scholarly journals Symmetric Circuits for Rank Logic

2022 ◽  
Vol 23 (1) ◽  
pp. 1-35
Author(s):  
Anuj Dawar ◽  
Gregory Wilsenach
Keyword(s):  
Fixed Point ◽  
Expressive Power ◽  
Broad Framework ◽  
The Individual ◽  
Rank Of A Matrix

Fixed-point logic with rank (FPR) is an extension of fixed-point logic with counting (FPC) with operators for computing the rank of a matrix over a finit field. The expressive power of FPR properly extends that of FPC and is contained in P, but it is not known if that containment is proper. We give a circuit characterization for FPR in terms of families of symmetric circuits with rank gates, along the lines of that for FPC given by Anderson and Dawar in 2017. This requires the development of a broad framework of circuits in which the individual gates compute functions that are not symmetric (i.e., invariant under all permutations of their inputs). This framework also necessitates the development of novel techniques to prove the equivalence of circuits and logic. Both the framework and the techniques are of greater generality than the main result.

scholarly journals Dynamics in Braess Paradox with Nonimpulsive Commuters

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Arianna Dal Forno ◽  
Ugo Merlone ◽  
Viktor Avrutin
Keyword(s):  
Fixed Point ◽  
Global Attractivity ◽  
Unique Fixed Point ◽  
Social Dilemma ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Cost Difference ◽  
Braess Paradox ◽  
The Individual ◽  
The Cost

In Braess paradox the addiction of an extra resource creates a social dilemma in which the individual rationality leads to collective irrationality. In the literature, the dynamics has been analyzed when considering impulsive commuters, i.e., those who switch choice regardless of the actual difference between costs. We analyze a dynamical version of the paradox with nonimpulsive commuters, who change road proportionally to the cost difference. When only two roads are available, we provide a rigorous proof of the existence of a unique fixed point showing that it is globally attracting even if locally unstable. When a new road is added the system becomes discontinuous and two-dimensional. We prove that still a unique fixed point exists, and its global attractivity is numerically evidenced, also when the fixed point is locally unstable. Our analysis adds a new insight in the understanding of dynamics in social dilemma.

On fixed-point logic with counting

2000 ◽  
Vol 65 (2) ◽  
pp. 777-787 ◽  
Author(s):  
Jörg Flum ◽  
Martin Grohe
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Planar Graphs ◽  
Expressive Power ◽  
Descriptive Complexity ◽  
Main Motivation ◽  
First Order ◽  
Finite Structures ◽  
Tree Width

One of the fundamental results of descriptive complexity theory, due to Immerman [13] and Vardi [18], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered finite structures by a logic in a similar way.The most obvious shortcoming of fixed-point logic itself on unordered structures is that it cannot count. Immerman [14] responded to this by adding counting constructs to fixed-point logic. Although it has been proved by Cai, Fürer, and Immerman [1] that the resulting fixed-point logic with counting, denoted by IFP+C, still does not capture all of polynomial time, it does capture polynomial time on several important classes of structures (on trees, planar graphs, structures of bounded tree-width [15, 9, 10]).The main motivation for such capturing results is that they may give a better understanding of polynomial time. But of course this requires that the logical side is well understood. We hope that our analysis of IFP+C-formulas will help to clarify the expressive power of IFP+C; in particular, we derive a normal form. Moreover, we obtain a problem complete for IFP+C under first-order reductions.

scholarly journals Meaningful call combinations and compositional processing in the southern pied babbler

2016 ◽  
Vol 113 (21) ◽  
pp. 5976-5981 ◽  
Author(s):  
Sabrina Engesser ◽  
Amanda R. Ridley ◽  
Simon W. Townsend
Keyword(s):  
Expressive Power ◽  
Evolutionary Origins ◽  
Naturally Occurring ◽  
Dangerous Situation ◽  
Vocal System ◽  
Group Members ◽  
The Individual ◽  
Work Supports ◽  
Viable Mechanism ◽  
Order Structures

Language’s expressive power is largely attributable to its compositionality: meaningful words are combined into larger/higher-order structures with derived meaning. Despite its importance, little is known regarding the evolutionary origins and emergence of this syntactic ability. Although previous research has shown a rudimentary capability to combine meaningful calls in primates, because of a scarcity of comparative data, it is unclear to what extent analog forms might also exist outside of primates. Here, we address this ambiguity and provide evidence for rudimentary compositionality in the discrete vocal system of a social passerine, the pied babbler (Turdoides bicolor). Natural observations and predator presentations revealed that babblers produce acoustically distinct alert calls in response to close, low-urgency threats and recruitment calls when recruiting group members during locomotion. On encountering terrestrial predators, both vocalizations are combined into a “mobbing sequence,” potentially to recruit group members in a dangerous situation. To investigate whether babblers process the sequence in a compositional way, we conducted systematic experiments, playing back the individual calls in isolation as well as naturally occurring and artificial sequences. Babblers reacted most strongly to mobbing sequence playbacks, showing a greater attentiveness and a quicker approach to the loudspeaker, compared with individual calls or control sequences. We conclude that the sequence constitutes a compositional structure, communicating information on both the context and the requested action. Our work supports previous research suggesting combinatoriality as a viable mechanism to increase communicative output and indicates that the ability to combine and process meaningful vocal structures, a basic syntax, may be more widespread than previously thought.

