The Challenge of Unifying Semantic and Syntactic Inference Restrictions

While syntactic inference restrictions don't play an important role for SAT, they are an essential reasoning technique for more expressive logics, such as first-order logic, or fragments thereof. In particular, they can result in short proofs or model representations. On the other hand, semantically guided inference systems enjoy important properties, such as the generation of solely non-redundant clauses. I discuss to what extend the two paradigms may be unifiable

Knowledge and action: how should we combine their logics?

International audienceKnowledge and action: how should we combine their logics

Scavenger 0.1: A Theorem Prover Based on Conflict Resolution

This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving new clauses by (a first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201

SCL with Theory Constraints

We lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution inferences by a partial model assumption instead of an a priori fixed order as done for instance in hierarchic superposition. The model representation consists of ground background theory literals and ground foreground first-order literals. One major advantage of the model guided approach is that clauses generated by SCL(T) enjoy a non-redundancy property that makes expensive testing for tautologies and forward subsumption completely obsolete. SCL(T) is a semi-decision procedure for pure clause sets that are clause sets without first-order function symbols ranging into the background theory sorts. Moreover, SCL(T) can be turned into a decision procedure if the considered combination of a first-order logic modulo a background theory enjoys an abstract finite model property.Comment: 22 page

{SCL} with Theory Constraints

We lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution inferences by a partial model assumption instead of an a priori fixed order as done for instance in hierarchic superposition. The model representation consists of ground background theory literals and ground foreground first-order literals. One major advantage of the model guided approach is that clauses generated by SCL(T) enjoy a non-redundancy property that makes expensive testing for tautologies and forward subsumption completely obsolete. SCL(T) is a semi-decision procedure for pure clause sets that are clause sets without first-order function symbols ranging into the background theory sorts. Moreover, SCL(T) can be turned into a decision procedure if the considered combination of a first-order logic modulo a background theory enjoys an abstract finite model property

The Challenge of Unifying Semantic and Syntactic Inference Restrictions

International audienceWhile syntactic inference restrictions don't play an important role for SAT, they are an essential reasoning technique for more expressive logics, such as first-order logic, or fragments thereof. In particular, they can result in short proofs or model representations. On the other hand, semantically guided inference systems enjoy important properties, such as the generation of solely non-redundant clauses. I discuss to what extend the two paradigms may be unifiable

Ordered Resolution with Straight Dismatching Constraints

International audienceWe present a sound and complete ordered resolution calculus for first-order clauses with straight dismatching constraints. The extended clause language is motivated by our first-order theorem proving approach through approximation and refinement. Using a clause language with straight dismatching constraints, single refinement steps do not result in a worst-case quadratic blowup in the number of clauses anymore. The refinement steps can now be represented by replacing one input clause with two equivalent clauses. We show soundness and completeness of ordered resolution with straight dismatching constraints. All needed operations on straight dismatching constraints take linear or linear logarithmic time in the size of the constraint

Decidability of the Monadic Shallow Linear First-Order Fragment with Straight Dismatching Constraints

The monadic shallow linear Horn fragment is well-known to be decidable and has many application, e.g., in security protocol analysis, tree automata, or abstraction refinement. It was a long standing open problem how to extend the fragment to the non-Horn case, preserving decidability, that would, e.g., enable to express non-determinism in protocols. We prove decidability of the non-Horn monadic shallow linear fragment via ordered resolution further extended with dismatching constraints and discuss some applications of the new decidable fragment.Comment: 29 pages, long version of CADE-26 pape