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Satisfiability Degree Theory for Temporal Logic∗ SATISFIABILITY DEGREE THEORY FOR TEMPORAL LOGIC∗ Jian Luo, Guiming Luo and Mo Xia Key Laboratory for Information System Security, Ministry of Education, Tsinghua National Laboratory for Information Science and Technology, School of Software, Tsinghua University, Beijing 100084, China Keywords: Satisfiability degree, Propositional logic, Temporal logic, Reasoning. Abstract: The truth value of propositional logic is not capable of representing the real word full of complexity and diver- sity. The requirements of the proposition satisfiability are reviewed in this paper. Every state is labeled with a vector, which is defined by the proposition satisfiability degree. The satisfiability degree for temporal logic is proposed based on the vector of satisfiability degree. It is used to interpret the truth degree of the temporal logic instead of true or false. A sound and precise reasoning system for temporal logic is established and the computation is given. One example of a leadership election is included to show that uncertain information can be quantized by the satisfiability degree. 1 INTRODUCTION Sometimes, given the premise is true, we want to deduce the truth degree of a considered conclu- The idea of temporal logic (Mattolini and Nesi, 2000) sion. The conditional satisfiability degree was pro- is that a formula is not statically true or false in a posed in (Luo and Yin, 2009), to quantitatively repre- time model. Instead, the models of temporal logi- sent the deductive reasoning, which is based on if the cal contain several states and a formula can be true satisfiability degree of premise is given, we deduce in some states and false in others. The formulas may the satisfiability degree of the conclusion. change their truth values as the system evolves from There are many algorithms to compute the sat- state to state, but the truth values of the formulas are isfiability degree, such as the backtracking algo- true or false. Sometimes, a state partially satisfies rithm (Yin et al., 2009), the satisfiability degree com- a formula, so it is not absolutely true or false, and putation based on CNF (Hu et al., 2009), the algo- the semantic of the temporal logic, which is based rithm based on binary decision diagrams (Luo and the classical Boolean logic, cannot interpret this case. Yin, 2009) and the propositional matrix search algo- Thus, the world requires new ways to express uncer- rithm (Luo and Luo, 2010). Once a propositional for- tainty. Many studies have used non-classical logic, mula is given, its satisfiability degree can be precisely such as fuzzy logic (Bergmann, 2008), probabilistic computed using those algorithms. Thus, this paper logic (Raedt and Kersting, 2003), modal logic, etc. only focuses on the performance and properties of sat- Satisfiability degree, a new precise logic presen- isfiability degree based on the temporal logic. tation, was proposed in (Luo and Yin, 2009). It de- Because the temporal logic is based on proposi- scribes the extent to which a proposition is true based tional logic and temporal connectives, the truth value on the truth table by finding out the proportion of sat- of a temporal formula can be precise interpreted by isfiable interpretations. Unlike fuzzy logic and proba- satisfiability degree. Thus, if there are only several bilistic logic, satisfiability degree does not need mem- models are available for the concerned formula, we bership function or distribution function and it is de- can choose the model with maximum satisfiability termined by the proposition itself. Moreover, satis- degree. Sometimes, a model checker may not find fiability degree extends the concepts of satisfaction counterexamples but it does means the system cannot and contradictory propositions in Boolean logic and be applied to some domains, but satisfiability degree truth values of propositions are precisely interpreted can provide us a quantitative analysis of model check- as their satisfiability degrees. ing (Kang and Park, 2005). ∗This work is supported by the Funds NSFC 60973049, 60635020, and TNList cross-discipline foundations. Luo J., Luo G. and Xia M.. 497 SATISFIABILITY DEGREE THEORY FOR TEMPORAL LOGIC. DOI: 10.5220/0003672804970500 In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FCTA-2011), pages 497-500 ISBN: 978-989-8425-83-6 Copyright c 2011 