Predicative proof theory of PDL and basic applications

Propositional dynamic logic (PDL) is presented in Schütte-style mode as one-sided semiformal tree-like sequent calculus Seq pdl ω with standard cut rule and the omega-rule with principal formulas [P * ]A. The omega-rule-free derivations in Seq pdl ω are finite (trees) and sequents deducible by these finite derivations are valid in PDL. Moreover the cut-elimination theorem for Seq pdl ω is provable in Peano Arithmetic (PA) extended by transfinite induction up to Veblen's ordinal ϕ ω (0). Hence (by the cutfree subformula property) such predicative extension of PA proves that any given [P * ]-free sequent is valid in PDL iff it is deducible in Seq pdl ω by a finite cut-and omega-rule-free derivation, while PDL-validity of arbitrary star-free sequents is decidable in polynomial space. The former also implies a Herbrand-style conclusion that e.g. a given formula S = P * A ∨ Z for star-free A and Z is valid in PDL iff there is a k ≥ 0 and a cut-and omega-rule-free derivation of sequent A, P 1 A, · · · , P k A, B where P i A is an abbreviation for P · · · P i times A. This eventually leads to PSPACE-decidability of PDL-validity of S, provided that P is atomic and A is in a suitable basic conjunctive normal form. Furthermore we consider star-free formulas A in dual basic disjunctive normal form, and corresponding expansions S = P * A ∨ Z whose PDL-validity problem is known to be EXPTIME-complete. We show that cutfree-derivability in Seq pdl ω (hence PDL-validity) of such S is equivalent to plain validity of a suitable "transparent" quantified boolean formula S. Hence EXPTIME = PSPACE holds true iff the validity problem for any S involved is solvable by a polynomial-space deterministic TM. This may reduce the former problem to a more transparent complexity problem in quantified boolean logic. The whole proof can be formalized in PA extended by transfinite induction along ϕ ω (0)-actually in the corresponding primitive recursive weakening, PRA ϕ ω (0)

Optimal methods for reasoning about actions and plans in multi-agent systems

Cet travail présente une solution au problème du décor inférenciel. Nous réalisons cela en donnant une éducation polynomiale d'un fragment du calcul des situations vers la logique épistémique dynamique (DEL). En suite, une nouvelle méthode de preuve pour DEL, dont la complexité algorithmique est inférieure à celle de la méthode de Reiter pour le calcul de situations, est proposée. Ce travail présente aussi une nouvelle logique pour raisonner sur les actions. Cette logique permet d'exprimer formellement "qu'il existe une suite d'action conduisant au but". L'idée étant que, avec la quantification sur les actions, la planification devient un problème de validité. Une axiomatisation et quelques résultats d'expressivité sont donnés, ainsi qu'une méthode de preuve basée sur les tableaux sémantiques.This work presents a solution to the inferential frame problem. We do so by providing a polynomial reduction from a fragment of situation calculus to espistemic dynamic logic (DEL). Then, a novel proof method for DEL, such that the computational complexity is much lower than that of Retier's proof method for situation caluculs, is proposed. This work also presents a new logic for reasoning about actions. This logic allows to formally express that "there exists a sequence of actions that leads to the goal". The idea is that, with quantification over actions, planning can become a validity problem. An axiomatisation and some expressivity results are provided, as well as a proof method based on sematic tableaux

Propositional Logics Complexity and the Sub-Formula Property

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M-imply) is PSPACE-complete. He showed a polynomially bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Goedel translation from S4 into Intuitionistic Logic, the PSPACE- completeness of M-imply is drawn. The sub-formula principle for a deductive system for a logic L states that whenever F1,...,Fk proves A, there is a proof in which each formula occurrence is either a sub-formula of A or of some of Fi. In this work we extend Statman result and show that any propositional (possibly modal) structural logic satisfying a particular formulation of the sub-formula principle is in PSPACE. If the logic includes the minimal purely implicational logic then it is PSPACE-complete. As a consequence, EXPTIME-complete propositional logics, such as PDL and the common-knowledge epistemic logic with at least 2 agents satisfy this particular sub-formula principle, if and only if, PSPACE=EXPTIME. We also show how our technique can be used to prove that any finitely many-valued logic has the set of its tautologies in PSPACE.Comment: In Proceedings DCM 2014, arXiv:1504.0192

Well structured program equivalence is highly undecidable

We show that strict deterministic propositional dynamic logic with intersection is highly undecidable, solving a problem in the Stanford Encyclopedia of Philosophy. In fact we show something quite a bit stronger. We introduce the construction of program equivalence, which returns the value $\mathsf{T}$ precisely when two given programs are equivalent on halting computations. We show that virtually any variant of propositional dynamic logic has $\Pi_1^1$-hard validity problem if it can express even just the equivalence of well-structured programs with the empty program \texttt{skip}. We also show, in these cases, that the set of propositional statements valid over finite models is not recursively enumerable, so there is not even an axiomatisation for finitely valid propositions.Comment: 8 page

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Visibly Linear Dynamic Logic

We introduce Visibly Linear Dynamic Logic (VLDL), which extends Linear Temporal Logic (LTL) by temporal operators that are guarded by visibly pushdown languages over finite words. In VLDL one can, e.g., express that a function resets a variable to its original value after its execution, even in the presence of an unbounded number of intermediate recursive calls. We prove that VLDL describes exactly the $\omega$-visibly pushdown languages. Thus it is strictly more expressive than LTL and able to express recursive properties of programs with unbounded call stacks. The main technical contribution of this work is a translation of VLDL into $\omega$-visibly pushdown automata of exponential size via one-way alternating jumping automata. This translation yields exponential-time algorithms for satisfiability, validity, and model checking. We also show that visibly pushdown games with VLDL winning conditions are solvable in triply-exponential time. We prove all these problems to be complete for their respective complexity classes.Comment: 25 Page

Compositional Set Invariance in Network Systems with Assume-Guarantee Contracts

This paper presents an assume-guarantee reasoning approach to the computation of robust invariant sets for network systems. Parameterized signal temporal logic (pSTL) is used to formally describe the behaviors of the subsystems, which we use as the template for the contract. We show that set invariance can be proved with a valid assume-guarantee contract by reasoning about individual subsystems. If a valid assume-guarantee contract with monotonic pSTL template is known, it can be further refined by value iteration. When such a contract is not known, an epigraph method is proposed to solve for a contract that is valid, ---an approach that has linear complexity for a sparse network. A microgrid example is used to demonstrate the proposed method. The simulation result shows that together with control barrier functions, the states of all the subsystems can be bounded inside the individual robust invariant sets.Comment: Submitted to 2019 American Control Conferenc