1,250 research outputs found
Strategy-Stealing Is Non-Constructive
In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing. This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists? We prove that this problem is PSPACE-Complete already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing arguments in the current literature
Let Models Speak Ciphers: Multiagent Debate through Embeddings
Discussion and debate among Large Language Models (LLMs) have gained
considerable attention due to their potential to enhance the reasoning ability
of LLMs. Although natural language is an obvious choice for communication due
to LLM's language understanding capability, the token sampling step needed when
generating natural language poses a potential risk of information loss, as it
uses only one token to represent the model's belief across the entire
vocabulary. In this paper, we introduce a communication regime named CIPHER
(Communicative Inter-Model Protocol Through Embedding Representation) to
address this issue. Specifically, we remove the token sampling step from LLMs
and let them communicate their beliefs across the vocabulary through the
expectation of the raw transformer output embeddings. Remarkably, by deviating
from natural language, CIPHER offers an advantage of encoding a broader
spectrum of information without any modification to the model weights. While
the state-of-the-art LLM debate methods using natural language outperforms
traditional inference by a margin of 1.5-8%, our experiment results show that
CIPHER debate further extends this lead by 1-3.5% across five reasoning tasks
and multiple open-source LLMs of varying sizes. This showcases the superiority
and robustness of embeddings as an alternative "language" for communication
among LLMs
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton
cycle, which is applicable to a wide variety of graphs, including relatively
sparse graphs. In contrast to previous criteria, ours is based on only two
properties: one requiring expansion of ``small'' sets, the other ensuring the
existence of an edge between any two disjoint ``large'' sets. We also discuss
applications in positional games, random graphs and extremal graph theory.Comment: 19 page
First-order logic with self-reference
We consider an extension of first-order logic with a recursion operator that
corresponds to allowing formulas to refer to themselves. We investigate the
obtained language under two different systems of semantics, thereby obtaining
two closely related but different logics. We provide a natural deduction system
that is complete for validities for both of these logics, and we also
investigate a range of related basic decision problems. For example, the
validity problems of the two-variable fragments of the logics are shown
coNexpTime-complete, which is in stark contrast with the high undecidability of
two-variable logic extended with least fixed points. We also argue for the
naturalness and benefits of the investigated approach to recursion and
self-reference by, for example, relating the new logics to Lindstrom's Second
Theorem
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on two properties only: one requiring expansion of "small” sets, the other ensuring the existence of an edge between any two disjoint "large” sets. We also discuss applications in positional games, random graphs and extremal graph theor
PYCSP3: Modeling Combinatorial Constrained Problems in Python
In this document, we introduce PYCSP, a Python library that allows us to
write models of combinatorial constrained problems in a simple and declarative
way. Currently, with PyCSP, you can write models of constraint satisfaction
and optimization problems. More specifically, you can build CSP (Constraint
Satisfaction Problem) and COP (Constraint Optimization Problem) models.
Importantly, there is a complete separation between modeling and solving
phases: you write a model, you compile it (while providing some data) in order
to generate an XCSP3 instance (file), and you solve that problem instance by
means of a constraint solver. In this document, you will find all that you need
to know about PYCSP, with more than 40 illustrative models
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