Show that exist a finite set of clauses F in first-order logic that Res*(F) is infinite

I’m kind of desperate at this point about this question.

A predicate-logic resolution derivation of a clause $ C$ from a set of clauses $ F$ is a sequence of clauses $ C_1,\dots,C_m$ , with $ C_m = C$ such that each $ C_i$ is either a clause of $ F$ (possibly with the variables renamed) or follows by a resolution step from two preceding clauses $ C_j ,C_k$ , with $ j, k < i$ . We write $ \operatorname{Res}^*(F)$ for the set of clauses $ C$ such that there is a derivation of $ C$ from $ F$ .

The question is to give an example of a finite set of clauses $ F$ in first-order logic such that $ \operatorname{Res}^*(F)$ is infinite.

Any help would be appreciated!

Show that infinite decidable sets $A$ and $B$ exist

I am stuck in this problem, so any help is appreciated. The problem asks to show that there exists decidable sets $ A$ and $ B$ such that $ A \leq_{m}^{p} B$ but $ B \not \leq_{m}^{p} A$ , and that $ A$ , $ B$ and $ \bar{A}$ and $ \bar{B}$ are infinite.

Here, $ \leq_{m}^{p}$ refers to many-one polynomial time reducibility…. I have a hunch that this may have something to do with letting $ A$ be a decidable set such that $ A \in EXP$ , but $ B \in P$ , so that the reduction cannot be done in polynomial time.

Can one determinize finite automata over infinite trees?

I’m currently considering deterministic, nondeterministic, universal, and alternating automata over infinite words and trees, with Büchi, co-Büchi, Muller, Rabin, Streett, or parity acceptance conditions.

I know that over words, all these automata accept the same languages, except deterministic and universal Büchi automata, as well as deterministic and nondeterministic co-Büchi automata.

I also know that over trees, nondeterministic and alternating Muller, Rabin, Streett, and parity automata accept the same languages, and strictly more languages than nondeterministic and alternating Büchi automata.

But I hardly know anything about deterministic tree automata, and I’m having a hard time finding more in the literature. And this captures my main question, whether one can determinize nondeterministic automata over infinite trees.

After all, they are definitely used. For example, in reactive LTL synthesis, we often convert the formula to a nondeterministic Büchi word automaton, then to a deterministic parity word automaton, from which we derive a deterministic parity tree automaton.

Also note that I didn’t mention the co-Büchi case when it comes to trees. I just know that, again in reactive LTL synthesis, we can alternatively convert the negated formula to a nondeterministic Büchi word automaton, then to a universal co-Büchi word automaton for the original formula, from which we can derive a universal co-Büchi tree automaton. But that alone doesn’t really say anything about the relation of deterministic parity tree automata and universal co-Büchi tree automata, or does it?

Find an infinite set of strings that are compressable better than in $O(\log n)$ space

The task is to find an infinite set of strings $ a_1,a_2\ldots$ , where $ |a_{i+1}|>|a_i|$ and to find a compression algorithm $ f$ for these strings, such that $ |f(a_i)|=o(\log_2 |a_i|)$ with $ i\to\infty$ .

I have considered a set of strings with minimal entropy: $ a_1=b, a_2=bb$ , etc. The optimal compression algorithm for these strings is $ f : a_i\mapsto |a_i|$ , but this compresses each string only to $ O(\log i)$ space.

Sending infinite frames in One-Bit Sliding WIndow Protocol?

I am reading Computer Networks by Andrew S. Tanenbaum and I wonder if there is a mistake in the protocol as I can’t find any solution for the following scenario.

Suppose the transmitter (A) has only one frame (X) to send, and the receiver (B) has nothing to send. B receives X and sends a frame with an empty info field and with an acknowledgement from X to A. But with A, this triggers the event frame_arrival, which in turn triggers A to send an “acknowledgement” to B, again without any packet (because there was only one). And so that ping-pong of totally useless frames goes on and on. Is that an error in the protocol, in the pseudocode of the protocol, or am I wrong?

I suppose that sequence number frame_expected of A will not agree with the second acknowledgement of B.

One-bit sliding protocol

Can I use Two Immovable Rods as an infinite ladder or monkey bars?

There are already several questions concerning immovable rods, but I thought I’d add this to the list.

Consider a stealthy and inventive assassin. She’s on a mission to infiltrate the castle and kill a visiting duke. The assassin has to cross a wide moat and scale the castle wall, but for unspecified reasons she can’t use her normal tools.

Luckily, our intrepid antihero has gotten her hands on not one, but two immovable rods.

Can she fix one rod in place, hang from it one-handed, and fix the other rod ahead of her, repeating this process to create a set of monkey bars in order to cross the moat?

Once across, can she do a similar process, but vertically, in order to create a ladder?

Is there necessarily an infinite number of inputs to any given output in a crypto hash function? [migrated]

This might be a very easy question. Let’s consider cryptograhic hash functions with the usual properties, weak and strong collision resistance and preimage resistance.

For any given output, obviously there are multiple inputs. But is that necessarily an infinite number of preimages, for any given hash value?

How would I go about giving a formal proof that there exists no crypto hash function h() such that there is a given value v = h(m*) for which the possible set of inputs m* is finite? Would this necessarily break collision resistance?

What is the most optimal build for keeping an infinite Crab Swarm apocalypse at bay?

My friends and I were discussing a meme we saw when our imaginations took us way too far, and now I’m curious about how many Crab Swarms it would take to kill the most efficient Crab Swarm killer, and who the most efficient Crab Swarm killer could be.

Setup:

You are an adventurer who happened upon some hijinks and now suddenly, you’re in the middle of a Crab Swarm apocalypse. That is,

  • You’re in the center of a 20sqx20sq (100ft x 100ft) flat square plain;
  • You have one week to prepare;
    • For purposes of this theoretical, you may assume you have any necessary resources in infinite amounts.
  • and, after that week, Crab Swarms begin to appear from all directions in an infinite stream.
    • There is nothing special about any individual Crab Swarm; each is exactly as described.
    • They are all hostile against you, specifically, and will do anything within their crablike powers to murder you.
    • The stream will not be stopped and cannot be halted until you are dead.

Specifically, I am interested in two scenarios: a level 5 adventurer (because Crab Swarms are CR4, so one level 5 adventurer should theoretically be able to defeat a Crab Swarm); and any arbitrary level 20 adventurer (for whom 250 CR4 Crab Swarms would make a CR20 encounter). What are the most effective builds at these levels for eliminating Crab Swarms and/or prolonging survival?

Caveats:

Spells like Teleport or maintaining indefinite amounts of Rope Tricks, while technically valid for the definition of prolonging survival, are not in the spirit of the scenario, and shouldn’t be considered. Running away is not an option.

By “murder”, I don’t necessarily believe that killing is required. Simply teleporting them to another Plane via skills like, say, Initiate of the Seventhfold Veil’s Violet Veil skill is an equally valid strategy (as well as being hilarious in concept).

I am open to basically any valid Pathfinder solution to this problem, from published books. Psionics, Path of War, whatever, bring it on.