Metric Temporal Logic over Data WordsSeminar Oxford Women in Computer Science

Karin QuaasUniversity of Leipzig

3.3.2015

Verification

- Program P , Specification S- Does the program P satisfy the specification S?

- Verification: Can one automatically verify whether P satisfies S?

- P ...abstract model: finite automata, timed automata, one-counter automata etc.- S...logical formula: FOL, LTL, MTL, etc.

Other Usage

- You may also use the logics I introduce for representing knowledge in a database

Linear Time Temporal Logic (LTL)

P ... a finite set of propositional variables.

! ::= true | p | ¬! | ! ! ! | !U! | X!

where p " P .

Syntactical abbreviations:!1 # !2

··= ¬!1 ! !2, !1 $ !2··= ¬(¬!1 ! ¬!2), F! ··= trueU!, G! ··= ¬F¬!

U ... Until; X ... Next; F ... Finally; G ... Globally

Examples:P = {red, green, button pressed, . . . }

"1 = redUbutton pressed "2 = button pressed # Xgreen"3 = button pressed # Fgreen "4 = G(green ! red)

w =

Verification Problems for LTL

The Satisfiability ProblemGiven: An LTL formulaQuestion: Does there exist a word that satisfies the formula?

The Model Checking ProblemGiven: An LTL formula, and an abstract model that generates/recognizes wordsQuestion: Does every word generated/recognized by the model satisfy the formula?

The Path Checking ProblemGiven: An LTL formula, a wordQuestion: Does the word satisfy the formula?

- Satisfiability and model checking of Kripke structures is PSPACE-complete[Sistla & Clarke, 1985]

- Path checking for LTL is in NC [Kuhtz & Finkbeiner, 2009]

LTL cannot express quantitative information

Examples:

• "3 = button pressed # Fgreen

w =

Finally...but when?

• "2 = button pressed # Xgreen

w =

When the next event occurs...but when?

Goal:We want to express that green holds within 40 seconds after button pressed.

Verification Problems for MTL over Timed Words

The Satisfiability ProblemGiven: An MTL formulaQuestion: Does there exist a timed word that satisfies the formula?

The Model Checking ProblemGiven: An MTL formula, and a timed automaton (recognizing timed words)Question: Does every timed word recognized by the timed automaton

satisfy the formula?

The Path Checking ProblemGiven: An MTL formula, a timed wordQuestion: Does the timed word satisfy the formula?

- Satisfiability is EXPSPACE-complete for timed words over N[Alur & Henzinger, 1993]

- Satisfiability and model checking is decidable [Ouaknine & Worrell, 2005]- Path checking for MTL is in NC [Bundala & Ouaknine, 2014]

Timed Words vs Data Words

Timed words are a special case of data words: monotonically increasing

w =0.3 92.11.8 10.3 12.0 32.0 40.5 68.1 70.2

(Timed word)

Many sequences of data are not monotonically increasing.

MTL over Data Words

Example:

• "6 = F%red $ X>0true $ F=0green&

w =15 2313 18 30 37 30 27 27

Goal:We want to express, e.g., that the invention of LTL led to an increase inthe number of “coding women”, and that the number of “coding women” inthe year when LTL was invented is equal to the number when MTL was invented.

MTL over Data Words

Example:

• "6 = F%red $ X>0true $ F=0green&

w =15 2313 18 30 37 30 27 27

Syntax:

P ... a finite set of propositional variables.

! ::= true | p | ¬! | ! ! ! | !UI! | XI!

where p " P , I ' Z is an interval.

Syntactical abbreviations: FI! ··= trueUI!, GI! ··= ¬FI¬!

Verification Problems for MTL over Data Words

The Satisfiability ProblemGiven: An MTL formulaQuestion: Does there exist a data word that satisfies the formula?

The Model Checking ProblemGiven: An MTL formula, and a one-counter machine (simulating data words)Question: Does every run of the one-counter machine satisfy the formula?

The Path Checking ProblemGiven: An MTL formula, a data wordQuestion: Does the data word satisfy the formula?

- Model checking is undecidable, decidable for deterministic one-countermachines [Q, 2013]

- Satisfiability is undecidable [Carapelle, Feng, Gil, & Q 2014]- Path checking for MTL is in P-complete [Feng, Lohrey, & Q, 2015 (not pub.)]

Summary

SAT MC PCLTL PSPACE( c. PSPACE( c. NCMTL(R!0) not prim.-rec. not prim-rec. NCMTL(Z) undec. undec. P( c.

Quantitative Logics and Automata

- DFG Research Training Group QuantLA in Leipzig/Dresden- Quantitative Logics and Automata, and their applications in verification,knowledge representation, natural language processing, and semi-structureddata (XML)

- O!ers 8 doctoral scholarships starting from October 1, 2015- Application deadline: May 15, 2015