A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [2007]. There the authors show important results on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system.The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n Ω(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [Dantchev et al. 2007]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs.

We study the performance of DPLL algorithms on param-eterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combi-natorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a-clique requires () steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas.

According to the Circularity Gap Report 2020, a mere 8.6% of the global economy wascircular in 2019. The Global Status Report 2018 declares that building construction and operationsaccounted for 36% of global final energy use and 39% of energy–related carbon dioxide (CO2)emissions. The Paris Agreement demands that the building and construction sector decarbonizesglobally by 2050. This requires strategies that minimize the environmental impact of buildingsand practices extending the lifecycle of their constituents within a circular resource flow. To ensurethat eective measures are applied, a suitable method is needed to assess compliance in materials,processes, and design strategies within circular economy principles. The study’s assumption is thatsynthetic and reliable indicators for that purpose could be based on reversibility and durabilityfeatures. The paper provides an overview of building design issues within the circular economyperspective, highlighting the diculty in finding circular technologies which are suitable to enhancebuildings’ service life while closing material loops. The results identify reversibility and durability aspotential indicators for assessing circular building technologies. The next research stage aims to furtherdevelop the rating of circularity requirements for both building technologies and entire buildings.

Abstract. During the last decade, an active line of research in proof complexity has been into the space complexity of proofs and how space is related to other measures. By now these aspects of resolution are fairly well understood, but many open problems remain for the related but stronger polynomial calculus (PC/PCR) proof system. For instance, the space complexity of many standard "benchmark formulas" is still open, as well as the relation of space to size and degree in PC/PCR. We prove that if a formula requires large resolution width, then making XOR substitution yields a formula requiring large PCR space, providing some circumstantial evidence that degree might be a lower bound for space. More importantly, this immediately yields formulas that are very hard for space but very easy for size, exhibiting a size-space separation similar to what is known for resolution. Using related ideas, we show that if a graph has good expansion and in addition its edge set can be partitioned into short cycles, then the Tseitin formula over this graph requires large PCR space. In particular, Tseitin formulas over random 4-regular graphs almost surely require space at least Ω`√n´. Our proofs use techniques recently introduced in . Our final contribution, however, is to show that these techniques provably cannot yield non-constant space lower bounds for the functional pigeonhole principle, delineating the limitations of this framework and suggesting that we are still far from characterizing PC/PCR space.

We introduce a new way of composing proofs in rule-based proof systems that generalizes tree-like and dag-like proofs. In the new definition, proofs are directed graphs of derived formulas, in which cycles are allowed as long as every formula is derived at least as many times as it is required as a premise. We call such proofs circular. We show that, for all sets of standard inference rules, circular proofs are sound. For Frege we show that circular proofs can be converted into tree-like ones with at most polynomial overhead. For Resolution the translation can no longer be a Resolution proof because, as we show, the pigeonhole principle has circular Resolution proofs of polynomial size. Surprisingly, as proof systems for deriving clauses from clauses, Circular Resolution turns out to be equivalent to Sherali-Adams, a proof system for reasoning through polynomial inequalities that has linear programming at its base. As corollaries we get: 1) polynomial-time (LP-based) algorithms that find circular Resolution proofs of constant width, 2) examples that separate circular from dag-like Resolution, such as the pigeonhole principle and its variants, and 3) exponentially hard cases for circular Resolution. the hypotheses, and the formula that needs to be proved displays strictly positive balance. With this interpretation of flows, circular proofs have the appealing flavour of a network in which demands are fulfilled by the hypotheses, and flow towards the conclusions, which produce surplus. Accordingly, and in analogy with the theory of classical network flows, it makes no difference whether the flows are required to be integers or real numbers, and valid flow assignments can be found efficiently, when they exist, by linear programming techniques.While proof-graphs with unrestricted cycles are, in general, unsound, we show that circular proofs are sound. We offer two very different proofs of this fact. The first one is combinatorial in nature and is phrased in the style of traditional soundness proofs in standard proof systems. Concretely, given a truth assignment that falsifies the conclusion, the soundness proof constructs a path of falsified formulas until it reaches a hypothesis, and does so by induction on the total flow-sum of the flow assignment that satisfies the flow-balance condition. The second proof is (semi-)algebraic and is phrased in the style of the duality theorem for linear programming. Concretely, we phrase the existence of a flow assignment that satisfies the flow-balance condition as the feasibility of a linear program, and observe that the infeasibility of its dual witnesses the soundness of the proof. Proof complexity of circular proofsWith all the definitions in place, we proceed to studying the power of circular proofs from the perspective of propositional proof complexity. For Frege systems, which operate with arbitrary propositional formulas through the standard textbook inference rules, we show that circularity adds no power: the circular, dag-like and tree-like variants of Frege polynomially ...