    • An Efficient Decision Procedure for Propositional Projection Temporal Logic

      2019, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    • A Proof System for a Unified Temporal Logic

      2019, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

    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).

    View full text