SCITEPRESS (Science and Technology Publications, Lda.) FCTA 2011 -InternationalConferenceonFuzzyComputationTheoryandApplications 2 SATISFIABILITY DEGREE Definition 2. Let M = (S;R;V) be a model. A path r in M is an infinite sequence s0 · s1 · s2 ::: of states Thus, we use satisfiability degree to define the truth such that, for 8i ≥ 0;(si;si+1) 2 R. Consider the path r = s0 · s1 · s2 :::It represents a values of propositions in states instead of ”true” or i false. Sometimes, it is difficult to choose some atomic possible future of our system. We write r for the 3 descriptions, because some properties cannot be sub- suffix starting at si, e.g.r is s3 · s4 ::: divided into. We define a new temporal model on n ∗ propositions set and use satisfiability degree to inter- 2.2 Satisfiability Degree for CTL pret the truth values of those propositions. The syntax of CTL∗ involves two classes of formulas, state formula and path formula. 2.1 Temporal Model • state formula, which are evaluated in states: Definition 1. Let S be a set of n propositions. A j ::= p j :j j j ^ j j A[a] j E[a] (2) model M over S is a triple M = (S;R;V) where: Where p 2 S and a is any path formula; and • S is a set of states; • path formula, which are evaluated along paths: • R ⊂ S × S is a total transition relation, i.e., any a ::= j j :a j a^a j a U a j G a j F a j X a (3) state s 2 S, there is a state s0 2 S, s.t. (s;s0) 2 R; The semantics (Huth and Ryan, 2005) of CTL∗ was defined by a satisfaction relation, denoted by , • V : Sn ×S ! [0;1]n is a function that mapping the which is characterized as the least relation on the satisfiability degree of each proposition in s . paths. Since the interpretation of a CTL∗ formula Thus, our model has a collection of state S, a rela- varies over states, we denote the satisfiability degree tion R, saying how the system can move from state of a CTL∗ formula j at state s according to the model M to state, and, associated with each state s, one has the M as fs (j). n-dimension vector V(s), components of which are Definition 3. For a model M = (S;R;V), the satisfi- satisfiability degree, denoted as f (p;s): ability degree of a CTL∗ formula at state s is defined T inductively as: V(s) = ( f (p1;s); f (p2;s);::: f (pn;s)) (1) M • 8 j 2 P; fs (j) = f (j;s), where P is the set of Where f (pi;s) is the satisfiability degree of Boolean formulas. pi in state s. We can express all the in- M M • fs (:j) = 1 − fs (j) formation about a model M using a directed M M M graph and Figure 1 bellow shows us an example, • fs (j1 ^j2) = min( fs (j1); fs (j2)), j1 ^j2 2= P M M ∗ where S = fs0;s1;s2g;S = fp;q;rg and V(s0) = • fs (X a) = fs∗ (a), where s is the next state. (0:5;0:6;0:2);V(s1) = (0:2;0:3;0:1) and V(s2) = M M • fs (F a) = sup fs∗ (a), where r start with s and (0:8;0;1). 8s∗2r s∗ is any state on r. M M 0.5 s0 • f (G a) = inf f ∗ (a), where r start with s and s ∗ s 0.6 8s 2r ∗ 0.2 s is any state on r. M • fs (a1Ua2) M M M = sup ( fs (a2); min ( fs (a1); fs (a2))), where 0 0≤i≤ j i j 8s j2r 0.2 0.8 r = s0 · s1 · s2 :::, and s0 = s, starts with s. 0.3 0 M M 0.1 s1 s2 1 • fs (A[a]) = inf fs (a), where r starts with s. 8r M M Figure 1: A temporal model based on satisfiability degree. • fs (E[a]) = sup fs (a), where r starts with s. 8r The truth value of each propositional formula is Note that, if all the propositions in set S are atomic determined by the state s with respect to the satisfia- propositions, then their satisfiability degree are 0 or bility degree vector V(s). Suppose M = (S;R;V) be a 1. That is the classical temporal model referred to the model, s 2 S, and j a CTL∗ formula. We have the fol- paper (Huth and Ryan, 2005). Therefore, the models low conclusions: If M;s j= j if, and only if we have we define are more expressive than that of classical f M(j) = 1. temporal models. s 498 SATISFIABILITY DEGREETHEORYFORTEMPORALLOGIC 3 A REASONING SYSTEM Theorem 2. Inference rules (5)-(12) are sound. Proof: the rule L: is trivial. Let w1;w2 2 W be in- A signed temporal formula can is a tuple (j; f ) where terpretations such that p(w1) = 1, and q(w2) = 1, then f is the satisfiability degree of j . The proposed rea- w = (w1;w2) 2 W × W; p ^ q(w) = 1. By the Defini- soning system is a pair n = (A;L) where A is a set of tion of satisfiability degree, we have temporal logic axioms, and L is a collection of infer- jWp × Wqj ence rules.
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