scholarly journals On the expressive power of monadic least fixed point logic

2006 ◽  
Vol 350 (2-3) ◽  
pp. 325-344 ◽  
Author(s):  
Nicole Schweikardt
Keyword(s):  
Fixed Point ◽  
Expressive Power

scholarly journals Cognitive cascades: How to model (and potentially counter) the spread of fake news

PLoS ONE ◽  
2022 ◽  
Vol 17 (1) ◽  
pp. e0261811
Author(s):  
Nicholas Rabb ◽  
Lenore Cowen ◽  
Jan P. de Ruiter ◽  
Matthias Scheutz
Keyword(s):  
Public Opinion ◽  
Social Science ◽  
Network Science ◽  
Cognitive Model ◽  
Population Level ◽  
Expressive Power ◽  
Belief Function ◽  
Computational Social Science ◽  
Model Sets ◽  
The Individual

Understanding the spread of false or dangerous beliefs—often called misinformation or disinformation—through a population has never seemed so urgent. Network science researchers have often taken a page from epidemiologists, and modeled the spread of false beliefs as similar to how a disease spreads through a social network. However, absent from those disease-inspired models is an internal model of an individual’s set of current beliefs, where cognitive science has increasingly documented how the interaction between mental models and incoming messages seems to be crucially important for their adoption or rejection. Some computational social science modelers analyze agent-based models where individuals do have simulated cognition, but they often lack the strengths of network science, namely in empirically-driven network structures. We introduce a cognitive cascade model that combines a network science belief cascade approach with an internal cognitive model of the individual agents as in opinion diffusion models as a public opinion diffusion (POD) model, adding media institutions as agents which begin opinion cascades. We show that the model, even with a very simplistic belief function to capture cognitive effects cited in disinformation study (dissonance and exposure), adds expressive power over existing cascade models. We conduct an analysis of the cognitive cascade model with our simple cognitive function across various graph topologies and institutional messaging patterns. We argue from our results that population-level aggregate outcomes of the model qualitatively match what has been reported in COVID-related public opinion polls, and that the model dynamics lend insights as to how to address the spread of problematic beliefs. The overall model sets up a framework with which social science misinformation researchers and computational opinion diffusion modelers can join forces to understand, and hopefully learn how to best counter, the spread of disinformation and “alternative facts.”

The expressive power of fixed-point logic with counting

1996 ◽  
Vol 61 (1) ◽  
pp. 147-176 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Fixed Point ◽  
Polynomial Time ◽  
Natural Extension ◽  
Expressive Power ◽  
Generic Model ◽  
Point Counting ◽  
Ordered Structures ◽  
Relational Structures ◽  
Positive Assessment

AbstractWe study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count.To this end an isomorphism preserving (‘generic’) model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the arithmetical side of the algorithms involved.In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics and the corresponding pebble games.The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants with the following properties:• The invariants themselves can be evaluated in polynomial time.• A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants.• The invariant r classifies all finite relational structures exactly up to equivalence with respect to the logic We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the : It corresponds exactly to the polynomial time computable families (φn)n ∈ ω in these logics.Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindström quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting.

Beyond first-order logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Fixed Point ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Compactness Theorem ◽  
First Order Logic ◽  
First Order ◽  
Skolem Theorem ◽  
Additional Rule ◽  
Second Order Logic

We consider various extensions of first-order logic. Informally, a logic 𝓛 is an extension of first-order logic if every sentence of first-order logic is also a sentence of 𝓛. We also require that 𝓛 is closed under conjunction and negation and has other basic properties of a logic. In Section 9.4, we list the properties that formally define the notion of an extension of first-order logic. Prior to Section 9.4, we provide various natural examples of such extensions. In Sections 9.1–9.3, we consider, respectively, second-order logic, infinitary logics, and logics with fixed-point operators. We do not provide a thorough treatment of any one of these logics. Indeed, we could easily devote an entire chapter to each. Rather, we define each logic and provide examples that demonstrate the expressive power of the logics. In particular, we show that none of these logics has compactness. In the final Section 9.4, we prove that if a proper extension of first-order logic has compactness, then the Downward Löwenhiem–Skolem theorem must fail for that logic. This is Lindstrom’s theorem. The Compactness theorem and Downward Löwenheim–Skolem theorem are two crucial results for model theory. Every property of first-order logic from Chapter 4 is a consequence of these two theorems. Lindström’s theorem implies that the only extension of first-order logic possessing these properties is first-order logic itself. Second-order logic is the extension of first-order logic that allows quantification of relations. The symbols of second-order logic are the same symbols used in first-order logic. The syntax of second-order logic is defined by adding one rule to the syntax of first-order logic. The additional rule makes second-order logic far more expressive than first-order logic. Specifically, the syntax of second-order logic is defined as follows. Any atomic first-order formula is a formula of second-order logic. Moreover, we have the following four rules: (R1) If φ is a formula then so is ￢φ. (R2) If φ and ψ are formulas then so is φ ∧ ψ. (R3) If φ is a formula, then so is ∃x φ for any variable x.

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