A note on tractability and artificial intelligence

The recognition that human minds/brains are finite systems with limited resources for computation has led researchers in Cognitive Science to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. As also artificial intelligence (AI) in its attempt to recreate intelligence and capacities inspired by the human mind is dealing with finite systems, transferring the Tractable Cognition thesis into this new context and adapting it accordingly may give rise to insights and ideas that can help in progressing towards meeting the goals of the AI endeavor

When almost is not even close: Remarks on the approximability of HDTP

A growing number of researchers in Cognitive Science advocate the thesis that human cognitive capacities are constrained by computational tractability. If right, this thesis also can be expected to have far-reaching consequences for work in Artificial General Intelligence: Models and systems considered as basis for the development of general cognitive architectures with human-like performance would also have to comply with tractability constraints, making in-depth complexity theoretic analysis a necessary and important part of the standard research and development cycle already from a rather early stage. In this paper we present an application case study for such an analysis based on results from a parametrized complexity and approximation theoretic analysis of the Heuristic Driven Theory Projection (HDTP) analogy-making framework

Amortized Circuit Complexity, Formal Complexity Measures, and Catalytic Algorithms

We study the amortized circuit complexity of boolean functions. Given a circuit model F and a boolean function f : {0,1}n → {0,1}, the F-amortized circuit complexity is defined to be the size of the smallest circuit that outputs m copies of f (evaluated on the same input), divided by m, as m → ∞. We prove a general duality theorem that characterizes the amortized circuit complexity in terms of 'formal complexity measures'. More precisely, we prove that the amortized circuit complexity in any circuit model composed out of gates from a finite set is equal to the pointwise maximum of the family of 'formal complexity measures' associated with F. Our duality theorem captures many of the formal complexity measures that have been previously studied in the literature for proving lower bounds (such as formula complexity measures, submodular complexity measures, and branching program complexity measures), and thus gives a characterization of formal complexity measures in terms of circuit complexity. We also introduce and investigate a related notion of catalytic circuit complexity, which we show is 'intermediate' between amortized circuit complexity and standard circuit complexity, and which we also characterize (now, as the best integer solution to a linear program). Finally, using our new duality theorem as a guide, we strengthen the known upper bounds for non-uniform catalytic space, introduced by Buhrman et. al [1] (this is related to, but not the same as, our notion of catalytic circuit size). Potechin [2] proved that for any boolean function f : {0,1}n → {0,1}, there is a catalytic branching program computing m=22n-1 copies of f with total size O(mn) - that is, linear size per copy - refuting a conjecture of Girard, Koucký and McKenzie [3]. Potechin then asked if the number of copies m can be reduced while retaining the amortized upper bound. We make progress on this question by showing that if f has degree d when represented as polynomial over F2, then there is a catalytic branching program computing m=2(n\≤d) copies of f with total size O(mn).</p

Guest Column

We survey lower-bound results in complexity theory that have been obtained via newfound interconnections between propositional proof complexity, boolean circuit complexity, and query/communication complexity. We advocate for the theory of total search problems (TFNP) as a unifying language for these connections and discuss how this perspective suggests a whole programme for further research.</jats:p

Separations in proof complexity and TFNP

It is well-known that Resolution proofs can be efficiently simulated by Sherali– Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS). These results have consequences for total NP search problems. First, we characterise the classes PPADS, PPAD, SOPL by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, PLS ̸⊆ PPP, SOPL ̸⊆ PPA, and EOPL ̸⊆ UEOPL. In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical TFNP classes introduced in the 1990s

Further collapses in TFNP

We show EOPL = PLS ∩ PPAD. Here the class EOPL consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse CLS = PLS ∩ PPAD by Fearnley et al. (STOC 2021). We also prove a companion result SOPL = PLS ∩ PPADS, where SOPL is the class associated with the Sink-of-Potential-Line problem