A TV screen shows images of the U.S. President Donald Trump, left, and North Korean leader Kim Jong Un during a news program at the Seoul Railway Station in Seoul, South Korea, Tuesday, May 2, 2017. South Koreans are bewildered by President Donald Trump’s recent use of the term “smart cookie” to refer to current leader Kim Jong Un, and by Trump’s assertion that he’d be “honored” by a possible meeting. (AP Photo/Ahn Young-joon)

A TV screen shows images of the U.S. President Donald Trump, left, and North Korean leader Kim Jong Un during a news program at the Seoul Railway Station in Seoul, South Korea, Tuesday, May 2, 2017. South Koreans are bewildered by President Donald Trump’s recent use of the term “smart cookie” to refer to current leader Kim Jong Un, and by Trump’s assertion that he’d be “honored” by a possible meeting. (AP Photo/Ahn Young-joon)

SEOUL, South Korea (AP) — U.S. presidents looking to halt North Korea’s push for nuclear weapons over the years normally haven’t had very kind words to describe its dictators. George W. Bush, for instance, called the late Kim Jong Il a “pygmy.”

So it’s little wonder that South Koreans are bewildered by President Donald Trump’s use of the term “smart cookie” to refer to current leader Kim Jong Un, and by Trump’s assertion that he’d be “honored” by a possible meeting. Some South Korean media described Trump as a “rugby ball,” an expression that suggests unpredictability.

Trump’s swing from hints of military action to praise for Kim highlights an odd reality: South Koreans, not easily rattled by their nuclear bomb testing neighbor to the north, find themselves increasingly baffled by the new leader of their strongest ally and military protector, the United States.

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The worry is that Trump, a neophyte when dealing with North Korea, perhaps the most successful geopolitical manipulator of big powers in Asia, maybe the world, might fall prey to something the North has long coveted: A bilateral peace deal with the United States that would formally end the Korean War, which stopped in 1953 only after a cease-fire, and remove the nearly 30,000 U.S. troops helping guard the South from North Korean adventurism.

By cutting Seoul out of those negotiations, many here fear, Washington could leave South Korea open to North Korea’s vision of a unified Korean Peninsula with the autocratic Kim family running things from Pyongyang.

There is little chance this would be accomplished peacefully.

This explains the disorientation in Seoul as the Trump administration has bounced from warnings of a potential military confrontation if the North doesn’t change course to talk of restarting negotiations with Pyongyang.

And now, in a CBS interview that aired Sunday, comes a note of admiration.

Trump said that Kim, who assumed office in his 20s, has held power despite efforts by “a lot of people” to take it away.

“So, obviously, he’s a pretty smart cookie,” Trump said.

In comments to Bloomberg News about Kim, Trump said: “If it would be appropriate for me to meet with him, I would absolutely, I would be honored to do it.”

Many in South Korea did a double take.

The Monday headline of a Segye Times article read: “Rugby ball Trump abruptly praises Kim Jong Un as a ‘pretty smart cookie.’”

South Korea’s Yonhap news agency produced a story Tuesday that tried, in part, to explain the phrase “smart cookie.”

It’s becoming almost a cottage industry here: Translating Trump’s Tweets and pronouncements about the North Korean nuclear standoff.

There isn’t universal condemnation of the effect.

Some in Seoul, even at high levels of government, have privately expressed optimism that, after decades of failing to stop North Korea’s march to a nuclear arsenal, an unorthodox approach from the United States could be just what’s needed.

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Yonhap, in another story, questioned whether Trump’s comments on Kim might be a strategy to justify future U.S. moves to strengthen pressure against North Korea.

But the concern in Seoul is less about the approach than about being cut out of the negotiations.

It will be South Koreans, after all, who must live most directly with any settlement with North Korea; it is they who would suffer again, and die in their millions, in another war.

North Korea hasn’t responded to the “smart cookie” comments, but in a commentary in the state-run Minju Joson newspaper, it called Trump’s previous language “bellicose hysteria.”

“The Trump administration would be well advised to learn how humbly the preceding administrations were put in the awkward position of lowering the fist of pressure they had raised before” North Korea, the commentary said.

Some analysts believe that North Korea already has shorter-range nuclear-armed missiles that can target South Korea, Japan and spots in the Pacific. But the North is probably years away from mastering the technology needed to build a missile that can reach the U.S. mainland and mount a miniaturized warhead on it.

The bigger point, as Bradley Martin, an author, journalist and longtime Asia commentator, wrote in an Asia Times piece on Monday that urged Americans not to panic over North Korea, is that it has its sights on Seoul, not Washington.

“They believe they can beat the South Koreans — perhaps even without using their nuclear weapons — if only they can maneuver the U.S. to stay out of that fight,” Martin wrote.

It’s hard to think of something that would give more truth to Trump’s “smart cookie” description than if Kim Jong Un is able, after all these years, to drive a wedge between Seoul and Washington.

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Foster Klug, AP’s bureau chief in Seoul, has covered the Koreas since 2005. Follow him on Twitter at twitter.com/apklug