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Volume 21, Issue 2
2025
1. Strong negation in the theory of computable functionals TCF
We incorporate strong negation in the theory of computable functionals TCF, a common extension of Plotkin's PCF and Gödel's system T, by defining simultaneously strong negation AN of a formula A and strong negation PN of a predicate P in TCF. As a special case of the latter, we get strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of double negation elimination for strong negation in TCF. We introduce the so-called tight formulas of TCF i.e., formulas implied by the weak negation of their strong negation, and the relative tight formulas. We present various case-studies and examples, which reveal the naturality of our definition of strong negation in TCF and justify the use of TCF as a formal system for a large part of Bishop-style constructive mathematics.
2. Safety and Liveness of Quantitative Properties and Automata
Safety and liveness stand as fundamental concepts in formal languages, playing a key role in verification. The safety-liveness classification of boolean properties characterizes whether a given property can be falsified by observing a finite prefix of an infinite computation trace (always for safety, never for liveness). In the quantitative setting, properties are arbitrary functions from infinite words to partially-ordered domains. Extending this paradigm to the quantitative domain, where properties are arbitrary functions mapping infinite words to partially-ordered domains, we introduce and study the notions of quantitative safety and liveness. First, we formally define quantitative safety and liveness, and prove that our definitions induce conservative quantitative generalizations of both the safety-progress hierarchy and the safety-liveness decomposition of boolean properties. Consequently, like their boolean counterparts, quantitative properties can be min-decomposed into safety and liveness parts, or alternatively, max-decomposed into co-safety and co-liveness parts. We further establish a connection between quantitative safety and topological continuity and provide alternative characterizations of quantitative safety and liveness in terms of their boolean analogs. Second, we instantiate our framework with the specific classes of quantitative properties expressed by automata. These quantitative automata contain finitely many states and rational-valued transition […]
3. A Game of Pawns
We introduce and study pawn games, a class of two-player zero-sum turn-based graph games. A turn-based graph game proceeds by placing a token on an initial vertex, and whoever controls the vertex on which the token is located, chooses its next location. This leads to a path in the graph, which determines the winner. Traditionally, the control of vertices is predetermined and fixed. The novelty of pawn games is that control of vertices changes dynamically throughout the game as follows. Each vertex of a pawn game is owned by a pawn. In each turn, the pawns are partitioned between the two players, and the player who controls the pawn that owns the vertex on which the token is located, chooses the next location of the token. Control of pawns changes dynamically throughout the game according to a fixed mechanism. Specifically, we define several grabbing-based mechanisms in which control of at most one pawn transfers at the end of each turn. We study the complexity of solving pawn games, where we focus on reachability objectives and parameterize the problem by the mechanism that is being used and by restrictions on pawn ownership of vertices. On the positive side, even though pawn games are exponentially-succinct turn-based games, we identify several natural classes that can be solved in PTIME. On the negative side, we identify several EXPTIME-complete classes, where our hardness proofs are based on a new class of games called Lock & Key games, which may be of independent […]